Global Sobolev regularity for general elliptic equations of p-Laplacian type

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

“…In particular, the result in Theorem 1.3 is even new when applied to equations of special form (1.5) whose gradient estimates in the classical Morrey spaces are obtained in [24,25,33]. Our results also improve the L q gradient estimates established in [2,29] for equation (1.1) since the principal part A(x, z, ξ) is merely assumed to be continuous in the z variable instead of being Hölder continuous.…”

“…From this, we get the desired conclusion by choosing γ small first, then σ, and δ last. For the case 1 < p < 2, the proof is similar with some slight adjustments as follows, which can also be found in the proofs of Lemma 4.3 and Lemma 4.6 in [2]. By arguing as above but using (4.1) together with [29, Lemma 3.1], then in place of (4.5) and (4.6) we now have for every τ 1 , τ 2 > 0 small that…”

“…The class of these equations is the smallest one that is invariant with respect to dilations and rescaling of domains and that contains equations of the form (1.8). Given the invariant structure, we employ the direct argument in [1,2] to show that the gradient of the solution u can be approximated by a bounded gradient in L p norm (see Proposition 4.1). It is essential for us that this approximation is uniform with respect to the two parameters.…”

“…This two-weight inequality holds true for quite general weights and we believe that it is of independent interest. We end the introduction by pointing out that our method of establishing interior gradient estimates in weighted Morrey spaces could be combined with the boundary techniques used in [2,3,24,25] to derive global estimates for Reifenberg flat domains and homogeneous Dirichlet boundary condition. Furthermore, by following [32] it might be possible to weaken the assumption u ∈ L ∞ in Theorem 1.3 by assuming only that u ∈ BMO.…”

Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data

“…On the other hand, interior W 1,q estimates for the nonhomogeneous quasilinear equation (1.1) was investigated in [17] using the perturbation method by Caffarelli-Peral [5] together with the two-parameter scaling technique introduced in [10] to deal with a specific parabolic equation. When A has sufficiently small BMO oscillation in x and is Lipschitz continuous in the z variable, it was established in [17] that: |F| 1 p−1 ∈ L q loc =⇒ ∇u ∈ L q loc for any q > p. This result was extended in [2,16,18,19] to cover more general situations. In particular, the authors of [2] derived the corresponding global estimates for Reifenberg flat domains and for zero Dirichlet boundary data.…”