On Dirichlet problem in Morrey spaces

In recent years many papers (see e.g. \cite{AS} \cite{CFG} \cite{Si} \cite{HK} ) have been devoted to the study of the continuity properties of solutions to equations of the form \begin{equation} \label{delta} - \Delta u + Vu = f \end{equation} or , more generally , \begin{equation} \label{ell} Lu + Vu = f \end{equation} where $ L $ is an uniformly elliptic operator in divergence form defined by \begin{equation} L \equiv - \frac{ \partial}{\partial x_{j}} \left ( a_{ij} \frac{\partial}{\partial x_{i}} \right ) \end{equation} The novelty, starting with the pioneering work of Aizenman \& Simon (see \cite{AS} ) , is the assumption made on the potential term $ V $ and the known term $ f $ which is the belonging to the {\sl Stummel-Kato} class and not the requirement of high integrability , $ V,f \in L^{p} (\Omega) \; p>n/2 $ , as in previous works (see e.g. \cite{LU}, \cite{Sta} ). We recall that the Stummel-Kato class $ S(\Omega) $ is the set of the locally integrable functions $ f $ such that \begin{equation} \lim_{\varepsilon \rightarrow 0} \sup_{x \in \Omega} \int_{ \{ y \in \Omega : |x-y| < \varepsilon \} } f(y)|x-y|^{2-n} dy =0 \end{equation} In the paper \cite{D} we studied the relation of the {\sl Stummel-Kato class } with the scale of {\sl Morrey spaces} $ L^{1,\lambda} (\Omega) (0< \lambda <n) $ (for definitions see section $~\ref{definizioni}$ ). More precisely if $ S(\Omega) $ denotes the Stummel-Kato class we have \begin{equation} L^{1,\lambda} (\Omega) \subset S(\Omega) \subset L^{1,\mu} (\Omega) \; 0 \leq \mu \leq n-2 < \lambda < n \end{equation} In the same paper , assuming $ V $ in $ L^{1,\lambda} (\Omega) (n-2 < \lambda <n) $ and $ f \equiv 0 $ we proved the local h\"older continuity of the solutions of equation ($~\ref{ell}$) . The present paper is concerned with the Dirichlet problem for equation \begin{equation} \label{equa} Lu- \left( b_{j} u \right)_{x_{j}} =(f_{j} )_{x_{j}} \end{equation} in an open bounded set $ \Omega \subset R^{n} $ where $ L $ has the same meaning as above and $ b_{j} , f_{j} $ belong to certain Morrey spaces. Our aim is to show that the solution to this Dirichlet problem can have certain regularity properties even in the case when the coefficients $ b_{j} , f_{j} $ do not possess high integrability pursuing the study begun in the above mentioned papers. Because it is well known that $ L^{p} (\Omega) (p>n) $ is properly contained in $ L^{2,\mu}(\Omega) (n-2< \mu <n) $ we improve the classical results on the $ L^{p} $ and h\"older regularity of the solution to the Dirichlet problem for equation ($~\ref{equa}$) (see \cite{Sta} ). The proof is based on the study of a {\sl non convolution integral operator} obtained from the representation formula for weak solution by means of the gradient of the Green function $ g(x,y) $ of $L. $ We study the boundedness properties of this operator in Theorem $~\ref{teooper}$ using an idea of Hedberg (see \cite{He} lemma p.506). Let us point out that though there exist pointwise estimates for $ \nabla g $ (see [G.W] Th. 3.3) these are valid under some regularity assumptions on the coefficient $ a_{ij} . $ Our technique instead allows to handle the case of discontinuous coefficients and it relies on an integral estimate for the Green function. We wish to thank prof. C. Kenig for a helpful conversation about this last topic. We also wish to thank prof. F. Chiarenza for some useful discussions.