The dependence of local regularity of solutions on the summability of coefficients and nonhomogenous term

Abstract: In this paper, we mainly discuss the local regularity of the solution to the following problem \begin{align*} \begin{cases} -\dive({\bf{A}}(x)\nabla u(x))=f(x),&~x\in\Omega,\ u(x)=0,&~x\in\partial\Omega, \end{cases} \end{align*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$. In particular, we are concerned with the connection between the regularity of the solution $u$ and the integrability of the coefficient matrix ${\bf{A}}(x)$ as well as the nonhomogeneous term $f$. To be more precise, our first result is to prove that the maximum norm of $u$ can be controlled by $|f|{s}$ with $f\in L^s(\Omega),~s>\frac{nq}{2q-n},~q>\frac{n}{2}$. Meanwhile, we construct some counterexamples to illustrate the index $\frac{nq}{2q-n}$ being sharp. Subsequently, we give an improved upper bound for the maximum norm of $u$. Namely, there exists a positive constant $C$ such that $$|u|{\infty}\leq C|f|{\frac{nq}{2q-n}}\left[\log\left(\frac{|f|{s}}{~~~~~|f|_{\frac{nq}{2q-n}}}+1\right)+1\right].$$ Specially, the main difference of our approach compared to the arguments of [\ref{CUR}, \ref{XU}] is to construct two classes of truncation functions to remove the assumption of the boundedness of $u$. Finally, based on the previous results and Moser iteration argument, we derive the Harnack inequality of $u$ from which the H\"older continuity of the solution follows. In addition, we also find that the Lebesgue space $L^{\frac{n}{2}}(\Omega)$ to which the inverse of the smallest eigenvalue $\lambda(x) $ of the matrix {\bf{A}}(x) belongs is essentially sharp in order to establish local boundedness and the H\"older continuity of the solution.