Global Lorentz and Lorentz–Morrey estimates below the natural exponent for quasilinear equations

Abstract

Abstract. Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the formwhere div A(x, ∇u) is modelled after the p-Laplacian, p > 1. The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the p-capacity. The vector field datum f is allowed to have low degrees of integrability and thus solutions may not have finite L p energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.

Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

“…We now present an extension lemma which can be found in [44] (see also [4,Lemma 3.2] for the details):…”

“…Recently in [10], the authors obtained the following Calderón-Zygmund type relation provided the nonlinearity satisfies a small BMO condition and the domain is suitably flat in the sense of Reifenberg: 4) for any 1 < q − ≤ q(·) ≤ q + < ∞ with q(·) being log-Hölder continuous. In particular, they cannot take q − = 1, which would recover (1.3).…”

“…dx, we apply the Sobolev-Poincaré inequality with s := n + 1 n (see [36] or [4,Theorem 3.3] and references therein for the details) to obtain…”

Weighted Lorentz estimates for nonlinear elliptic obstacle problems with partially regular nonlinearities

“…The regularity in Lorentz spaces concerning partial differential equations was originated from Talenti's work [28] based on symmetrization. Since then, there is a lot of papers to study the Lorentz regularity of various problems of PDEs; see some recent references in [2,4,5,23], and we also refer the reader to Xiao [32], who characterized a nonnegative Radon measure μ on R d to produce a continuous map I α from the Lorentz space L (p,1) to the Lebesgue space L p μ . We would like to mention that Baroni [4,5] showed the Lorentz estimates for evolutionary p-Laplacian systems and obstacle parabolic p-Laplacian, respectively, by using the large-M-inequality principle introduced by Acerbi and Mingione [1].…”

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

“…In the other cases this is not easy and requires to tilt the estimates around the natural growth exponent. For instance this was done in [3] for nonlinear elliptic equations which made use of estimates below the natural growth exponent and Hodge decompositions. In the parabolic case such tools can be replaced by the methods in [5] and might be done in future work.…”