Singular behavior of the Dirichlet problem in Hölder spaces of the solutions to the Dirichlet problem in a cone

Abstract

In the present study we consider the solution of the Dirichlet problem in conical domain. For general elliptic problems in non Hilbertian Sobolev spaces built on $L^{p},1<p<\infty$ the theory of sums of operators developed by Dore-Venni $\left[8\right]$ provides an optimal result. Holder spaces, as opposed to LP spaces, are not UMD. Using the results of Da Prato-Grisvard $\left[6\right]$ and Labbas $\left[14\right]$ we cope with the singular behaviour of the solution in the framework of H$\ddot{\textrm{o}}$lder and little H$\ddot{\textrm{o}}$lder spaces.