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.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.