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A version of Olech’s lemma in a problem of the calculus of variations. (English) Zbl 0874.49013

The authors study variational problems of the following form: \((\text{P}_a)\) minimize \(\int_\Omega g(\nabla u(x))dx\), \(u\in\langle a,\cdot\rangle+ W^{1,1}_0(\Omega)\). While the existence and uniqueness of solutions for \((\text{P}_a)\) have been discussed by the first author [Nonlinear Anal., Theory Methods Appl. 20, No. 4, 337-341 (1993; Zbl 0784.49021); ibid. 343-347 (1993; Zbl 0784.49022)], the present paper is concerned with continuous dependence of the solution on the parameter \(a\). Among other things, the authors prove that if \((a,g^{**}(a))\) is an extreme point of \(\text{epi}(g^{**})\) (\(g^{**}\) denotes the convexification of \(g\)) and if \(a_k\to a\), then under some natural assumptions, the solutions \(u_k\) of \((\text{P}_{a_k})\) converge to the solution \(u\) of \((\text{P}_a)\) strongly in \(W^{1,1}_0(\Omega)\). Continuous dependence of the solutions may fail if \((a,g^{**}(a))\) is not an extreme point.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
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