Can one hear the shape of a drum

Fil: Kaufmann, Uriel. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina

Fractional Sobolev spaces with variable exponents and fractional p(X)-Laplacians

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Optimal Shape Design For Elliptic Systems

Roughly speaking, the basic idea behind the so-called minimax method is the following: Find a critical value of a functional ϕ ∈ C1 (X, ℝ) as a minimax (or maximin) value c ∈ ℝ of ϕ over a suitable class A of subsets of X: 

$$ c = \mathop {\inf }\limits_{A \in \mathcal{A}} \mathop {\sup }\limits_{u \in A} \phi \left( u \right). $$

The Mountain-Pass Theorem

Optimal shape design

In this paper we analyze possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a nonlocal eigenvalue problem of fractional type that approximate, when taking a suitable limit, the classical Steklov eigenvalue problem.

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Fractional eigenvalue problems that approximate Steklov eigenvalues

In this paper we deal with the stability of solutions of fractional $p-$Laplace problems with nonlinear sources when the fractional parameter $s$ goes to 1. We prove a general convergence result for general weak solutions which is applied to study the convergence of ground state solutions of $p-$fractional problems in bounded and unbounded domains as $s$ goes to 1. Moreover, our result applies to treat the stability of $p-$fractional eigenvalues as $s$ goes to 1.

Stability of solutions for nonlocal problems.

In this work, we study the asymptotic behavior of the curves of the Fucik spectrum for weighted second-order linear ordinary differential equations. We prove a Weyl type asymptotic behavior of the hyperbolic type curves in the spectrum in terms of some integrals of the weights. We present an algorithm which computes the intersection of the Fucik spectrum with rays through the origin, and we compare their values with the asymptotic ones.

Asymptotic behavior of the curves in the Fučík spectrum

We study the rate of convergence for (variational) eigenvalues of several non-linear problems involving oscillating weights and subject to different kinds of boundary conditions in bounded domains.

Eigenvalue homogenization for quasilinear elliptic equations with different boundary conditions

In this paper we continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption \begin{align} \begin{cases} u_t = \mathcal{L} u-u^p\quad& \mbox{in}\quad \mathbb R^N\times(0,\infty),\\ u(x,0) = u_0(x)\quad& \mbox{in}\quad \mathbb R^N, \end{cases} \end{align} where $p>1$, $u_0\ge0$ and bounded and $$ \mathcal{L} u(x,t)=\int J(x-y)\left(u(y,t)-u(x,t)\right)\,dy $$ with $J\in C_0^{\infty}(\mathbb R^N)$, radially symmetric, $J\geq 0$ with $\int J=1$. 
Our assumption on the initial datum is that $0\le u_0\in L^\infty(\mathbb R^N)$ and $$ |x|^{\alpha}u_0(x)\to A>0\quad\mbox{as}\quad|x|\to\infty. $$ 
This problem was studied in the supercritical and critical cases $p\ge 1+2/\alpha$. %See also \cite{PR,TW2} for the case $u_0\in L^\infty(\mathbb R^N)\cap L^1(\mathbb R^N)$, $p\ge 1+2/N$. 
In the present paper we study the subcritical case $1 0$.

Ariel Martin Salort

Papers

Fractional eigenvalue problems that approximate Steklov eigenvalues

Stability of solutions for nonlocal problems.

Asymptotic behavior of the curves in the Fučík spectrum

Eigenvalue homogenization for quasilinear elliptic equations with different boundary conditions

Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data: the subcritical case