A priori bounds for some classes of elliptic equations

We present an overview on some recent results concerning certain a priori bounds for the solutions of dfferent kinds of Dirichlet problems for second order linear elliptic partial differential equations. We start considering equations in divergence form with discontinuous coeffcients in unbounded domains. The main theorem, in this case, consists in a Lp-a priori bound, p > 1 (see [1, 2, 3]). Applications of this bound in the framework of non variational problems, in weighted and no-weigthed cases, are also given (cfr. [4, 5]). Successively, we show some a priori estimates for non-divergence struc- ture elliptic equations, whose smooth coeffcients satisfy a new condition generalizing Cordes' one, proved in [6, 7]. The bounds are achieved by means of a potential estimate obtained for the solutions of the same kind of problems, but with more regular datum. References [1] S. Monsurro, M. Transirico: A Lpestimate for weak solutions of elliptic equations, Abstr. Appl. Anal. vol. 2012 (2012), 15 pages. [2] S. Monsurro, M. Transirico: Dirichlet problem for divergence form elliptic equations with discontinuous coecients, Bound. Value Probl. vol. 2012 (2012), 20 pages. [3] S. Monsurro, M. Transirico: A priori bounds in Lp for solutions of elliptic equations in divergence form, Bull. Sci. Math. 137 (2013), 851-866. [4] S. Monsurro, M. Transirico: A W2;p-estimate for a class of elliptic operators, Int. J. Pure Appl. Math. (4) 83 (2013), 489-499.