Nonconvex variational problems with general singular perturbations

Abstract

We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem.