Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations

Abstract

Abstract. A weighted norm inequality of Muckenhoupt-Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.

Pointwise gradient estimates for a class of singular quasilinear equations with measure data

“…Thanks to [17,Proposition 2.6], (see also [18,Corollary 2.13]), for any η > 0 there exists δ 1 = δ 1 (n, p, Λ, η) be such that the following holds. If δ ≤ δ 1 , there exists a functioñ v ∈ W 1,∞ (B 6ρ 1 (x 3 )) such that…”

On the Problem of Maximal $$L^q$$-regularity for Viscous Hamilton–Jacobi Equations

“…It is worth observing that many results in the literature cover the case 1 < γ 2; that is, when the gradient term has at most natural growth. General results in the full range γ > 1, based on methods from nonlinear potential theory, appeared quite recently in [34,36,37]. Roughly speaking, properties (M)+(A) say that if f belongs to a sufficiently small Lebesgue space, then solutions should enjoy much better regularity than W 1,γ , namely, be in W 1,qγ (Q) (and even in W 2,q (Q), by standard Calderón-Zygmund theory).…”

Global Calderón--Zygmund theory for parabolic $p$-Laplacian system: the case $1

“…In all the above works, the comparison estimates can not be applied to the range 1 < p ≤ 2 − 1 n , in which case the distributional solutions may not even belong to W 1,1 loc . More recently, based on various tools developed for quasilinear equations with measure data and linear or nonlinear potential and Calderón-Zygmund theories, (see [6,18,32,33,41]), Phuc and the second author [37] (see also [39,38]) dealt with the singular case 3n−2 2n−1 < p ≤ 2 − 1 n for the first time. The key ingredients are some new local comparison estimates.…”