Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

Abstract

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L n,∞ . We also obtain the a priori global and local estimates for the L n,∞ -norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

Remarks on some quasilinear equations with gradient terms and measure data

“…see [9], [5], [16], [27], and one gets useful convergence properties, see [16,Theorem 4.1 and §5] for the proof:…”

“…As a consequence, any R-solution u of problem (2.1) satisfies |u| p−1 ∈ L σ (Ω), ∀σ ∈ [1, N/(N − p) . More precisely, u and |∇u| belong to some Marcinkiewicz spaces [5], [16], [27], and one gets useful convergence properties, see [16,Theorem 4.1 and §5] for the proof:…”

“…there exists c > 0 such that |B(x, r)\Ω| ≧ cr N for any x ∈ R N \Ω and r > 0, then (2.4) holds with s = N, and C depends also on the geometry of Ω. Then |∇u| ∈ L N,∞ (Ω), hence u ∈ BM O(Ω), see [17], [27].…”

The Calderón-Zygmund theory for elliptic problems with measure data

“…We shall use a delicate combination of some the arguments from the latter paper with the Morrey space estimates of Section 8, a direct comparison argument on certain Calderón-Zygmund type balls, and finally a modification of some ideas from [14,48]. A different, elegant approach to M n estimates based on a suitable version of Gehring's lemma in Marcinkiewicz spaces has been recently given in [44]. Let us emphasize here the fact that our technique is robust enough to catch the borderline case θ = p, and therefore to get the limiting regularity (1.31).…”

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

“…The following properties are well-known in case p < N, see [7], [25] and more delicate in case p = N, see [36] and [45], where they require more regularity on the domain, namely, R N \Ω is geometrically dense:…”