Consider the Fucik eigenvalue problem

where $u = u^+ - u^-$, and $u^\pm := \max{\pm u, 0}$.

In [1] 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 $\beta > \lambda_1(p)$. This definition is, in essence, the same as of Theorem 1.2 in [2] for the linear case $p=2$ (see also [3]), and it was pointed out in that works that for $p>1$ 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.


Proof. 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 [1]), i.e., $c(s) > c(s’)$ whenever $s < s’$; moreover, $c(s) \to \lambda_1(p)$ as $s \to +\infty$, see Proposition 4.4 in [1]. Thus, for each $\beta > \lambda_1(p)$ there exists unique $s \in \mathbb{R}$ such that $\beta = c(s)$. (See figure below). Notice that the $c(s)$ is constructed in [1] 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.

Let us show now that $\alpha^*(c(s)) = s + c(s)$ for any $c(s) = \beta > \lambda_1(p)$. 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 $\alpha^*(c(s)) < s + c(s)$ for some $s$. Then, by definition of $\alpha^*(c(s))$, there have to exist a function $u \in W_0^{1,p}$ such that

Due to the continuity and monotonicity of $c(s)$ (see Proposition 4.1 in [1]), there exists $s_0$ such that

or, equivalently,

which is, in fact, the main contradictory assumption in the proof of Theorem 3.1 in [1] (see also the proof of Lemma 5.3, (5.10) in [1]). Thus, proceeding exactly as in the proof of Theorem 3.1 in [1], we obtain a contradiction to the definition of $c(s_0)$.


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Last modified: 11-Aug-20


Bibliography

  1. 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.  2 3 4 5 6 7 8

  2. 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. 

  3. 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.