Published Paper

Inserted: 9 jan 2023
Last Updated: 22 feb 2024

Journal: Nonlinear Analysis
Volume: 238
Number: 113395
Year: 2024
Doi: 10.1016/j.na.2023.113395

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.

Keywords: Gamma-convergence, G-convergence, Elliptic equations, Localisation Method

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