Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ will be identified with \sgn(u). We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if $\Omega$ is radial, then least energy nodal solutions are foliated Schwarz symmetric, and they are nonradial in case $\Omega$ is a ball. The case $q=1$ requires special treatment since the formally associated energy functional is not differentiable, and many arguments have to be adjusted

The second eigenvalue of the fractional p−p-Laplacian

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem $$\inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}.$$ We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.Comment: 38 pages. The test function used in the proof of Theorem 3.1 needed to be slightly modified, in order to be admissible for $1<p<2$. We fixed this issu

Asymptotic behaviour of higher eigenfunctions of the p-Laplacian as p goes to 1

Subject of this thesis is the asymptotic behaviour of the higher eigenvalues of the p-Laplacian operator as p goes to 1. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if p is close enough to 1. Moreover, it is shown that second eigenfunctions can be obtained as limit of least energy nodal solutions of a p-superlinear problem

The fractional Cheeger problem

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.Comment: 33 pages, 2 figure