Relaxation and optimization for linear-growth convex integral\n functionals under PDE constraints

2016/03/03 by Adolfo Arroyo-Rabasa, Arroyo-Rabasa, Adolfo
Mathematics · #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1603.01310

Abstract

We give necessary and sufficient conditions for minimality of generalized\nminimizers for linear-growth functionals of the form \
mathcal F[u] :=\n
int_
Omega f(x,u(x))
,
textdx,
qquad u:
Omega
subset
mathbb R^N
to\n
mathbb R^d, where u is an integrable function satisfying a general PDE\nconstraint. Our analysis is based on two ideas: a relaxation argument into a\nsubspace of the space of bounded vector-valued Radon measures mathcal\nM(\Ω; mathbb R^d), and the introduction of a set-valued pairing in\n mathcal M(\Ω; mathbb R^N) \× rm L^\∞(\Ω; mathbb R^N). By\nthese means we are able to show an intrinsic relation between minimizers of the\nrelaxed problem and maximizers of its dual formulation also known as the\nsaddle-point conditions. In particular, our results can be applied to\nrelaxation and minimization problems in BV, BD.\n

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