Explicit estimates for solutions of mixed elliptic problems

Abstract

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second order equation in divergence form with discontinuous coefficient. Our concern is to estimate the solutions with explicit constants, for domains in $\mathbb{R}^n$ ($n\geq 2$) of class $C^{0,1}$. The existence of $L^\infty$ and $W^{1,q}$-estimates is assured for $q=2$ and any $q<n/(n-1)$ (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative $L^\infty$-estimates is based on the DeGiorgi technique developed by Stampacchia. By using the potential theory, we derive $W^{1,p}$-estimates for different ranges of the exponent $p$ depending on that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.