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.