We consider the problem of minimizing autonomous, multiple integrals like (P) min {∫Ω f (u, ∇u) dx: u ∈ u0 + W01,p(Ω)} where f: ℝ × ℝN → [0, ∞) is a continuous, possibly nonconvex function of the gradient variable ∇u. Assuming that the bipolar function f** of f is affine as a function of the gradient ∇u on each connected component of the sections of the detachment set D = {f** < f}, we prove attainment for (P) under mild assumptions on f and f**. We present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp.

A sharp attainment result for nonconvex variational problems / CELADA P.; G. CUPINI; GUIDORZI M.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 20:(2004), pp. 301-328. [10.1007/s00526-003-0238-5]

A sharp attainment result for nonconvex variational problems

CUPINI, GIOVANNI;
2004

Abstract

We consider the problem of minimizing autonomous, multiple integrals like (P) min {∫Ω f (u, ∇u) dx: u ∈ u0 + W01,p(Ω)} where f: ℝ × ℝN → [0, ∞) is a continuous, possibly nonconvex function of the gradient variable ∇u. Assuming that the bipolar function f** of f is affine as a function of the gradient ∇u on each connected component of the sections of the detachment set D = {f** < f}, we prove attainment for (P) under mild assumptions on f and f**. We present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp.
2004
20
301
328
CELADA P.; G. CUPINI; GUIDORZI M.
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/205273
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