Existence and regularity of minimizers of nonconvex integrals with $p-q$ growth

Celada, Pietro; Cupini, Giovanni; Guidorzi, Marcello

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 343-358.

We show that local minimizers of functionals of the form ${\int }_{\Omega }\left[f\left(Du\right(x\left)\right)+g(x,u(x\left)\right)\right]\mathrm{d}x$, $u\in u_0+W_0^{1,p}\left(\Omega \right)$, are locally Lipschitz continuous provided $f$ is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and $g$ is Lipschitz continuous in $u$. As a consequence of this, we obtain an existence result for a related nonconvex functional.

Classification : 49N60, 49J10
Mots clés : nonstandard growth, existence, Lipschitz continuity

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