Higher differentiability for a class of problems under p,q subquadratic growth

Abstract

We study the higher differentiability for nonlinear elliptic equation in divergence form A(x, Du) = b(x). The result covers the cases in which A(x, ξ) satisfies p, q growth, with 1 < p < 2 in ξ and a Sobolev dependence of with respect to x. By means of an a-priori estimate we ensure the W 2,p loc (Ω)-property for the solution of the boundary value problem.

Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions

“…In order to achieve the a priori estimate under the sharp integrability assumption on the independent term f , besides the use of the difference quotient method, we need to apply carefully the well known iteration Lemma on concentric balls to control the terms with critical integrability. We'd like to mention that previous higher differentiability result in the subquadratic non standard growth case has been recently obtained in [19] with independent term f ∈ L p p−1 . Especially, note that…”

Lipschitz bounds for integral functionals with $(p,q)$-growth conditions

“…is valid. The choice of τ (see (33)), ∆ ℓ and estimate (32) combined with a straightforward induction argument yield (35). Indeed, assuming J ℓ−1 ≤ τ ℓ−1 J 0 , we obtain…”

“…Currently, regularity theory for non-autonomous integrands with non-standard growth, e.g. p(x)-Laplacian or double phase functionals, are a very active field of research, see, e.g., the recent papers [4,9,12,18,16,17,19,20,21,23,28,27,30,35] and [2,11] for related results about the Lavrentiev phenomena.…”