• Abstract

The paper addresses the doubly elliptic eigenvalue problem

where $Ω$ is a bounded open subset of $\mathbb{R}^N$ $(N ≥ 2)$ with a $C^1$ boundary $Γ = Γ_0∪Γ_1,$$Γ_0∩Γ_1 = ∅,$ $Γ_1$ being nonempty and relatively open on $Γ.$ Moreover, $\mathcal{H}^{N-1}(\overline{\Gamma}_0 \cap \overline{\Gamma}_1)=0$ and $\mathcal{H}^{N−1} (Γ_0) > 0.$ We prove that $L^2 (Ω)×L^2 (Γ_1)$ admits a Hilbert basis constituted by eigenfunctions and we describe the behavior of the eigenvalues. Moreover, when $Γ$ is at least $C^2$ and $\overline{Γ}_0∩\overline{Γ}_1=∅,$ we give several qualitative properties of the eigenfunctions.

• General Summary

• Abstract

In this paper we obtain uncertainty and Strichartz type inequalities for fractional Hajłasz-Sobolev spaces in the metric setting.

• General Summary

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We present some relaxation and integral representation results for energy functionals in the setting of structured deformations, with special emphasis given to the case of multi-level structured deformations. In particular, we present an integral representation result for an abstract class of variational functionals in this framework via a global method for relaxation and identify, under quite general assumptions, the corresponding relaxed energy densities through the study of a related local Dirichlet-type problem.
Some applications to specific relaxation problems are also mentioned, showing that our global method approach recovers some previously established results.

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In this article, we consider the large time behavior of solutions $ρ$ to the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay. More precisely, we derive the time decay estimate of the solution to the Cauchy problem, provided by $ρ_0(0)∈L^1$ and $ρ_0∈C([−τ,0];H^1).$ The result is based on the combination of the weighted $L^2$ energy estimate and the $L^1$ estimate.

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In this paper, we analyze the effective behaviour of the solution of an elliptic problem in a two-phase composite material with non-standard imperfect contact conditions between its constituents. More specifically, we consider on the interface an equi-valued surface condition and a non-local flux condition involving a scaling parameter $α.$ We perform a homogenization procedure by using the periodic unfolding technique. As a result, we obtain two different effective models, depending on the scaling parameter $α.$ More precisely, in the case $α > −1,$ we are led to a standard Dirichlet problem for an elliptic equation, while in the case $α = −1,$ we get a bidomain system, consisting in the coupling of an elliptic equation with an algebraic one.

• General Summary

• Abstract

This paper examines the characterization of bounded variation (BV) and Sobolev functions by using some non-local functionals. We analyze their pointwise convergence and establish a connection with the Sobolev norm and the total variation measure. By investigating a wider class of non local functionals, we provide a deeper understanding of how these approximations capture the local properties of BV and Sobolev spaces, thereby reinforcing their applicability in the Euclidean setting.

• General Summary