Trace and extension operators for fractional Sobolev spaces with variable exponent

We show that, under certain regularity assumptions, there exists a linear extension operator

THE DISCRETE LAPLACIAN ACTING ON 3-FORMS

In the current paper, we study the discrete Laplacian acting on 3-forms. We establish a new criterion of essential self-adjointness using the Nelson lemma. Moreover, we give an upper bound on the infimum of the essential spectrum

Density properties for fractional Sobolev spaces with variable exponents

International audienceIn this article we show some density properties of smooth and compactly supported functions in fractional Sobolev spaces with variable exponents. The additional difficulty in this nonlocal setting is caused by the fact that the variable exponent Lebesgue spaces are not translation-invariant

THE DISCRETE LAPLACIAN OF A 3-SIMPLICIAL COMPLEX

In this paper, We introduce the notion of oriented tetrahedrons especially tetrahedrons in a connected oriented locally finite graph. This framework then permits to define the Laplace operator on this structure of the 3-simplicial complex. We develop the notion of X-completeness for the graphs, based on the cut-off functions. Moreover, We study essential self-adjointness of the discrete Laplacian from the X-completeness geometric hypothesis

THEORY OF CAPACITIES IN FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS

In this paper we develop a capacities theory connected with the fractional Sobolev spaces with variable exponents. Two kinds of capacities are studied: Sobolev capacity and relative capacity. Basic properties of capacities, including monotonicity, outer capacity and several results, are studies. We prove that both capacities is a Choquet capacity and all borel sets are capacitable. Mathematics Subject Classification: Primary 31B15, 46E35

A class of fractional parabolic reaction-diffusion systems with control of total mass: theory and numerics

In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems posed in a bounded domain of $\mathbb{R}^N$. The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide non-negativity of the solutions and uniform control of the total mass. The diffusion operators are of type $u_i\mapsto d_i(-\Delta)^s u_i$ where $0<s<1$. Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type $u_i\mapsto -d_i\Delta u_i$. On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case $s=1$

On the Continuous embeddings between the fractional Haj{\l}asz-Orlicz-Sobolev spaces

Let $G$ be an Orlicz function and let $\alpha, \beta, s$ be positive real numbers. Under certain conditions on the Orlicz function $G$, we establish some continuous embeddings results between the fractional order Orlicz-Sobolev spaces defined on metric-measure spaces $W_s^{\alpha, G}(X, d, \mu)$ and the fractional Haj{\l}asz-Orlicz-Sobolev spaces $M^{\beta, G}(X,d,\mu)$

STIELTJES CRITERION FOR DISCRETE LAPLACIAN ON 3-SIMPLICIAL COMPLEX

In this paper, we continue the discussion of the question of essential self-adjointness for the discrete Laplacian acting on 3-simplicial complex. We establish a new criterion using Stieltjes vectors approach