Gamma-convergence of quadratic functionals perturbed by bounded linear functionals

TL;DR

This paper investigates the $ ext{Gamma}$-convergence of quadratic functionals with bounded linear perturbations, revealing their limits as sums of quadratic, linear, and constant terms, with the constant linked to a Radon measure.

Contribution

It establishes that the constant part of the $ ext{Gamma}$-limit can be represented by a Radon measure on large classes of open sets, extending classical convergence theories.

Findings

The $ ext{Gamma}$-limit decomposes into quadratic, linear, and constant parts.

The constant term $ u( ext{Omega})$ can be characterized as a Radon measure.

An example shows the limits of the localization method in $ ext{Gamma}$-convergence.

Abstract

Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak topology of $H^1_0(\Omega)$, can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of $G$- and $H$-convergence completely characterises the quadratic and linear parts of the $\Gamma$-limit and shows that their coefficients do not depend on $\Omega$. The constant, which instead depends on $\Omega$ and will be denoted by $-\nu(\Omega)$, plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that $\nu$ coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets $\Omega$. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in $\Gamma$-convergence cannot be easily improved.