On Lipschitz regularity for bounded minimizers of functionals with (p,q) growth

Abstract

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p\ge 2$. This relation improves existing restrictions when $p\le N-1$, moreover our result is sharp in the range $N>\frac{p(2+p)}{2}+1$. The standard Lipschitz regularity takes the form $W_{\text{loc}}^{1,\infty }-W_{\text{loc}}^{1,p}$, whereas we obtain $W_{\text{loc}}^{1,\infty }-L_{\text{loc}}^{\infty }$ regularity estimate and then make use of existing sharp $L_{\text{loc}}^{\infty }$ bounds to obtain the required conclusion.

On completion of this paper, we found that this result is proved by M. Bildhauer, M. Fuchs (Zap. Nauchn. Sem. POMI, 288, 2002). The methods are mostly similar. Instead of DeGiorgi iteration, we use Moser iteration. We use a quadratic term for the regularization and bounded slope condition for Lipschitz regularity of the regularized minimizer. This paper is not meant for publication