On some $L^{p}$-estimates for solutions of elliptic equations in unbounded domains

Abstract

In this review article we present an overview on some a priori estimates in $L^p$, $p>1$, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^p$-bounds, $p>2$, for the solution of the associated Dirichlet problem, obtained in correspondence with two different sign assumptions. Then, putting together the above mentioned bounds and using a duality argument, we extend the estimate also to the case $1<p<2$, for each sign assumption, and for a data in $L^p\cap L^2$.