New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

Abstract

In this paper, our work is aimed to show the fractional maximal gradient estimates and point-wise gradient estimates for quasilinear divergence form elliptic equations with general Dirichlet boundary data:in terms of the Riesz potentials, where Ω is a Reifenberg flat domain of R n (n ≥ 2), the nonlinearity A is a monotone Carathéodory vector valued function, g belongs to some W 1,p (Ω; R) for p > 1 and f ∈ L p p−1 (Ω; R n ). Our proofs of gradient regularity results are established in the weighted Lorentz spaces. Here, we generalize our earlier results concerning the "good-λ" technique and the study of so-called cut-off fractional maximal functions. Moreover, as an application of point-wise gradient estimates, we also prove the existence of solutions for a generalized quasilinear elliptic equation containing the Riesz potential of gradient term.

Lorentz estimates for quasilinear elliptic double obstacle problems involving a Schrödinger term

“…This means the same conclusions of these lemmas can be obtained by replacing B by B ϱ ( ξ ) ∩ Ω for all ξ ∈ ∂ Ω and 0 < ϱ ≤ diam(Ω). For the proof, we use the same technique as in 52 …”

“…We sketch here the important and different points. For more details, we refer to 37,52 for the original idea and 47 for the shorter one.…”

“…More of the regularity results are obtained in this respect, Choe 27 obtained C 0, α and C 1, α ‐estimates for solutions and an extensive literature on regularity for solutions/minima to a class of variational integrals of more general types. For instance, for nonlinear elliptic double obstacle problems, the interior Hölder estimate for problems nonstandard growth studied in Bildhauer et al, 23 global L q estimates by Byun et al 24,25 Associated with nonlinear elliptic equations (without obstacles), there have been various results pertaining to regularity theory and Calderón‐Zygmund, together with some technical issues discussed in other works 28‐37 and the further references to be continued.…”

“…Our technique is somewhat the improvement of a surprising approach introduced in studies 28,29,34 that dealt with the Calderón‐Zygmund estimates for quasilinear elliptic/parabolic systems. The approach then inspired a lot of other research on the regularity theory of certain partial differential equations 35‐37,47‐50 . Motivated by such effective method, in our proposed technique, global gradient estimates of solutions to double obstacle problems () are preserved under fractional maximal operators.…”

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

“…It is well known that regularity results and potential estimates for general non-degenerate case were obtained in many recent studies by Mingione and his collaborators [12,14,26,34]; the particular case τ = 1 for non-degenerate p-Laplacian equations presented in [31,32] for example. And there has been a substantial amount of contributions for degenerate case [3,7,37,40,41,[44][45][46], etc.…”