The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{ align*} \begin{ cases} \mathrm{ div}(|\nabla u|^{ p-2}\nabla u) &= f+ \ \mathrm{ div}(|\mathbf{ F}|^{ p-2}\mathbf{ F}) \qquad \text{ in} \ \ \Omega, \\ \hspace{ 1.2cm} u &=\ g \hspace{ 3.1cm} \text{ on} \ \ \partial \Omega, \end{ cases} \end{ align*} in Lorentz space, with given data $\mathbf{ F} \in L^p(\Omega;\mathbb{ R}^n)$, $f \in L^{ \frac{ p}{ p-1}}(\Omega)$, $g \in W^{ 1,p}(\Omega)$ for $p>1$ and $\Omega \subset \mathbb{ R}^n$ ($n \ge 2$) satisfying a Reifenberg flat domain condition or a $p$-capacity uniform thickness condition, which are considered in several recent papers. To better specify our result, the proofs of regularity estimates involve fractional maximal operators and valid for a more general class of quasilinear nonhomogeneous elliptic equations with mixed data. This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good-$\lambda$ approach technique proposed in our preceding works~\cite{ MPT2018,PNCCM,PNJDE,PNCRM}, to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.
Categories: math.AP