Table of Contents
International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 845760, 16 pages
http://dx.doi.org/10.1155/2014/845760
Research Article

Explicit Estimates for Solutions of Mixed Elliptic Problems

Núcleo de Investigadores Científicos, Universidad Central del Ecuador, Ciudadela Universitaria Avenida América, 290-4799 Quito, Ecuador

Received 25 October 2013; Accepted 13 January 2014; Published 31 March 2014

Academic Editor: Zhenhua Guo

Copyright © 2014 Luisa Consiglieri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 coefficients. Our concern is to estimate the solutions with explicit constants, for domains in () of class . The existence of and estimates is assured for and any (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive estimates for different ranges of the exponent depending on the fact 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.