Jekyll2020-08-11T13:23:47+02:00http://localhost:4000/Vladimir BobkovGreetings on my homepage ;-)Alternative definition of the first nontrivial Fucik curve2020-08-11T02:00:00+02:002020-08-11T13:23:23+02:00http://localhost:4000/blog/2020/08/Alternative-definition-of-the-first-nontrivial-Fucik-curve<p>Consider the Fucik eigenvalue problem</p> <script type="math/tex; mode=display">% <![CDATA[ \left\{ \begin{aligned} -\Delta u &= \alpha (u^+)^{p-1} - \beta (u^-)^{p-1}, &&x \in \Omega,\\ u&=0, &&x \in \partial\Omega, \end{aligned} \right. %]]></script> <p>where $u = u^+ - u^-$, and $u^\pm := \max{\pm u, 0}$.</p> <p>In [<sup id="fnref:cuesta"><a href="#fn:cuesta" class="footnote">1</a></sup>] it is proved that the first nontrivial curve of the Fucik spectrum can be described as a set of points $(s + c(s), c(s))$, where $s \in \mathbb{R}$ and $c(s)$ defined by</p> <script type="math/tex; mode=display">c(s) = \inf_{\gamma \in \Gamma} \max_{u \in \gamma[-1,1]} \left(\int_{\Omega}|\nabla u|^p \, dx - s \int_{\Omega}|u^+|^p \, dx \right).</script> <p>Here</p> <script type="math/tex; mode=display">\Gamma := \{\gamma \in C([-1,1], S):~ \gamma(-1) = -\varphi_1,~ \gamma(1) = \varphi_1 \},</script> <p>where $S :=\{w \in W_0^{1,p}:~ \|w\|_{L^p}=1\}$ and $\varphi_1$ is the first eigenfunction.</p> <p>There is another characterization of the first nontrivial curve of the Fucik spectrum. Namely, consider</p> <script type="math/tex; mode=display">\alpha^*(\beta) := \inf\left\{ \frac{\int_{\Omega}|\nabla u^-|^p \, dx}{\int_{\Omega}|u^-|^p \, dx}:~ u \in W_0^{1,p},~ u^\pm \not\equiv 0,~ \frac{\int_{\Omega}|\nabla u^+|^p \, dx}{\int_{\Omega}|u^+|^p \, dx} = \beta \right\}.</script> <p>Note that the admissible set for this minimization problem is nonempty for all $\beta &gt; \lambda_1(p)$. This definition is, in essence, the same as of Theorem 1.2 in [<sup id="fnref:CTV"><a href="#fn:CTV" class="footnote">2</a></sup>] for the linear case $p=2$ (see also [<sup id="fnref:molle"><a href="#fn:molle" class="footnote">3</a></sup>]), and it was pointed out in that works that for $p&gt;1$ this definition is also ok. Let us prove this fact explicitly.</p> <p><strong>Proposition.</strong> The set of points $(\alpha^*(\beta), \beta)$ is the first nontrivial curve of the Fucik spectrum.</p> <hr /> <!--more--> <p><em>Proof.</em> The main idea is to switch between the parametrizations: $c(s)$ parametrized by diagonals, while $\alpha^*(\beta)$ is parametrized by horizontal lines. Note that $c(s)$ is strictly decreasing (see Propositions 4.1 in [<sup id="fnref:cuesta:1"><a href="#fn:cuesta" class="footnote">1</a></sup>]), i.e., $c(s) &gt; c(s’)$ whenever $s &lt; s’$; moreover, $c(s) \to \lambda_1(p)$ as $s \to +\infty$, see Proposition 4.4 in [<sup id="fnref:cuesta:2"><a href="#fn:cuesta" class="footnote">1</a></sup>]. Thus, for each $\beta &gt; \lambda_1(p)$ there exists unique $s \in \mathbb{R}$ such that $\beta = c(s)$. (See figure below). Notice that the $c(s)$ is constructed in [<sup id="fnref:cuesta:3"><a href="#fn:cuesta" class="footnote">1</a></sup>] only for $s \geq 0$ and then the constructed part is reflected with respect to the bisector $\alpha = \beta$. However, it doesn’t cause troubles.</p> <p>Let us show now that $\alpha^*(c(s)) = s + c(s)$ for any $c(s) = \beta &gt; \lambda_1(p)$. Note first that the eigenfunction which corresponds to $(\alpha, \beta) = (s+c(s), c(s))$ is always an admissible point for $\alpha^*(c(s))$, and hence $\alpha^*(c(s)) \leq s + c(s)$. Suppose, by contradiction, that $\alpha^*(c(s)) &lt; s + c(s)$ for some $s$. Then, by definition of $\alpha^*(c(s))$, there have to exist a function $u \in W_0^{1,p}$ such that</p> <script type="math/tex; mode=display">% <![CDATA[ \alpha^*(c(s)) \leq \frac{\int_{\Omega}|\nabla u^-|^p \, dx}{\int_{\Omega}|u^-|^p \, dx} < s+ c(s) \quad \text{and} \quad \frac{\int_{\Omega}|\nabla u^+|^p \, dx}{\int_{\Omega}|u^+|^p \, dx} = \beta = c(s). %]]></script> <center><img src="\images\FigFucik1.png" width="50%" style="margin:10px 0px" /> <figcaption> </figcaption> </center> <p>Due to the continuity and monotonicity of $c(s)$ (see Proposition 4.1 in [<sup id="fnref:cuesta:4"><a href="#fn:cuesta" class="footnote">1</a></sup>]), there exists $s_0$ such that</p> <script type="math/tex; mode=display">% <![CDATA[ \frac{\int_{\Omega}|\nabla u^-|^p \, dx}{\int_{\Omega}|u^-|^p \, dx} = s_0 + c(s) < s_0 + c(s_0) \quad \text{and} \quad \frac{\int_{\Omega}|\nabla u^+|^p \, dx}{\int_{\Omega}|u^+|^p \, dx} = \beta < c(s_0), %]]></script> <p>or, equivalently,</p> <script type="math/tex; mode=display">% <![CDATA[ \int_{\Omega}|\nabla u^-|^p \, dx < (s_0 + c(s_0))\int_{\Omega}|u^-|^p \, dx \quad \text{and} \quad \int_{\Omega}|\nabla u^+|^p \, dx < c(s_0) \int_{\Omega}|u^+|^p \, dx, %]]></script> <p>which is, in fact, the main contradictory assumption in the proof of Theorem 3.1 in [<sup id="fnref:cuesta:5"><a href="#fn:cuesta" class="footnote">1</a></sup>] (see also the proof of Lemma 5.3, (5.10) in [<sup id="fnref:cuesta:6"><a href="#fn:cuesta" class="footnote">1</a></sup>]). Thus, proceeding exactly as in the proof of Theorem 3.1 in [<sup id="fnref:cuesta:7"><a href="#fn:cuesta" class="footnote">1</a></sup>], we obtain a contradiction to the definition of $c(s_0)$.</p> <hr /> <p>The PDF-version of this post can be found <a href="/pdf/posts/Alternative-difinition-Fucik-curve.pdf">here</a>.</p> <hr /> <p>Last modified: 11-Aug-20</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:cuesta"> <p>Cuesta, M., De Figueiredo, D., Gossez, J. P. (1999). The beginning of the Fucik spectrum for the p-Laplacian. Journal of Differential Equations, 159(1), 212-238.&nbsp;<a href="#fnref:cuesta" class="reversefootnote">&#8617;</a>&nbsp;<a href="#fnref:cuesta:1" class="reversefootnote">&#8617;<sup>2</sup></a>&nbsp;<a href="#fnref:cuesta:2" class="reversefootnote">&#8617;<sup>3</sup></a>&nbsp;<a href="#fnref:cuesta:3" class="reversefootnote">&#8617;<sup>4</sup></a>&nbsp;<a href="#fnref:cuesta:4" class="reversefootnote">&#8617;<sup>5</sup></a>&nbsp;<a href="#fnref:cuesta:5" class="reversefootnote">&#8617;<sup>6</sup></a>&nbsp;<a href="#fnref:cuesta:6" class="reversefootnote">&#8617;<sup>7</sup></a>&nbsp;<a href="#fnref:cuesta:7" class="reversefootnote">&#8617;<sup>8</sup></a></p> </li> <li id="fn:CTV"> <p>Conti, M., Terracini, S., Verzini, G. (2005). On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formulae. Calculus of Variations and Partial Differential Equations, 22(1), 45-72.&nbsp;<a href="#fnref:CTV" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:molle"> <p>Molle, R., Passaseo, D. (2015). Variational properties of the first curve of the Fucik spectrum for elliptic operators. Calculus of Variations and Partial Differential Equations, 54(4), 3735-3752.&nbsp;<a href="#fnref:molle" class="reversefootnote">&#8617;</a></p> </li> </ol> </div>Consider the Fucik eigenvalue problem where $u = u^+ - u^-$, and $u^\pm := \max{\pm u, 0}$. In  it is proved that the first nontrivial curve of the Fucik spectrum can be described as a set of points $(s + c(s), c(s))$, where $s \in \mathbb{R}$ and $c(s)$ defined by Here where $S :=\{w \in W_0^{1,p}:~ \|w\|_{L^p}=1\}$ and $\varphi_1$ is the first eigenfunction. There is another characterization of the first nontrivial curve of the Fucik spectrum. Namely, consider Note that the admissible set for this minimization problem is nonempty for all $\beta &gt; \lambda_1(p)$. This definition is, in essence, the same as of Theorem 1.2 in  for the linear case $p=2$ (see also ), and it was pointed out in that works that for $p&gt;1$ this definition is also ok. Let us prove this fact explicitly. Proposition. The set of points $(\alpha^*(\beta), \beta)$ is the first nontrivial curve of the Fucik spectrum. Cuesta, M., De Figueiredo, D., Gossez, J. P. (1999). The beginning of the Fucik spectrum for the p-Laplacian. Journal of Differential Equations, 159(1), 212-238.&nbsp;&#8617; Conti, M., Terracini, S., Verzini, G. (2005). On a class of optimal partition problems related to the Fucik spectrum and to the monotonicity formulae. Calculus of Variations and Partial Differential Equations, 22(1), 45-72.&nbsp;&#8617; Molle, R., Passaseo, D. (2015). Variational properties of the first curve of the Fucik spectrum for elliptic operators. Calculus of Variations and Partial Differential Equations, 54(4), 3735-3752.&nbsp;&#8617;Counterexample to a lower bound for the first Dirichlet-Neumann eigenvalue2020-04-24T02:00:00+02:002020-08-11T12:33:43+02:00http://localhost:4000/blog/2020/04/Counterexample-to-a-lower-bound-for-the-first-Dirichlet-Neumann-eigenvalue%20copy<p>In this post we would like to show the impossibility of having one certain lower bound for the first eigenvalue $\nu_1(\Omega)$ of the following problem:</p> <script type="math/tex; mode=display">% <![CDATA[ \left\{ \begin{aligned} -\Delta u &= \nu u, &&x \in \Omega,\\ u&=0, &&x \in \Gamma_0,\\ \frac{\partial u}{\partial n}&=0, &&x \in \partial \Omega \setminus \Gamma_0. \end{aligned} \right. %]]></script> <p>Here $\Omega \subset \mathbb{R}^N$ is a bounded domain whose boundary $\partial \Omega$ consists of two connected components: the outer one - $\Gamma_0$, and the inner one - $\partial \Omega \setminus \Gamma_0$.</p> <p>Let $\Omega^\#$ be the spherical shell (i.e., difference of two balls centred at the same point) of the same measure as $\Omega$ and such that the outer boundary of $\Omega^\#$ has the same $(N-1)$-measure as $\Gamma_0$.</p> <p>It was proved by Payne &amp; Weinberger [<sup id="fnref:PW"><a href="#fn:PW" class="footnote">1</a></sup>] that in the planar case $N=2$, $\nu_1(\Omega)$ satisfies the following reversed Faber-Krahn inequality:</p> <script type="math/tex; mode=display">\nu_1(\Omega) \leq \nu_1(\Omega^\#).</script> <p>This result was generalized to the $N$-dimensional case, as well as to the $p$-Laplacian version of the problem, by Anoop &amp; Kumar [<sup id="fnref:AK"><a href="#fn:AK" class="footnote">2</a></sup>], under the assumption that $\Gamma_0$ is a sphere.</p> <p>In Section 4 of the same paper [<sup id="fnref:AK:1"><a href="#fn:AK" class="footnote">2</a></sup>], it is asked whether there exists a universal constant $C&gt;0$ such that the following lower bound for $\nu_1(\Omega)$ takes place:</p> <script type="math/tex; mode=display">C \nu_1(\Omega^\#) \leq \nu_1(\Omega).</script> <p>Let us show that such $C$ does not exists, in general. To this end, we will construct a sequence of equimeasurable domains $\Omega_n$ with equimeasurable outer boundary, such that $\nu_1(\Omega_n) \to 0$.</p> <hr /> <!--more--> <p>We construct $\Omega_n$ as follows. Let the outer boundary $\Gamma_0$ of $\Omega_n$ be a sphere $\partial B$. Now we consider a smaller concentric spherical shell $\Theta_n$ with a cylindrical hole $H_n$ of radius $1/n$ in it, as depicted in the following figure:</p> <center><img src="\images\CouterexampleIII-1.png" width="50%" style="margin:10px 0px" /> <figcaption> </figcaption> </center> <p>Clearly, the boundary of $\Theta_n \setminus H_n$ is connected. Moreover, we assume that the inner boundary of $\Theta_n$ is of a fixed radius. We set $\Omega_n = B \setminus (\Theta_n \setminus H_n)$, and we assume the width of $\Theta_n$ to be such that $\Omega_n$ is of a constant measure with respect to $n$.</p> <p>Note that $\nu_1(\Omega)$ has the following variational characterization:</p> <script type="math/tex; mode=display">\nu_1(\Omega) = \inf\left\{ \frac{\int_\Omega |\nabla u|^2 \, dx}{\int_\Omega |u|^2 \, dx}: u \in W^{1,2}(\Omega) \setminus \{0\},~ u=0 ~\text{on}~ \Gamma_0 \right\}.</script> <p>Let us take the following admissible function for this minimization problem. We set $u_n = 0$ in the “outer” part of $\Omega_n$, and $u_n=C=const$ in the “inner” part of $\Omega_n$. In the cylinder $H_n$ we assume $u_n$ to be, e.g., a linear function like $u_n(x_1,\dots,x_N) = ax_1+b$, where $a$ and $b$ are adjusted in such a way that $u_n$ is continuous. See the figure below.</p> <center><img src="\images\CouterexampleIII-2.png" width="50%" style="margin:10px 0px" /> <figcaption> </figcaption> </center> <p>Thus, we get</p> <script type="math/tex; mode=display">\nu_1(\Omega_n) \leq \frac{\int_{\Omega_n} |\nabla u_n|^2 \, dx}{\int_{\Omega_n} |u_n|^2 \, dx} = \frac{a^2 |H_n|}{\int_{H_n} |ax_1 + b|^2 \, dx + C^2 \times (\text{measure of the "inner" part of}~ \Omega_n)}.</script> <p>Clearly, the numerator goes to zero, while the denominator is bounded away from zero. Thus, the desired result follows.</p> <hr /> <p>Last modified: 11-Aug-20</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:PW"> <p>Payne, L. E., &amp; Weinberger, H. F. (1961). Some isoperimetric inequalities for membrane frequencies and torsional rigidity. Journal of Mathematical Analysis and Applications, 2(2), 210-216.&nbsp;<a href="#fnref:PW" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:AK"> <p>Anoop, T. V., &amp; Kumar, K. A. (2020). On reverse Faber-Krahn inequalities. Journal of Mathematical Analysis and Applications, 485(1), 123766.&nbsp;<a href="#fnref:AK" class="reversefootnote">&#8617;</a>&nbsp;<a href="#fnref:AK:1" class="reversefootnote">&#8617;<sup>2</sup></a></p> </li> </ol> </div>In this post we would like to show the impossibility of having one certain lower bound for the first eigenvalue $\nu_1(\Omega)$ of the following problem: Here $\Omega \subset \mathbb{R}^N$ is a bounded domain whose boundary $\partial \Omega$ consists of two connected components: the outer one - $\Gamma_0$, and the inner one - $\partial \Omega \setminus \Gamma_0$. Let $\Omega^\#$ be the spherical shell (i.e., difference of two balls centred at the same point) of the same measure as $\Omega$ and such that the outer boundary of $\Omega^\#$ has the same $(N-1)$-measure as $\Gamma_0$. It was proved by Payne &amp; Weinberger  that in the planar case $N=2$, $\nu_1(\Omega)$ satisfies the following reversed Faber-Krahn inequality: This result was generalized to the $N$-dimensional case, as well as to the $p$-Laplacian version of the problem, by Anoop &amp; Kumar , under the assumption that $\Gamma_0$ is a sphere. In Section 4 of the same paper , it is asked whether there exists a universal constant $C&gt;0$ such that the following lower bound for $\nu_1(\Omega)$ takes place: Let us show that such $C$ does not exists, in general. To this end, we will construct a sequence of equimeasurable domains $\Omega_n$ with equimeasurable outer boundary, such that $\nu_1(\Omega_n) \to 0$. Payne, L. E., &amp; Weinberger, H. F. (1961). Some isoperimetric inequalities for membrane frequencies and torsional rigidity. Journal of Mathematical Analysis and Applications, 2(2), 210-216.&nbsp;&#8617; Anoop, T. V., &amp; Kumar, K. A. (2020). On reverse Faber-Krahn inequalities. Journal of Mathematical Analysis and Applications, 485(1), 123766.&nbsp;&#8617;&nbsp;&#8617;2Hadamard shape derivative formula for quasilinear problems. Forgotten references2020-03-25T01:00:00+01:002020-04-24T22:25:52+02:00http://localhost:4000/blog/2020/03/Hadamard-shape-derivative-formula-for-quasilinear-problems-Forgotten-references%20copy<p>In a recently published (but submitted a long time ago) article [<sup id="fnref:BK"><a href="#fn:BK" class="footnote">1</a></sup>] by <em>S. Kolonitskii</em> and myself, we studied the dependence of the least critical levels of the energy functional</p> <script type="math/tex; mode=display">E[u] = \frac{1}{p} \int_\Omega |\nabla u|^p \, dx - \int_\Omega F(u) \, dx</script> <p>upon domain perturbations driven by a family of diffeomorphisms</p> <script type="math/tex; mode=display">% <![CDATA[ \Phi_t(x) = x + t R(x), \quad R \in C^1(\mathbb{R}^N, \mathbb{R}^N), \quad |t|<\delta. %]]></script> <p>Here $F$ satisfies certain rather classical conditions.</p> <p>Let us take an arbitrary minimizer $v_0$ of $E$ over the Nehari manifold $\mathcal{N}(\Omega)$ and consider a function $v_t(y) := v_0(\Phi_t^{-1}(y))$, $y \in \Omega_t$. One of the main results of our paper is the following Hadamard-type formula:</p> <script type="math/tex; mode=display">% <![CDATA[ \left. \frac{\partial E[\alpha(v_t) v_t]}{\partial t} \right|_{t=0} = - \frac {p-1} p \int_{\partial \Omega} \left| \frac{\partial v_0}{\partial n} \right|^p \left<R, n\right> \, d\sigma, %]]></script> <p>where $\alpha(v_t) \in \mathbb{R}$ is a normalization coefficient such that $\alpha(v_t) v_t \in \mathcal{N}(\Phi_t(\Omega))$, and $n$ is the outward unit normal vector to $\partial \Omega$.</p> <p>In the particular case $F(u) = |u|^q$, $q \in [1, p^*)$, it can easily be checked that the finding of the least critical level of $E$ can be restated as the finding of minimum for the problem</p> <hr /> <!--more--> <script type="math/tex; mode=display">\mu_q(\Omega) = \min_{u \in W_0^{1,p}(\Omega) \setminus \{0\}} J(u) := \min_{u \in W_0^{1,p}(\Omega) \setminus \{0\}} \frac{\int_\Omega |\nabla u|^p \, dx}{\left(\int_\Omega |u|^q \, dx\right)^\frac{p}{q}}.</script> <p>As a corollary of the proof of the Hadamard formula above, we obtain the following fact. If $u_0$ is a minimizer of $\mu_q(\Omega)$ normalized such that $|u_0|_{L^q(\Omega)} = 1$, and $u_t(y) := u_0(\Phi_t^{-1}(y))$, $y \in \Omega_t$, then</p> <p>\begin{equation}\label{eq:had2} \left. \frac{\partial J(u_t)}{\partial t} \right|_{t=0} = - (p-1) \int_{\partial \Omega} \left| \frac{\partial u_0}{\partial n} \right|^p \left&lt;R, n\right&gt; \, d\sigma. \end{equation}</p> <p><strong>Short historical remark.</strong> If $q=p$, then $\mu_p(\Omega)$ is the first eigenvalue of the $p$-Laplacian, and the formula \eqref{eq:had2} (in fact, the actual derivative of $\mu_p(\Omega)$) was established by <em>Garcia Melian &amp; Sabina de Lis</em> [<sup id="fnref:garcia"><a href="#fn:garcia" class="footnote">2</a></sup>] in (2001) for $p&gt;1$, and by <em>Lamberti</em> [<sup id="fnref:lamberti"><a href="#fn:lamberti" class="footnote">3</a></sup>] in (2003) for $p \geq 2$. In the case $p=2$, this formula goes back to <em>Hadamard</em> (1908), see, e.g., [<sup id="fnref:garabedian"><a href="#fn:garabedian" class="footnote">4</a></sup>].</p> <p>However it appeared recently that the nonlinear case has been studied much earlier in a work of <em>Roppongi</em> [<sup id="fnref:rop"><a href="#fn:rop" class="footnote">5</a></sup>] who established the Hadamard formula \eqref{eq:had2} for $p \geq 2$ and $q \geq p$ in (1994). In fact, the article of <em>Roppongi</em> is build upon the article of <em>Osawa</em> [<sup id="fnref:osawa"><a href="#fn:osawa" class="footnote">6</a></sup>] who considered the case $p=2$, $q&gt;2$. Moreover, <em>Osawa</em> also treated the Hadamard formula under the Robin boundary condition and the Neumann boundary condition. The reader will be able to find easily a few more related works of this Japanese group by looking at citations of <em>Osawa’s</em> article. Later, in (2017), the Hadamard formula \eqref{eq:had2} was independently stated in [<sup id="fnref:CFR"><a href="#fn:CFR" class="footnote">7</a></sup>] by referring to the arguments from [<sup id="fnref:garcia:1"><a href="#fn:garcia" class="footnote">2</a></sup>].</p> <p>It is quite a pity that we didn’t know these references before and didn’t include them in our paper. Moreover, it seems that these works of the Japanese group are forgotten and have not being cited in related contemporary research. It makes sense to revive them.</p> <hr /> <p>The PDF-version of this post can be found <a href="/pdf/posts/Hadamard-forgotten-references.pdf">here</a>.</p> <hr /> <p>Last modified: 24-Apr-20</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:BK"> <p>Bobkov, V., &amp; Kolonitskii, S. (2020). On qualitative properties of solutions for elliptic problems with the $p$-Laplacian through domain perturbations. Communications in Partial Differential Equations, 45(3), 230-252.&nbsp;<a href="#fnref:BK" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:garcia"> <p>Garcia Melian, J., &amp; Sabina de Lis, J. (2001). On the perturbation of eigenvalues for the $p$-Laplacian. Comptes Rendus de l’Acad'emie des Sciences-Series I-Mathematics, 332(10), 893-898.&nbsp;<a href="#fnref:garcia" class="reversefootnote">&#8617;</a>&nbsp;<a href="#fnref:garcia:1" class="reversefootnote">&#8617;<sup>2</sup></a></p> </li> <li id="fn:lamberti"> <p>Lamberti, P. D. (2003). A differentiability result for the first eigenvalue of the $p$-Laplacian upon domain perturbation. Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, 1(2), 741-754.&nbsp;<a href="#fnref:lamberti" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:garabedian"> <p>Garabedian, P. R., &amp; Schiffer, M. (1952). Convexity of domain functionals. Journal d’Analyse Math'ematique, 2(2), 281-368.&nbsp;<a href="#fnref:garabedian" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:rop"> <p>Roppongi, S. (1994). The Hadamard variation of the ground state value of some quasi-linear elliptic equations. Kodai Mathematical Journal, 17(2), 214-227.&nbsp;<a href="#fnref:rop" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:osawa"> <p>Osawa, T. (1992). The Hadamard variational formula for the ground state value of $-\Delta u = \lambda |u|^{p-1} u$. Kodai Mathematical Journal, 15(2), 258-278.&nbsp;<a href="#fnref:osawa" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:CFR"> <p>Carroll, T., Fall, M. M., &amp; Ratzkin, J. (2017). On the rate of change of the sharp constant in the Sobolev-Poincare inequality. Mathematische Nachrichten, 290(14-15), 2185-2197.&nbsp;<a href="#fnref:CFR" class="reversefootnote">&#8617;</a></p> </li> </ol> </div>In a recently published (but submitted a long time ago) article  by S. Kolonitskii and myself, we studied the dependence of the least critical levels of the energy functional upon domain perturbations driven by a family of diffeomorphisms Here $F$ satisfies certain rather classical conditions. Let us take an arbitrary minimizer $v_0$ of $E$ over the Nehari manifold $\mathcal{N}(\Omega)$ and consider a function $v_t(y) := v_0(\Phi_t^{-1}(y))$, $y \in \Omega_t$. One of the main results of our paper is the following Hadamard-type formula: where $\alpha(v_t) \in \mathbb{R}$ is a normalization coefficient such that $\alpha(v_t) v_t \in \mathcal{N}(\Phi_t(\Omega))$, and $n$ is the outward unit normal vector to $\partial \Omega$. In the particular case $F(u) = |u|^q$, $q \in [1, p^*)$, it can easily be checked that the finding of the least critical level of $E$ can be restated as the finding of minimum for the problem Bobkov, V., &amp; Kolonitskii, S. (2020). On qualitative properties of solutions for elliptic problems with the $p$-Laplacian through domain perturbations. Communications in Partial Differential Equations, 45(3), 230-252.&nbsp;&#8617;A Picone-type inequality2019-08-28T02:00:00+02:002019-08-30T11:14:57+02:00http://localhost:4000/blog/2019/08/A-Picone-type-inequality<p>The Picone inequality (identity) is a well-known tool with wide applications in PDEs, see, e.g., [<sup id="fnref:AH"><a href="#fn:AH" class="footnote">1</a></sup>]. In my paper with M. Tanaka [<sup id="fnref:BT"><a href="#fn:BT" class="footnote">2</a></sup>], Proposition 8, we found one form of the Picone inequality which appears to be quite useful for studying problmes with the $(p,q)$-Laplacian. In this post, I would like to provide a slight generalization of Proposition 8 from [<sup id="fnref:BT:1"><a href="#fn:BT" class="footnote">2</a></sup>].</p> <p><strong>Theorem.</strong> Let $1 &lt; q &lt; p &lt; \infty$ and $\alpha, \beta &gt; 0$. Assume that $u&gt;0$ and $\varphi \geq 0$ are some differentiable functions in a domain $\Omega$. Then</p> <script type="math/tex; mode=display">|\nabla u|^{p-2} \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\alpha C} |\nabla \varphi|^p</script> <p>and</p> <script type="math/tex; mode=display">|\nabla u|^{q-2} \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\beta}|\nabla (\varphi^{p/q})|^q,</script> <p>where $C = 1$ if $p \leq q+1$, and $C= \frac{(q-1)^{p-2} (p-q)}{(p-2)^{p-2}}$ if $p \geq q+1$.</p> <p>In particular, if $\mu&gt;0$, then</p> <script type="math/tex; mode=display">(|\nabla u|^{p-2} + \mu |\nabla u|^{q-2}) \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\alpha C} |\nabla \varphi|^p + \frac{\mu}{\beta}|\nabla (\varphi^{p/q})|^q.</script> <hr /> <!--more--> <p><em>Proof.</em> First, by standard calculations we get</p> <p>\begin{align} \notag &amp; |\nabla u|^{p-2} \nabla u \nabla \left( \frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}} \right)\newline \notag &amp; = p |\nabla u|^{p-2} \nabla u \nabla \varphi \frac{\varphi^{p-1}}{\alpha u^{p-1} + \beta u^{q-1}} - |\nabla u|^{p} \varphi^p \frac{\alpha (p-1) u^{p-2} + \beta (q-1) u^{q-2}}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^2}\newline \label{eq:picone:1} &amp; \leq p |\nabla u|^{p-1} |\nabla \varphi| \frac{\varphi^{p-1}}{\alpha u^{p-1} + \beta u^{q-1}} - |\nabla u|^{p} \varphi^p \frac{\alpha (p-1) u^{p-2} + \beta (q-1) u^{q-2}}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^2}. \end{align}</p> <p>Applying to the first term Young’s inequality</p> <script type="math/tex; mode=display">ab =\frac{a}{\rho^{\frac{p-1}{p}}}\,\rho^{\frac{p-1}{p}} b \leq \frac{|a|^p}{p\rho^{p-1}} + \frac{\rho (p-1) |b|^{\frac{p}{p-1}}}{p}</script> <p>with $a= |\nabla \varphi|$, $b= |\nabla u|^{p-1} \frac{\varphi^{p-1}}{\alpha u^{p-1} + \beta u^{q-1}}$, and any $\rho&gt;0$, we obtain</p> <p>\begin{align} \notag \eqref{eq:picone:1} &amp; \leq \frac{|\nabla \varphi|^p}{\rho^{p-1}} + \frac{\rho(p-1) |\nabla u|^{p} \varphi^p}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^{\frac{p}{p-1}}} - |\nabla u|^{p} \varphi^p \frac{\alpha (p-1) u^{p-2} + \beta (q-1) u^{q-2}}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^2}\newline \notag &amp; = \frac{|\nabla \varphi|^p}{\rho^{p-1}} + \frac{\rho(p-1) |\nabla u|^{p} \varphi^p}{\left( \alpha u^{p-1} + \beta u^{q-1} \right)^{\frac{p}{p-1}}} \left[ 1 - \frac{\alpha (p-1) + \beta (q-1) u^{q-p}}{\rho(p-1) \left(\alpha + \beta u^{q-p}\right)^{\frac{p-2}{p-1}}} \right]. \end{align}</p> <p>Analysing the function</p> <script type="math/tex; mode=display">g(u) = \frac{\alpha (p-1) + \beta (q-1) u^{q-p}}{\rho(p-1) \left(\alpha + \beta u^{q-p}\right)^{\frac{p-2}{p-1}}},</script> <p>we see that $g(u) \geq 1$ for all $u &gt; 0$ in the following cases:</p> <p>\begin{align} \notag &amp;1)\quad p \leq q+1 \quad \text{and} \quad \rho \leq \alpha^\frac{1}{p-1},\newline \notag &amp;2) \quad p \geq q+1 \quad \text{and} \quad \rho \leq \frac{(q-1)^\frac{p-2}{p-1}(p-q)^\frac{1}{p-1}}{(p-2)^\frac{p-2}{p-1}}\alpha^\frac{1}{p-1}. \end{align}</p> <p>Under the assumptions 1) or 2) we get</p> <script type="math/tex; mode=display">|\nabla u|^{p-2} \nabla u \nabla \left( \frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}} \right) \leq \frac{|\nabla \varphi|^p}{\rho^{p-1}}.</script> <p>Thus, taking $\rho$ as the maximal admissible value in the cases 1) and 2), we obtain the constat $C_1$.</p> <p>Arguing in the similar way, for $\psi=\varphi^{p/q}$ (note that $p/q&gt;1$ and $\varphi^p=\psi^q$) we obtain</p> <p>\begin{align} \notag &amp; |\nabla u|^{q-2} \nabla u \nabla \left( \frac{\psi^q}{\alpha u^{p-1} + \beta u^{q-1}} \right) \newline \notag &amp;\le \frac{|\nabla \psi|^q}{\rho^{q-1}} + \frac{\rho(q-1) |\nabla u|^{q} \psi^q}{\left( \alpha u^{p-1} + \beta u^{q-1} \right)^{\frac{q}{q-1}}} \left[ 1 - \frac{\alpha (p-1) u^{p-q} + \beta (q-1)}{\rho(q-1) \left(\alpha u^{p-q} + \beta \right)^{\frac{q-2}{q-1}}} \right]. \end{align}</p> <p>Analysing the function</p> <script type="math/tex; mode=display">g(u) = \frac{\alpha (p-1) u^{p-q} + \beta (q-1)}{\rho(q-1) \left(\alpha u^{p-q} + \beta \right)^{\frac{q-2}{q-1}}},</script> <p>we see that $g(u) \geq 1$ for all $u &gt; 0$ provided $\rho \leq \beta^\frac{1}{q-1}$. Thus, taking $\rho = \beta^\frac{1}{q-1}$, we get</p> <script type="math/tex; mode=display">|\nabla u|^{q-2} \nabla u \nabla \left( \frac{\psi^q}{\alpha u^{p-1} + \beta u^{q-1}} \right) \le \frac{|\nabla \psi|^q}{\rho^{q-1}} = \frac{|\nabla \psi|^q}{\beta}.</script> <p>Combining the above inequalities, we complete the proof.</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:AH"> <p>Allegretto, W., &amp; Huang, X. Y. (1998). A Picone’s identity for the $p$-Laplacian and applications. Nonlinear Analysis: Theory, Methods &amp; Applications, 32(7), 819-830.&nbsp;<a href="#fnref:AH" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:BT"> <p>Bobkov, V., &amp; Tanaka, M. (2015). On positive solutions for $(p,q)$-Laplace equations with two parameters. Calculus of Variations and Partial Differential Equations, 54(3), 3277-3301.&nbsp;<a href="#fnref:BT" class="reversefootnote">&#8617;</a>&nbsp;<a href="#fnref:BT:1" class="reversefootnote">&#8617;<sup>2</sup></a></p> </li> </ol> </div>The Picone inequality (identity) is a well-known tool with wide applications in PDEs, see, e.g., . In my paper with M. Tanaka , Proposition 8, we found one form of the Picone inequality which appears to be quite useful for studying problmes with the $(p,q)$-Laplacian. In this post, I would like to provide a slight generalization of Proposition 8 from . Theorem. Let $1 &lt; q &lt; p &lt; \infty$ and $\alpha, \beta &gt; 0$. Assume that $u&gt;0$ and $\varphi \geq 0$ are some differentiable functions in a domain $\Omega$. Then and where $C = 1$ if $p \leq q+1$, and $C= \frac{(q-1)^{p-2} (p-q)}{(p-2)^{p-2}}$ if $p \geq q+1$. In particular, if $\mu&gt;0$, then Allegretto, W., &amp; Huang, X. Y. (1998). A Picone’s identity for the $p$-Laplacian and applications. Nonlinear Analysis: Theory, Methods &amp; Applications, 32(7), 819-830.&nbsp;&#8617; Bobkov, V., &amp; Tanaka, M. (2015). On positive solutions for $(p,q)$-Laplace equations with two parameters. Calculus of Variations and Partial Differential Equations, 54(3), 3277-3301.&nbsp;&#8617;&nbsp;&#8617;2$p$-Laplacian in polar coordinates2019-04-07T02:00:00+02:002019-04-07T13:33:37+02:00http://localhost:4000/blog/2019/04/p-Laplacian-in-polar-coordinates<p>Several times I found myself looking for an explicit expression of the $p$-Laplacian in polar coordinates. Usual Laplace operator considered in polar coordinates can be very useful if one works with radial domains. So, in some problems it can be quite natural to be interested in the corresponding expression for the $p$-Laplacian. However, such an expression appears to be quite bulky, which makes it complicated to apply. Nevertheless, to find easily this expression in future, I decided to post it here.</p> <p>Let $u = u(x,y) = u(r,\theta)$, where $r&gt;0$ and $\theta \in (-\pi, \pi)$. Then we have</p> <p>\begin{align} \notag \Delta_p u(x,y) &amp;= \left(u_r^2 + \frac{u_\theta^2}{r^2} \right)^\frac{p-4}{2} \newline \notag &amp;\times \left( (p-1) u_r^2 u_{rr} + \frac{u_r^3}{r} + \frac{2(p-2) u_r u_\theta u_{r\theta}}{r^2} + \frac{u_r^2 u_{\theta \theta}}{r^2} + \frac{u_{rr} u_\theta^2}{r^2} - \frac{(p-3) u_r u_\theta^2}{r^3} + \frac{(p-1) u_\theta^2 u_{\theta \theta}}{r^4} \right). \end{align}</p> <p>In particular, if $u = u(x,y) = u(r)$, then $u_\theta = u_{\theta \theta}=0$, and we get the usual</p> <script type="math/tex; mode=display">\Delta_p u(x,y) = (p-1) |u_r|^{p-2} u_{rr} + \frac{|u_r|^{p-2} u_r}{r}.</script> <hr /> <!--more--> <p>To obtain the claimed expression, let us write all necessary partial derivatives in polar coordinates:</p> <script type="math/tex; mode=display">u_{x} = u_r \cos \theta - u_\theta \frac{\sin \theta}{r},</script> <script type="math/tex; mode=display">u_y = u_r \sin \theta + u_\theta \frac{\cos \theta}{r},</script> <script type="math/tex; mode=display">u_{xx} = u_{rr} \cos^2 \theta - u_{r\theta} \frac{2 \cos \theta \sin \theta}{r} + u_{\theta \theta} \frac{\sin^2 \theta}{r^2} + u_r \frac{\sin^2 \theta}{r} + u_{\theta} \frac{2 \cos \theta \sin \theta}{r^2},</script> <script type="math/tex; mode=display">u_{yy} = u_{rr} \sin^2 \theta + u_{r\theta} \frac{2 \cos \theta \sin \theta}{r} + u_{\theta \theta} \frac{\cos^2 \theta}{r^2} + u_r \frac{\cos^2 \theta}{r} - u_{\theta} \frac{2 \cos \theta \sin \theta}{r^2},</script> <script type="math/tex; mode=display">u_{xy} = u_{rr} \cos \theta \sin \theta + u_{r\theta} \frac{\cos^2 \theta - \sin^2 \theta}{r} - u_{\theta \theta} \frac{\cos \theta \sin \theta}{r^2} - u_r \frac{\cos \theta \sin \theta}{r} - u_{\theta} \frac{\cos^2 \theta - \sin^2 \theta}{r^2}.</script> <p>Then, we substitute these expressions into the pointwise formula of the $p$-Laplacian in $2D$ (see, e.g., <a href="https://jyx.jyu.fi/bitstream/handle/123456789/54658/Lindqvistplaplacenotes.pdf?sequence=1">here</a>):</p> <script type="math/tex; mode=display">\Delta_p u = \text{div}\left(|\nabla u|^{p-2} \nabla u\right) = |\nabla u|^{p-4} \left( |\nabla u|^2 \Delta u + (p-2) u_x^2 u_{xx} + (p-2) u_y^2 u_{yy} + 2(p-2) u_{x}u_y u_{xy}. \right).</script> <p>After some standard simplifications, we arrive at the desired expression.</p>Several times I found myself looking for an explicit expression of the $p$-Laplacian in polar coordinates. Usual Laplace operator considered in polar coordinates can be very useful if one works with radial domains. So, in some problems it can be quite natural to be interested in the corresponding expression for the $p$-Laplacian. However, such an expression appears to be quite bulky, which makes it complicated to apply. Nevertheless, to find easily this expression in future, I decided to post it here. Let $u = u(x,y) = u(r,\theta)$, where $r&gt;0$ and $\theta \in (-\pi, \pi)$. Then we have \begin{align} \notag \Delta_p u(x,y) &amp;= \left(u_r^2 + \frac{u_\theta^2}{r^2} \right)^\frac{p-4}{2} \newline \notag &amp;\times \left( (p-1) u_r^2 u_{rr} + \frac{u_r^3}{r} + \frac{2(p-2) u_r u_\theta u_{r\theta}}{r^2} + \frac{u_r^2 u_{\theta \theta}}{r^2} + \frac{u_{rr} u_\theta^2}{r^2} - \frac{(p-3) u_r u_\theta^2}{r^3} + \frac{(p-1) u_\theta^2 u_{\theta \theta}}{r^4} \right). \end{align} In particular, if $u = u(x,y) = u(r)$, then $u_\theta = u_{\theta \theta}=0$, and we get the usualOptimality of an upper bound for the number of ordered factorizations of odd numbers2019-03-29T01:00:00+01:002019-08-30T11:01:18+02:00http://localhost:4000/blog/2019/03/Optimality-of-upper-bound-for-number-of-factorizations<p>Let us take any odd natural $n$ and denote by $H(n)$ the number of ways how $n$ can be represented as the product of factors larger or equal than $3$, where the order of factors matters. For instance, if $n=27$, then $H(n)=4$, since</p> <script type="math/tex; mode=display">n = 3 \cdot 3 \cdot 3 = 9 \cdot 3 = 3 \cdot 9 = 27.</script> <p>It is known that $H(n) &lt; n^{\eta}$, where $\eta = 1.37779\dots$ is the unique positive real zero of $\left(1-\frac{1}{2^s}\right) \zeta(s)=2$, see Theorem 5 in [<sup id="fnref:CLM"><a href="#fn:CLM" class="footnote">1</a></sup>]. Our aim is to show that this upper bound is optimal.</p> <p><strong>Lemma.</strong> For any $\varepsilon&gt;0$ there exist infinitely many odd $n$ such that $H(n) &gt; n^{\eta-\varepsilon}$.</p> <p><em>Proof.</em> We will argue along the same lines as in Section 3 of [<sup id="fnref:CL"><a href="#fn:CL" class="footnote">2</a></sup>], where the similar optimality was obtained for all (even) numbers. First, we notice that</p> <script type="math/tex; mode=display">\left(1-\frac{1}{2^s}\right) \zeta(s) = \sum_{m=1,~ m ~\text{odd}}^{\infty} \frac{1}{m^s}.</script> <hr /> <!--more--> <p>Since this expression is a decreasing function of $s$, we take any $\varepsilon&gt;0$ and find that</p> <script type="math/tex; mode=display">\sum_{m=1,~ m ~\text{odd}}^{\infty} \frac{1}{m^{\eta-\varepsilon}} > 2,</script> <p>and there exists some odd $b$ such that</p> <script type="math/tex; mode=display">\sum_{m=1,~ m ~\text{odd}}^{b} \frac{1}{m^{\eta-\varepsilon}} > 2.</script> <p>Recalling the monotonicity property and the definition of $\eta$, we can find some $\gamma \in (\eta-\varepsilon, \eta)$ such that</p> <script type="math/tex; mode=display">\sum_{m=1,~ m ~\text{odd}}^{b} \frac{1}{m^{\gamma}} = 2,</script> <p>which we rewrite as</p> <script type="math/tex; mode=display">\sum_{m=3,~ m ~\text{odd}}^{b} \frac{1}{m^{\gamma}} = 1.</script> <p>Let us now consider for each odd $m \in [3,b]$ and sufficiently large $t&gt;0$ the function</p> <script type="math/tex; mode=display">c_m = c_m(t) = \text{floor}\left(\frac{t}{m^\gamma}\right).</script> <p>Notice that</p> <script type="math/tex; mode=display">\frac{t}{m^\gamma} -1 \leq c_m \leq \frac{t}{m^\gamma},</script> <p>which yields, in particular, that</p> <p>\begin{equation}\label{eq:summ} \left(\frac{\sum_m c_m}{t}\right)^{\sum_m c_m} \geq \left(\sum_{m=3,~ m ~\text{odd}}^{b} \frac{1}{m^{\gamma}}-\frac{b}{t}\right)^{\sum_m c_m} = \left(1-\frac{b}{t}\right)^{\sum_m c_m} \geq \left(1-\frac{b}{t}\right)^{t} = e^{-b} + o(1). \end{equation}</p> <p>This inequality will be used slightly below.</p> <p>Let us define</p> <script type="math/tex; mode=display">n = \prod_{m=3,~ m ~\text{odd}}^b m^{c_m}.</script> <p>Since $m$ is odd, we see that $n$ is odd. Moreover, clearly, $H(n)$ is bounded from below by the number of multiset permutations</p> <script type="math/tex; mode=display">v(n) = \frac{(\sum_m c_m)!}{\prod_m c_m!}.</script> <p>Let us estimate $v(n)$ by Stirling’s bounds $\sqrt{2\pi} n^{n+1/2} e^{-n} \leq n! \leq e n^{n+1/2} e^{-n}$:</p> <script type="math/tex; mode=display">v(n) \geq \frac{\sqrt{2\pi}}{e^b} \sqrt{\frac{\sum_m c_m}{\prod_m c_m}} \prod_i\left(\frac{\sum_m c_m}{c_i}\right)^{c_i}.</script> <p>First, recalling that $c_i \leq \frac{t}{i^\gamma}$ and using \eqref{eq:summ}, we get</p> <script type="math/tex; mode=display">\prod_i \left(\frac{\sum_m c_m}{c_i}\right)^{c_i} \geq \prod_i \left(\frac{\sum_m c_m}{t}\right)^{c_i} \left(\prod_i i^{c_i}\right)^\gamma = \left(\frac{\sum_m c_m}{t}\right)^{\sum_m c_m} n^\gamma \geq (e^{-b} + o(1)) n^\gamma.</script> <p>Second, arguing exactly as in [<sup id="fnref:CL:1"><a href="#fn:CL" class="footnote">2</a></sup>], we deduce that</p> <script type="math/tex; mode=display">\sqrt{\frac{\sum_m c_m}{\prod_m c_m}} \geq c_b \frac{1}{(\ln n)^\frac{b-5}{4}}</script> <p>for some $c_b &gt; 0$.</p> <p>Combining the lattter two inequalities, we deduce that</p> <script type="math/tex; mode=display">H(n) \geq v(n) \geq C \frac{n^\gamma}{(\ln n)^\frac{b-5}{4}} > n^{\eta-\varepsilon}</script> <p>for some $C&gt;0$ and all sufficiently large $t$. The proof is complete.</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:CLM"> <p>Chor, B., Lemke, P., &amp; Mador, Z. (2000). On the number of ordered factorizations of natural numbers. Discrete Mathematics, 214(1-3), 123-133.&nbsp;<a href="#fnref:CLM" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:CL"> <p>Coppersmith, D., &amp; Lewenstein, M. (2005). Constructive bounds on ordered factorizations. SIAM Journal on Discrete Mathematics, 19(2), 301-303.&nbsp;<a href="#fnref:CL" class="reversefootnote">&#8617;</a>&nbsp;<a href="#fnref:CL:1" class="reversefootnote">&#8617;<sup>2</sup></a></p> </li> </ol> </div>Let us take any odd natural $n$ and denote by $H(n)$ the number of ways how $n$ can be represented as the product of factors larger or equal than $3$, where the order of factors matters. For instance, if $n=27$, then $H(n)=4$, since It is known that $H(n) &lt; n^{\eta}$, where $\eta = 1.37779\dots$ is the unique positive real zero of $\left(1-\frac{1}{2^s}\right) \zeta(s)=2$, see Theorem 5 in . Our aim is to show that this upper bound is optimal. Lemma. For any $\varepsilon&gt;0$ there exist infinitely many odd $n$ such that $H(n) &gt; n^{\eta-\varepsilon}$. Proof. We will argue along the same lines as in Section 3 of , where the similar optimality was obtained for all (even) numbers. First, we notice that Chor, B., Lemke, P., &amp; Mador, Z. (2000). On the number of ordered factorizations of natural numbers. Discrete Mathematics, 214(1-3), 123-133.&nbsp;&#8617; Coppersmith, D., &amp; Lewenstein, M. (2005). Constructive bounds on ordered factorizations. SIAM Journal on Discrete Mathematics, 19(2), 301-303.&nbsp;&#8617;Pleijel’s type estimate for the $p$-Laplacian2019-01-10T01:00:00+01:002020-02-18T15:18:55+01:00http://localhost:4000/blog/2019/01/Pleijels-type-estimate-for-the-p-Laplacian<p>Consider the sequence $\{\lambda_n(\Omega)\}$ of eigenvalues of the Dirichelet $p$-Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$ obtained via the Lusternik–Schnirelmann min-max approach. Let $\varphi_n$ be an eigenfunction associated to $\lambda_n(\Omega)$. We are interested in the estimates for the number of nodal domains of $\varphi_n$ which we denote as $\mu(\varphi_n)$.</p> <p>In the linear case $p=2$, the well-known Courant nodal domain theorem says that $\mu(\varphi_n) \leq n$ for all $n \geq 1$. Its generalization to the nonlinear case $p \neq 2$ obtained in [<sup id="fnref:DR"><a href="#fn:DR" class="footnote">1</a></sup>] asserts that</p> <script type="math/tex; mode=display">\mu(\varphi_n) \leq 2n-2 \quad \text{for all } n \geq 2,</script> <p>which implies</p> <script type="math/tex; mode=display">\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq 2.</script> <p>On the other hand, in the linear case $p=2$, there is a result of Pleijel [<sup id="fnref:pleijel"><a href="#fn:pleijel" class="footnote">2</a></sup>] on the following asymptotic refinement of the Courant nodal domain theorem:</p> <p>\begin{equation}\label{eq:P} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{4}{j_{0,1}^2} = 0.69166\ldots, \end{equation}</p> <p>see, e.g., <a href="/blog/2018/01/Pleijels-constant-for-the-disk.html">this post for a discussion</a>.</p> <p>The aim of the present post is to generalize the result of Pleijel to the $p$-Laplacian settings. Pleijel’s approach is purely variational and consists of two main ingredients: <em>the Faber-Krahn inequality</em> and <em>the Weyl law</em>.</p> <ol> <li><em>The Faber-Krahn inequality</em> is easily available for the $p$-Laplacian, and it can be formulated as</li> </ol> <script type="math/tex; mode=display">|\Omega|^\frac{p}{N} \lambda_1(\Omega) \geq |B_1|^\frac{p}{N} \lambda_1(B_1),</script> <p>where $B_1$ is a unit ball in $\mathbb{R}^N$; see, e.g., the discussion <a href="/blog/2017/10/Lower-bounds-for-the-first-eigenvalue-of-the-p-laplacian.html">here</a>. Therefore, noting that $\lambda_n(\Omega) = \lambda_1(\Omega_i)$ for any $i=1..\mu(\varphi_n)$ where $\Omega_i$ is a nodal domain of $\varphi_n$, we get</p> <hr /> <!--more--> <script type="math/tex; mode=display">|\Omega| \lambda_n(\Omega)^\frac{N}{p} \geq \mu(\varphi_n) |B_1| \lambda_1(B_1)^\frac{N}{p}.</script> <p>Equivalently,</p> <p>\begin{equation}\label{eq:FKP} \mu(\varphi_n) \leq \frac{|\Omega| \lambda_n(\Omega)^\frac{N}{p}}{|B_1| \lambda_1(B_1)^\frac{N}{p}}. \end{equation}</p> <ol> <li><em>The Weyl law</em> is used to estimate $\lambda_n(\Omega)$ in \eqref{eq:FKP} in terms of $n$. Unfortunately, this law is not available for the $p$-Laplacian in the required form; see the discussion in [<sup id="fnref:F"><a href="#fn:F" class="footnote">3</a></sup>] and [<sup id="fnref:APA"><a href="#fn:APA" class="footnote">4</a></sup>]. Instead, we will obtain the simplest explicit Weyl-type upper bound for $\lambda_n(\Omega)$, and this will be enough to get a Pleijel’s type result. Let $Q_h$ stands for the $N$-dimensional cube with the side length $h$. First, if $h \to 0$, then the number $m$ of cubes $Q_h$ disjointly inscribed in $\Omega$ is given by</li> </ol> <script type="math/tex; mode=display">m \approx \frac{|\Omega|}{h^N}.</script> <p>Second, by the variational characterization of $\lambda_n(\Omega)$, we can estimate</p> <script type="math/tex; mode=display">\lambda_n(\Omega) \leq \lambda_1(Q_{h_n}),</script> <p>where $h_n$ is such that there are $n$ disjoint cubes $Q_{h_n}$ inscribed in $\Omega$. We can assume that $h_n$ is maximal.</p> <p>Third, we know that</p> <script type="math/tex; mode=display">\lambda_1(Q_h) = \lambda_1(Q_1) h^{-p}.</script> <p>Combining the previous three facts, we get</p> <p>\begin{equation}\label{eq:WP} \lambda_n(\Omega) \leq \lambda_1(Q_{h_n}) = \lambda_1(Q_1) h_n^{-p} \approx \lambda_1(Q_1) \left(\frac{n}{|\Omega|}\right)^\frac{p}{N} \quad \text{as } n \to \infty. \end{equation}</p> <p>Finally, mixing \eqref{eq:FKP} and \eqref{eq:WP}, we deduce that</p> <p>\begin{equation}\label{eq:Plp} \boxed{\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{1}{|B_1|} \left(\frac{\lambda_1(Q_1)}{\lambda_1(B_1)} \right)^\frac{N}{p}.} \end{equation}</p> <p>Notice that this upper bound does not depend on $\Omega$. Below, we will discuss a possible way how to improve this bound.</p> <hr /> <p>All we need now is to get a ‘‘good’’ upper bound for $\lambda_1(Q_1)$ and a ‘‘good’’ lower bound for $\lambda_1(B_1)$.</p> <p>Let us start with an upper bound for $\lambda_1(Q_1)$. From Proposition 2.7 of [<sup id="fnref:BP"><a href="#fn:BP" class="footnote">5</a></sup>] we know that</p> <script type="math/tex; mode=display">% <![CDATA[ \lambda_1(Q_1) \leq \widetilde{\pi}_p^p N \quad \text{for} \quad p<2 %]]></script> <p>and</p> <script type="math/tex; mode=display">\lambda_1(Q_1) \leq \widetilde{\pi}_p^p N^\frac{p}{2} \quad \text{for} \quad p>2,</script> <p>where</p> <script type="math/tex; mode=display">\widetilde{\pi}_p = (p-1)^\frac{1}{p} \frac{2 \pi}{p \sin(\pi /p)} \equiv 2 (p-1)^\frac{1}{p} \int_0^1 \frac{ds}{(1-s^p)^\frac{1}{p}}.</script> <p>As lower estimates for $\lambda_1(B_1)$, we use the estimate</p> <script type="math/tex; mode=display">% <![CDATA[ \lambda_1(B_1) \geq N \left(\frac{p}{p-1}\right)^{p-1} \quad \text{for} \quad p<2, %]]></script> <p>see [<sup id="fnref:BEZ"><a href="#fn:BEZ" class="footnote">6</a></sup>] or [<sup id="fnref:BD2"><a href="#fn:BD2" class="footnote">7</a></sup>]; and</p> <script type="math/tex; mode=display">\lambda_1(B_1) \geq Np \quad \text{for} \quad p>2,</script> <p>see [<sup id="fnref:BD1"><a href="#fn:BD1" class="footnote">8</a></sup>] and, in general, <a href="/blog/2017/10/Lower-bounds-for-the-first-eigenvalue-of-the-p-laplacian.html">this post</a> for a discussion of lower bounds.</p> <p>Thus, substituting all these things into \eqref{eq:Plp}, we get</p> <p>\begin{equation}\label{eq:Pp&lt;2} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{\Gamma\left(\frac{N}{2}+1\right)\pi^\frac{N}{2} 2^N (p-1)^N}{p^\frac{(2p-1)N}{p} \sin(\pi /p)^N} \quad \text{for} \quad p&lt;2 \end{equation}</p> <p>and</p> <p>\begin{equation}\label{eq:Pp&gt;2} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{\Gamma\left(\frac{N}{2}+1\right)\pi^\frac{N}{2} 2^N N^\frac{(p-2)N}{2p} (p-1)^\frac{N}{p}}{p^\frac{(p+1)N}{p} \sin(\pi /p)^N} \quad \text{for} \quad p&gt;2. \end{equation}</p> <p>The corresponding plot is depicted below by the increasing line. We see that these upper bounds does not give us a Pleijel constant smaller than $1$ even in the dimension $N=2$, which is quite sad. Note that if $p \to 1$, then the bound \eqref{eq:Pp&lt;2} approaches $\frac{4}{\pi}=1.2732\dots$, while if $p \to \infty$, then the bound \eqref{eq:Pp&gt;2} approaches $\frac{8}{\pi}=2.5464\dots$, see the blue line on figure below.</p> <center><img src="\images\pleijel1.jpg" width="90%" border="15" /> <figcaption> $N=2$. </figcaption> </center> <hr /> <p>Let us now discuss a possible improvement of \eqref{eq:Plp} which concerns an improvement of the Weyl-type upper bound. For simplicity, let us fix $N=2$. First, we can inscribe in $\Omega$ not a square tiling, but a hexagonal tiling. If $H_r$ stands for a hexagon with the inradius $r$, and if $r \to 0$, then the number $m$ of $H_r$’s disjointly inscribed in $\Omega$ is given by</p> <script type="math/tex; mode=display">m \approx \frac{|\Omega|}{2 \sqrt{3} r^2}.</script> <p>Therefore, analogously to \eqref{eq:WP} we get</p> <script type="math/tex; mode=display">\lambda_n(\Omega) \leq \lambda_1(H_{r_n}) = \lambda_1(H_1) r_n^{-p} \approx \lambda_1(H_1) \left(\frac{2\sqrt{3} n}{|\Omega|}\right)^\frac{p}{2} \quad \text{as } n \to \infty,</script> <p>and hence, from \eqref{eq:FKP},</p> <p>\begin{equation}\label{eq:Plp1} \boxed{\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{2\sqrt{3}}{|B_1|} \left(\frac{\lambda_1(H_1)}{\lambda_1(B_1)} \right)^\frac{2}{p}.} \end{equation}</p> <p>Noting that $B_1 \subset H_1$, we get $\lambda_1(H_1) \leq \lambda_1(B_1)$, which yields</p> <script type="math/tex; mode=display">\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{2\sqrt{3}}{\pi} = 1.1026\dots</script> <p>Moreover, if $p \to \infty$, then, by the known result from [<sup id="fnref:JLM"><a href="#fn:JLM" class="footnote">9</a></sup>], $\lambda_1(H_1)^\frac{1}{p} \to 1$ and $\lambda_1(B_1)^\frac{1}{p} \to 1$, i.e., this upper estimate of the upper estimate \eqref{eq:Plp1} is sharp for $p \to \infty$. See the green line on the figure above.</p> <p>Thus, unfortunately, even if $n \to \infty$, we cannot show that $\mu(\varphi_n) \leq n$ for all $p&gt;1$ without getting a <em>substantial</em> improvement of the Weyl-type upper bound for $\lambda_n(\Omega)$. Such an improvement is clearly a prominent problem which needs to be studied much closer.</p> <hr /> <p>The PDF-version of this post can be found <a href="/pdf/posts/Pleijel-p-Laplacian.pdf">here</a>.</p> <hr /> <p>Last modified: 18-Feb-20</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:DR"> <p>Drábek, P., &amp; Robinson, S. B. (2002). On the generalization of the Courant nodal domain theorem. Journal of Differential Equations, 181(1), 58-71.&nbsp;<a href="#fnref:DR" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:pleijel"> <p>Pleijel, Å. (1956). Remarks on Courant’s nodal line theorem. Communications on pure and applied mathematics, 9(3), 543-550.&nbsp;<a href="#fnref:pleijel" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:F"> <p>Friedlander, L. (1989). Asymptotic behaviour of the eigenvalues of the $p$-laplacian. Communications in Partial Differential Equations, 14(8-9), 1059-1069.&nbsp;<a href="#fnref:F" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:APA"> <p>Azorero, J. G., &amp; Peral Alonso, I. (1988). Comportement asymptotique des valeurs propres du $p$-laplacien. Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 307(2), 75-78.&nbsp;<a href="#fnref:APA" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:BP"> <p>Bonder, J. F., &amp; Pinasco, J. P. (2008). Estimates for eigenvalues of quasilinear elliptic systems. Part II. Journal of Differential Equations, 245(4), 875-891.&nbsp;<a href="#fnref:BP" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:BEZ"> <p>Bueno, H., Ercole, G., &amp; Zumpano, A. (2009). Positive solutions for the $p$-Laplacian and bounds for its first eigenvalue. Advanced Nonlinear Studies, 9(2), 313-338.&nbsp;<a href="#fnref:BEZ" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:BD2"> <p>Benedikt, J., &amp; Drábek, P. (2013). Asymptotics for the principal eigenvalue of the $p$-Laplacian on the ball as p approaches 1. Nonlinear Analysis: Theory, Methods &amp; Applications, 93, 23-29.&nbsp;<a href="#fnref:BD2" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:BD1"> <p>Benedikt, J., &amp; Drábek, P. (2012). Estimates of the principal eigenvalue of the $p$-Laplacian. Journal of Mathematical Analysis and Applications, 393(1), 311-315.&nbsp;<a href="#fnref:BD1" class="reversefootnote">&#8617;</a></p> </li> <li id="fn:JLM"> <p>Juutinen, P., Lindqvist, P., &amp; Manfredi, J. J. (1999). The $\infty$-eigenvalue problem. Archive for rational mechanics and analysis, 148(2), 89-105.&nbsp;<a href="#fnref:JLM" class="reversefootnote">&#8617;</a></p> </li> </ol> </div>Consider the sequence $\{\lambda_n(\Omega)\}$ of eigenvalues of the Dirichelet $p$-Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$ obtained via the Lusternik–Schnirelmann min-max approach. Let $\varphi_n$ be an eigenfunction associated to $\lambda_n(\Omega)$. We are interested in the estimates for the number of nodal domains of $\varphi_n$ which we denote as $\mu(\varphi_n)$. In the linear case $p=2$, the well-known Courant nodal domain theorem says that $\mu(\varphi_n) \leq n$ for all $n \geq 1$. Its generalization to the nonlinear case $p \neq 2$ obtained in  asserts that which implies On the other hand, in the linear case $p=2$, there is a result of Pleijel  on the following asymptotic refinement of the Courant nodal domain theorem: \begin{equation}\label{eq:P} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{4}{j_{0,1}^2} = 0.69166\ldots, \end{equation} see, e.g., this post for a discussion. The aim of the present post is to generalize the result of Pleijel to the $p$-Laplacian settings. Pleijel’s approach is purely variational and consists of two main ingredients: the Faber-Krahn inequality and the Weyl law. The Faber-Krahn inequality is easily available for the $p$-Laplacian, and it can be formulated as where $B_1$ is a unit ball in $\mathbb{R}^N$; see, e.g., the discussion here. Therefore, noting that $\lambda_n(\Omega) = \lambda_1(\Omega_i)$ for any $i=1..\mu(\varphi_n)$ where $\Omega_i$ is a nodal domain of $\varphi_n$, we get Drábek, P., &amp; Robinson, S. B. (2002). On the generalization of the Courant nodal domain theorem. Journal of Differential Equations, 181(1), 58-71.&nbsp;&#8617; Pleijel, Å. (1956). Remarks on Courant’s nodal line theorem. Communications on pure and applied mathematics, 9(3), 543-550.&nbsp;&#8617;Higher-order variations of the $p$-Dirichlet energy2018-11-08T01:00:00+01:002018-11-08T16:01:43+01:00http://localhost:4000/blog/2018/11/Higher-order-variations-of-the-p-Dirichlet-energy<p>In this post, we would like to discuss some combinatorial aspects of the $p$-Laplacian. Namely, let $\int_\Omega |\nabla u|^p \, dx$ be the $p$-Dirichlet energy, where $u \in W^{1,p}(\Omega)$ and $p&gt;1$. Its first variation is given by</p> <script type="math/tex; mode=display">D^1 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1) = p \int_\Omega |\nabla u|^{p-2} (\nabla u, \nabla \xi_1) \, dx,</script> <p>where $\xi_1 \in W^{1,p}(\Omega)$.</p> <p>The second variation (if exists) is also easy to compute:</p> <p>\begin{align} \notag D^2 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2) &amp;= p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) \, dx \newline \notag &amp;+ p \int_\Omega |\nabla u|^{p-2} (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align}</p> <p>where $\xi_2 \in W^{1,p}(\Omega)$.</p> <p>Let us make some effort to calculate the third variation:</p> <p>\begin{align} \notag D^3 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2, \xi_3) &amp;= p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) \, dx \newline \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla \xi_2, \nabla \xi_3) \, dx \newline \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_2) (\nabla \xi_1, \nabla \xi_3) \, dx \newline \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_3) (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align}</p> <p>where $\xi_3 \in W^{1,p}(\Omega)$.</p> <p>We already start seeing some structure. So, let us now try to derive a general formula for the $n$-th variation of the energy functional. Our main result is the following one.</p> <p><strong>Proposition.</strong> Let $u \in W^{1,p}(\Omega)$. If for a natural $n \geq 1$ there exists $n$-th variation of the $p$-Direchlet energy of $u$ in direction $(\xi_1,\dots,\xi_n) \in (W^{1,p}(\Omega))^n$, then</p> <p>\begin{align} \notag &amp;D^n \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \dots, \xi_n) \newline \notag &amp;= \int_\Omega \left( \sum\limits_{i=0}^{\lfloor \frac{n}{2} \rfloor} |\nabla u|^{p-2(n-i)} \prod\limits_{j=0}^{n-i-1} (p-2j) \left[\sum\limits_{\sigma \in B(n,n-2i)} \prod\limits_{k=1}^{n-2i} (\nabla u, \nabla \xi_{\sigma(k)}) \left(\sum\limits_{\omega \in P(n,\sigma)} \prod\limits_{l=1}^{i} (\nabla \xi_{\omega(l,1)}, \nabla \xi_{\omega(l,2)}) \right) \right] \right) dx, \end{align}</p> <p>where</p> <ul> <li> <p>$B(n,n-2i)$ is the set of all possible $(n-2i)$-combinations of $\{1,2,\dots,n\}$ such that the ordering inside each $\sigma \in B(n,n-2i)$ is immaterial. Evidently, the cardinality of $B(n,n-2i)$ is ${n \choose n-2i}$. In particular, if $i=0$, then $card(B(n,n-2i)) = 1$.</p> </li> <li> <p>$P(n,\sigma)$ is the set of all possible partitions of the set $\{1,2,\dots,n\} \setminus \sigma$ into pairs such that the ordering of pairs and inside a pair is immaterial. Note that $card(\sigma)=n-2i$, and hence the number of pairs in each $\omega \in P(\sigma)$ is $i$. It is not hard to see that the cardinality of $P(\sigma)$ is $\frac{(2i)!}{2^i i!}$. We represent $\omega$ as a $i \times 2$-matrix $(\omega(s,t))_{s=1..i,~t=1,2}$. For instance, if $n=6$ and $\sigma = \{1,2\}$, then</p> </li> </ul> <script type="math/tex; mode=display">% <![CDATA[ P(\sigma) = \left\{ \begin{pmatrix} 3 & 5 \newline 4 & 6 \end{pmatrix}, \begin{pmatrix} 3 & 4 \newline 6 & 5 \end{pmatrix}, \begin{pmatrix} 3 & 4 \newline 4 & 6 \end{pmatrix} \right\}. %]]></script> <hr /> <!--more--> <p>Let us also calculate the fourth variation (just for fun) - either by straightforward computation, or by application of our general formula:</p> <p>\begin{align} \notag D^4 &amp;\left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2, \xi_3) \\ \notag &amp;= p(p-2)(p-4)(p-6) \int_\Omega |\nabla u|^{p-8} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) (\nabla u, \nabla \xi_4) \, dx \\ \notag &amp;+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla \xi_3, \nabla \xi_4) \, dx \\ \notag &amp;+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_3) (\nabla \xi_2, \nabla \xi_4) \, dx \\ \notag &amp;+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_4) (\nabla \xi_2, \nabla \xi_3) \, dx \\ \notag &amp;+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) (\nabla \xi_1, \nabla \xi_4) \, dx \\ \notag &amp;+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_4) (\nabla \xi_1, \nabla \xi_3) \, dx \\ \notag &amp;+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_3) (\nabla u, \nabla \xi_4) (\nabla \xi_1, \nabla \xi_2) \, dx \\ \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla \xi_1, \nabla \xi_2) (\nabla \xi_3, \nabla \xi_4) \, dx \\ \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla \xi_1, \nabla \xi_3) (\nabla \xi_2, \nabla \xi_4) \, dx \\ \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla \xi_1, \nabla \xi_4) (\nabla \xi_2, \nabla \xi_3) \, dx. \end{align}</p> <p>where $\xi_4 \in W^{1,p}(\Omega)$.</p> <hr /> <p>Visually, it could be easier to present this result as an $n$-th directional derivative of the $p$-th power of the norm of a vector. Namely, if $A, B_i \in \mathbb{R}^N$, then for any natural $n \geq 1$, we have</p> <p>\begin{align} \notag &amp;D^n \left(|A|^p\right) (B_1, \dots, B_n) \newline \notag &amp;= \sum\limits_{i=0}^{\lfloor \frac{n}{2} \rfloor} |A|^{p-2(n-i)} \prod\limits_{j=0}^{n-i-1} (p-2j) \left[\sum\limits_{\sigma \in B(n,n-2i)} \prod\limits_{k=1}^{n-2i} (A, B_{\sigma(k)}) \left(\sum\limits_{\omega \in P(n,\sigma)} \prod\limits_{l=1}^{i} (B_{\omega(l,1)}, B_{\omega(l,2)}) \right) \right]. \end{align}</p> <hr /> <p>The PDF-version of this post can be found <a href="/pdf/posts/Higher-order-variation.pdf">here</a>.</p> <hr /> <p>Last modified: 08-Nov-18</p> <hr />In this post, we would like to discuss some combinatorial aspects of the $p$-Laplacian. Namely, let $\int_\Omega |\nabla u|^p \, dx$ be the $p$-Dirichlet energy, where $u \in W^{1,p}(\Omega)$ and $p&gt;1$. Its first variation is given by where $\xi_1 \in W^{1,p}(\Omega)$. The second variation (if exists) is also easy to compute: \begin{align} \notag D^2 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2) &amp;= p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) \, dx \newline \notag &amp;+ p \int_\Omega |\nabla u|^{p-2} (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align} where $\xi_2 \in W^{1,p}(\Omega)$. Let us make some effort to calculate the third variation: \begin{align} \notag D^3 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2, \xi_3) &amp;= p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) \, dx \newline \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla \xi_2, \nabla \xi_3) \, dx \newline \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_2) (\nabla \xi_1, \nabla \xi_3) \, dx \newline \notag &amp;+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_3) (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align} where $\xi_3 \in W^{1,p}(\Omega)$. We already start seeing some structure. So, let us now try to derive a general formula for the $n$-th variation of the energy functional. Our main result is the following one. Proposition. Let $u \in W^{1,p}(\Omega)$. If for a natural $n \geq 1$ there exists $n$-th variation of the $p$-Direchlet energy of $u$ in direction $(\xi_1,\dots,\xi_n) \in (W^{1,p}(\Omega))^n$, then \begin{align} \notag &amp;D^n \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \dots, \xi_n) \newline \notag &amp;= \int_\Omega \left( \sum\limits_{i=0}^{\lfloor \frac{n}{2} \rfloor} |\nabla u|^{p-2(n-i)} \prod\limits_{j=0}^{n-i-1} (p-2j) \left[\sum\limits_{\sigma \in B(n,n-2i)} \prod\limits_{k=1}^{n-2i} (\nabla u, \nabla \xi_{\sigma(k)}) \left(\sum\limits_{\omega \in P(n,\sigma)} \prod\limits_{l=1}^{i} (\nabla \xi_{\omega(l,1)}, \nabla \xi_{\omega(l,2)}) \right) \right] \right) dx, \end{align} where $B(n,n-2i)$ is the set of all possible $(n-2i)$-combinations of $\{1,2,\dots,n\}$ such that the ordering inside each $\sigma \in B(n,n-2i)$ is immaterial. Evidently, the cardinality of $B(n,n-2i)$ is ${n \choose n-2i}$. In particular, if $i=0$, then $card(B(n,n-2i)) = 1$. $P(n,\sigma)$ is the set of all possible partitions of the set $\{1,2,\dots,n\} \setminus \sigma$ into pairs such that the ordering of pairs and inside a pair is immaterial. Note that $card(\sigma)=n-2i$, and hence the number of pairs in each $\omega \in P(\sigma)$ is $i$. It is not hard to see that the cardinality of $P(\sigma)$ is $\frac{(2i)!}{2^i i!}$. We represent $\omega$ as a $i \times 2$-matrix $(\omega(s,t))_{s=1..i,~t=1,2}$. For instance, if $n=6$ and $\sigma = \{1,2\}$, thenSobolev-type inequality with product of norms2018-10-19T02:00:00+02:002019-01-10T19:47:45+01:00http://localhost:4000/blog/2018/10/Sobolev-type-inequality-with-products-of-norms<p><strong>Lemma.</strong> <em>Let $\Omega \subset \mathbb{R}^N$ be a bounded open set, $N \geq 1$. Let $1 \leq q \leq p \leq \gamma$, and $\gamma \leq p^*$ if $N&gt;p$ and $\gamma&lt;\infty$ if $N &lt; p$. Then there exists $C=C(\Omega,q,p,\gamma)$ such that for any $u \in W_0^{1,p}(\Omega)$ the following inequality is satisfied:</em></p> <script type="math/tex; mode=display">\left(\int_\Omega |u|^q \, dx \right)^{\gamma-p} \left(\int_\Omega |u|^\gamma \, dx \right)^{p-q} \leq C \left(\int_\Omega |\nabla u|^p \, dx \right)^{\gamma-q}.</script> <p>In fact, this inequality easily follows from the general Sobolev inequality just by applying the latter one to each term on the lhs of the former one. (I’m almost sure there could be some investigation of an inequality of this type in the literature, but I was not able to find it.)</p> <p>There are a couple of things about this inequality:</p> <p>1) It is just beautiful that the exponents are permuted)</p> <p>2) If one considers the optimization problem for the constant $C$, then it is possible to see that a corresponding minimizer of</p> <script type="math/tex; mode=display">\inf_{u \in W_0^{1,p}(\Omega) \setminus \{0\}} \frac{\left(\int_\Omega |\nabla u|^p \, dx \right)^{\gamma-q}}{\left(\int_\Omega |u|^q \, dx \right)^{\gamma-p} \left(\int_\Omega |u|^\gamma \, dx \right)^{p-q}}</script> <p>exists and satisfies the convex-concave equation</p> <script type="math/tex; mode=display">-\Delta_p u = \lambda |u|^{q-2} u + |u|^{\gamma-2} u.</script> <p>In fact, such a minimization problem was explicitly considered in connection with the above equation in [<sup id="fnref:I"><a href="#fn:I" class="footnote">1</a></sup>].</p> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:I"> <p>Il’yasov, Y. (2005). On nonlocal existence results for elliptic equations with convex–concave nonlinearities. Nonlinear Analysis: Theory, Methods &amp; Applications, 61(1-2), 211-236.&nbsp;<a href="#fnref:I" class="reversefootnote">&#8617;</a></p> </li> </ol> </div>Lemma. Let $\Omega \subset \mathbb{R}^N$ be a bounded open set, $N \geq 1$. Let $1 \leq q \leq p \leq \gamma$, and $\gamma \leq p^*$ if $N&gt;p$ and $\gamma&lt;\infty$ if $N &lt; p$. Then there exists $C=C(\Omega,q,p,\gamma)$ such that for any $u \in W_0^{1,p}(\Omega)$ the following inequality is satisfied: In fact, this inequality easily follows from the general Sobolev inequality just by applying the latter one to each term on the lhs of the former one. (I’m almost sure there could be some investigation of an inequality of this type in the literature, but I was not able to find it.) There are a couple of things about this inequality: 1) It is just beautiful that the exponents are permuted) 2) If one considers the optimization problem for the constant $C$, then it is possible to see that a corresponding minimizer of exists and satisfies the convex-concave equation In fact, such a minimization problem was explicitly considered in connection with the above equation in . Bibliography Il’yasov, Y. (2005). On nonlocal existence results for elliptic equations with convex–concave nonlinearities. Nonlinear Analysis: Theory, Methods &amp; Applications, 61(1-2), 211-236.&nbsp;&#8617;Growth rate of divisor function for odd numbers2018-09-25T02:00:00+02:002018-09-25T23:40:01+02:00http://localhost:4000/blog/2018/09/Growth-rate-of-divisor-function-for-odd-numbers<p>Consider the standard divisor function $\sigma_\alpha(n)$ defined as</p> <script type="math/tex; mode=display">\sigma_\alpha(n) = \Sigma_{d | n} d^\alpha.</script> <p>Assume that $\alpha &gt; 1$. It was proved at least by Gronwall in [<sup id="fnref:G"><a href="#fn:G" class="footnote">1</a></sup>] (yeah, this is the same Gronwall who obtained <a href="https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality">Gronwall’s inequality</a>) that $\sigma_\alpha(n)$ satisfies</p> <script type="math/tex; mode=display">% <![CDATA[ \sigma_\alpha(n) < \zeta(\alpha) n^\alpha %]]></script> <p>for all $n \in \mathbb{N}$, and this upper bound is achieved. Here $\zeta(\alpha)$ is the Riemann Zeta function.</p> <p>Recently, thinking with Falko Baustian on one problem, we were partially interested in the asymptotic behaviour of $\sigma_\alpha(n)$ for odd numbers $n$. The natural question is whether we can improve the upper bound for odd $n$’s?</p> <p><strong>Proposition.</strong> Let $\alpha&gt;1$, and let $n \in \mathbb{N}$ be odd. Then</p> <script type="math/tex; mode=display">% <![CDATA[ \sigma_\alpha(n) < \left(1-\frac{1}{2^\alpha}\right) \zeta(\alpha) n^\alpha, %]]></script> <p>and this upper bound is achieved.</p> <p><em>Proof</em>. We will argue sketchely and along the same lines as in [<sup id="fnref:G:1"><a href="#fn:G" class="footnote">1</a></sup>], pp. 114-116. First, we decompose an arbitrary $n$ into prime factors as</p> <script type="math/tex; mode=display">n = p_{\lambda_1}^{\nu_1} \cdot \ldots \cdot p_{\lambda_k}^{\nu_k},</script> <p>where $p_i$ is a prime, $\lambda_i &lt; \lambda_{i+1}$, and $\nu_i &gt; 0$. Note that since $n$ is odd, we have $p_i \neq 2$.</p> <p>As in formula (5) of [<sup id="fnref:G:2"><a href="#fn:G" class="footnote">1</a></sup>], we have</p> <p>\begin{equation}\label{eq:sigma} \sigma_\alpha(n) = n^\alpha \Pi_{i=1}^k \frac{1-\frac{1}{p_{\lambda_i}^{\alpha(\nu_i+1)}}}{1-\frac{1}{p_{\lambda_i}^\alpha}}. \end{equation}</p> <p>Since the numerator and denominator in \eqref{eq:sigma} are less than one and since $n$ is odd, we can estimate $\sigma_\alpha(n)$ as</p> <script type="math/tex; mode=display">% <![CDATA[ \sigma_\alpha(n) < n^\alpha \Pi_{i=1}^k \frac{1}{1-\frac{1}{p_{\lambda_i}^\alpha}} < n^\alpha \Pi_{i=2}^\infty \frac{1}{1-\frac{1}{p_{i}^\alpha}} = n^\alpha \left(1-\frac{1}{2^\alpha}\right) \Pi_{i=1}^\infty \frac{1}{1-\frac{1}{p_{i}^\alpha}} = n^\alpha \left(1-\frac{1}{2^\alpha}\right) \zeta(\alpha), %]]></script> <p>where $p_i$ are consequtive primes $p_1 = 2$, $p_2=3$, etc. Therefore, the desired upper bound is obtained.</p> <hr /> <!--more--> <p>Let us now prove that the upper bound is achieved. For this end, we consider the admissible sequence</p> <script type="math/tex; mode=display">n_{k,\nu} = (p_2 \cdot \ldots \cdot p_k)^{\nu-1}.</script> <p>Note that</p> <script type="math/tex; mode=display">\Pi_{i=2}^k \frac{1-\frac{1}{p_{i}^{\alpha \nu}}}{1-\frac{1}{p_{i}^\alpha}} \to \frac{1-\frac{1}{2^\alpha}}{1-\frac{1}{2^{\nu \alpha}}} \frac{\zeta(\alpha)}{\zeta(\nu \alpha)}</script> <p>uniformly convergent with respect to $\nu$ as $k \to \infty$, see [<sup id="fnref:G:3"><a href="#fn:G" class="footnote">1</a></sup>], p. 115. Thus, noting that $1/2^{\nu \alpha} \to 0$ and $\zeta(\nu \alpha) \to 1$ as $\nu \to \infty$, we conclude from \eqref{eq:sigma} that</p> <script type="math/tex; mode=display">\frac{\sigma_\alpha(n_{k,k+1})}{n_{k,k+1}^\alpha} \to \left(1-\frac{1}{2^\alpha}\right) \zeta(\alpha).</script> <hr /> <h1 id="bibliography">Bibliography</h1> <div class="footnotes"> <ol> <li id="fn:G"> <p>Gronwall, T. H. (1913). Some asymptotic expressions in the theory of numbers. Transactions of the American Mathematical Society, 14(1), 113-122.&nbsp;<a href="#fnref:G" class="reversefootnote">&#8617;</a>&nbsp;<a href="#fnref:G:1" class="reversefootnote">&#8617;<sup>2</sup></a>&nbsp;<a href="#fnref:G:2" class="reversefootnote">&#8617;<sup>3</sup></a>&nbsp;<a href="#fnref:G:3" class="reversefootnote">&#8617;<sup>4</sup></a></p> </li> </ol> </div>Consider the standard divisor function $\sigma_\alpha(n)$ defined as Assume that $\alpha &gt; 1$. It was proved at least by Gronwall in  (yeah, this is the same Gronwall who obtained Gronwall’s inequality) that $\sigma_\alpha(n)$ satisfies for all $n \in \mathbb{N}$, and this upper bound is achieved. Here $\zeta(\alpha)$ is the Riemann Zeta function. Recently, thinking with Falko Baustian on one problem, we were partially interested in the asymptotic behaviour of $\sigma_\alpha(n)$ for odd numbers $n$. The natural question is whether we can improve the upper bound for odd $n$’s? Proposition. Let $\alpha&gt;1$, and let $n \in \mathbb{N}$ be odd. Then and this upper bound is achieved. Proof. We will argue sketchely and along the same lines as in , pp. 114-116. First, we decompose an arbitrary $n$ into prime factors as where $p_i$ is a prime, $\lambda_i &lt; \lambda_{i+1}$, and $\nu_i &gt; 0$. Note that since $n$ is odd, we have $p_i \neq 2$. As in formula (5) of , we have \begin{equation}\label{eq:sigma} \sigma_\alpha(n) = n^\alpha \Pi_{i=1}^k \frac{1-\frac{1}{p_{\lambda_i}^{\alpha(\nu_i+1)}}}{1-\frac{1}{p_{\lambda_i}^\alpha}}. \end{equation} Since the numerator and denominator in \eqref{eq:sigma} are less than one and since $n$ is odd, we can estimate $\sigma_\alpha(n)$ as where $p_i$ are consequtive primes $p_1 = 2$, $p_2=3$, etc. Therefore, the desired upper bound is obtained. Gronwall, T. H. (1913). Some asymptotic expressions in the theory of numbers. Transactions of the American Mathematical Society, 14(1), 113-122.&nbsp;&#8617;&nbsp;&#8617;2&nbsp;&#8617;3