Gradient estimates via non standard potentials and continuity

Abstract

Abstract. We consider elliptic problems with non standard growth conditions whose most prominent model example is the p(x)-Laplacean equationwith a measure data right-hand side µ. We prove pointwise gradient estimates in terms of a non standard version of the non-linear Wolff potential of the right-hand side measure, and moreover a characterization for C 1 -regularity of the solution, also in terms of the Wolff potential. The C 1 -regularity criterion is also related to the density of µ and the decay rate of its L n -norm on small balls. Moreover, from the pointwise gradient estimates the Calderón and Zygmund theory and several types of local estimates follow as a consequence.

Gradient estimates via non-linear potentials

“…We re-emphasize here that, when considering parabolic problems, the techniques presented here are the only one available for getting such estimates. The techniques presented in this paper are general enough to be applied in different contexts; an example is [12], where problems with non-standard growth of p(x)-type -see for instance [1] -are considered.…”

Elliptic interpolation estimates for non-standard growth operators

“…On the other hand, Bögelein and one of the authors generalized in [7] pointwise potential estimates for the gradient of the solution, which were originally established by Duzaar and Mingione in [11], to the non-standard growth situation. I.e.…”

“…We therefore provide a unified approach to both the pointwise potential estimates-the ones for u and the ones for Du-in a scale depending on the regularity of both γ(·) and a(·, z), that is, referring to (1.2), the regularity of both coefficient and exponent. Finally, as a byproduct of our approach, we generalize the result which Bögelein and one of the authors proved in [7], extending their gradient bound from the partial case p(·) ≥ 2 to the whole range p(·) ∈ (2 − (1.4)ˆΩ |Du| dx =: M < +∞.…”

“…The proofs of these theorems can be performed analogously to the one of [7,Theorem 1.4], taking into account the additional condition we impose here, and therefore we refer the reader to this paper, in particular Section 4.1.…”

“…However, the solution v of the homogenous problem shows at least local higher integrability properties and so does the solution w of the frozen homogeneous problem. We start with a higher integrability Lemma for v, which can be found in [37]; we refer the reader in particular to the discussion in [7,Remark 3.3] concerning the dependence of the constant.…”