Related papers: The \infty eigenvalue problem and a problem of opt…

In this paper, we consider the optimization problem for the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the $p$-Laplacian $\Delta_p$, $1< p< \infty$, over a family of doubly connected planar domains $\Omega= B \setminus \overline{P}$,…

We consider optimization problems of the first eigenvalue of elliptic operators with applications to two-phase optimal design problems (also known as topology optimization problems) of conductivity and elasticity relaxed by homogenization.…

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of the anisotropic laplacian perturbed by an integral of the unknown function. Using also some properties related to the associated \lq\lq…

We prove existence and uniqueness results for solutions to a class of optimal transportation problems with infinitely many marginals, supported on the real line. We also provide a characterization of the solution with an explicit formula.…

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via…

We study the first eigenvalue of the $p-$Laplacian (with $1<p<\infty$) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the…

We study a shape optimization problem associated with the first eigenvalue of a nonlinear spectral problem involving a mixed operator ($p-$Laplacian and Laplacian) with a constraint on the volume. First, we prove the existence of a…

We study a rather general class of optimal "ballistic" transport problems for matrix-valued measures. These problems naturally arise, in the spirit of \emph{Y. Brenier. Comm. Math. Phys. (2018) 364(2) 579-605}, from a certain dual…

We consider an eigenvalue problem of the form \begin{equation*} \left\{\begin{array}{rclll} -\Delta_{p} u -\Delta_{q} u&=& \lambda K(x)|u|^{p-2}u & \mbox{ in } \Omega^e u&=&0\qquad \quad &\mbox{ on } \partial \Omega u(x) &\to& 0 &\mbox{ as…

We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which…

We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First,…

We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of…

We collect some examples of optimal transports in order to explore the (in)stability of the identity map as an optimal transport. First, we consider density and domain perturbations near regular portions of domains. Second, we investigate…

In this paper we study an optimal shape design problem for the first eigenvalue of the fractional $p-$laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is…

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove…

We study a non-local eigenvalue problem related to the fractional Sobolev spaces for large values of p and derive the limit equation as p goes to infinity. Its viscosity solutions have many interesting properties and the eigenvalues exhibit…

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with…

We consider an elliptic operator in which the second-order term is very small in one direction. In this regime, we study the behaviour of the principal eigenfunction and of the principal eigenvalue. Our first result deals with the limit of…

The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…

In this paper we deal with a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and prove a reverse H\"older type inequality for the…