%0 Journal Article
%T Explicit Estimates for Solutions of Mixed Elliptic Problems
%A Luisa Consiglieri
%J International Journal of Partial Differential Equations
%D 2014
%R 10.1155/2014/845760
%X 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. 1. Introduction The knowledge of the data makes all the difference in the real-world applications of boundary value problems. Quantitative estimates are of extreme importance in any other area of science such as engineering, biology, geology, and even physics, to mention a few. In the existence theory to the nonlinear elliptic equations, fixed point arguments play a crucial role. The solution may exist such that it is estimated in an appropriate functional space, where the boundedness constant is frequently given in an abstract way. Their derivation is so complicated that it is difficult to express them, or they include unknown ones that are achieved by a contradiction proof, as, for instance, the Poincar¨¦ constant for nonconvex domains. The majority of works consider the same symbol for any constant that varies from line to line along the whole paper (also known as universal constant). In conclusion, the final constant of the boundedness appears completely unknown from the physical point of view. In presence of this, our first concern is to exhibit the dependence on the data of the boundedness constant. To this end, first (Section 3.1) we solve in the Dirichlet, mixed, and Neumann problems to an elliptic second-order equation in divergence form with discontinuous coefficient, and simultaneously we establish the quantitative estimates with explicit constants. Besides in Section 3.2 we derive estimative constants involving and measure data, via the technique of solutions obtained by limit approximation (SOLA) (cf. [1¨C4]). Dirichlet, Neumann, and mixed problems with respect to uniformly elliptic equation in divergence form are
%U http://www.hindawi.com/journals/ijpde/2014/845760/