Lorentz improving estimates for the \(p\)-Laplace equations with mixed data (Q2201732)

Let \(\Omega \subset \mathbb{R}^n\), with \(n\geq 2\), be an open bounded domain, and let \(p\in (1,n]\). The authors investigate the regularity of weak solutions of the following nonhomogeneous quasilinear elliptic Dirichlet problem with mixed data \[ (1) \ \ \ \left\{\begin{array}{ll} -\text{div}(\mathbb{A}(x,\nabla u))=f+\text{div}(\mathbb{B}(x,{\mathbf F})) \ \ \ \ &\text{in} \ \ \Omega,\\[3mm] u=g &\text{on} \ \ \partial\Omega. \end{array}\right. \] Here, \(\mathbb{A},\mathbb{B}:\Omega\times \mathbb{R}^n\rightarrow \mathbb{R}^n\) are Carath\'eodory vector vauled functions satisfying, for some constants \(\Lambda_1,\Lambda_2>0\), \[ |\mathbb{A}(x,z)|+|\mathbb{B}(x,z)|\leq \Lambda_1 |z|^{p-1}, \ \ \ \langle \mathbb{A}(x,z_1)-\mathbb{A}(x,z_2),z_1-z_2\rangle \geq \Lambda_2(|z_1|^2+|z_2|^2)^{\frac{p-2}{2}}|z_1-z_2|^2 \] for all \(x\in \Omega\) and for all \(z,z_1,z_2\in\mathbb{R}\); the data \(f,{\mathbf F}\) and \(g\) have the following regularity \[f\in L^{\frac{p-1}{p}}(\Omega), \ \ \ {\mathbf F}\in L^p(\Omega,\mathbb{R}^n), \ \ \ g\in W^{1,p}(\Omega);\] and the domain \(\Omega\) satisfies one of the following conditions: \(-\) \ the complement \(\mathbb{R}^n\setminus \Omega\) is uniformly \(p\)-thick; \(-\) \ \(\Omega\) is a Reifenberg flat domain. Under the above assumptions, the authors establish a good-\(\lambda\) Theorem and a Global Lorentz Regularity Estimate for the weak solutions of problem (1). These regularity results are given in terms of fractional maximal operators. They extend to the case of mixed data some previous results of the authors and also improve the regularity of the weak solutions established there.