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

Fractional order Orlicz-Sobolev spaces

Eigenvalues and minimizers for a non-standard growth non-local operator

We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev–Slobodeckiĭ norm. We compare it to the fractional Sobolev space obtained by the K-method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.

/pdf/a-note-on-homogeneous-sobolev-spaces-of-fractional-order-3sb5t7cc3q.pdf

A note on homogeneous Sobolev spaces of fractional order

In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter $s\uparrow 1$ in the spirit of the celebrated result of Bourgain-Brezis-Mironescu. We then deduce some consequences such as $\Gamma-$convergence of the modulars and convergence of solutions for some fractional versions of the $\Delta_g$ operator as the fractional parameter $s\uparrow 1$.

In this article, we prove modular and norm Polya–Szego inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes ...

/pdf/a-polya-szego-principle-for-general-fractional-orlicz-2gezbr53jd.pdf

Ariel Martin Salort

Papers

Fractional order Orlicz-Sobolev spaces

Eigenvalues and minimizers for a non-standard growth non-local operator

A note on homogeneous Sobolev spaces of fractional order

Fractional order Orlicz-Sobolev spaces

A Pólya–Szegö principle for general fractional Orlicz–Sobolev spaces