## 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

### Keywords:

extremality; strong convergence; weak convergence

### Citations:

Zbl 0784.49021; Zbl 0784.49022
Full Text: