Elliptic second order equations. (English) Zbl 0726.35036

From the author’s introduction: “These lectures deal primarily with elliptic second-order equations in non-divergence form; for the sake of simplicity we consider through most of the lectures linear equations with leading terms only, namely the equation (1) \(a_{ij}(x)D_{ij}u=f.\)
We emphasize that no smoothness assumptions are made on the coefficients. Our main concern here is to describe some qualitative properties of the solutions of (1), such as Harnack inequality, Hölder regularity theory and Calderon-Zygmund \(L^ p\) estimates.
(...). We shall prove here the same type of results, but asking the coefficients and the right-hand side to be under control in \({\mathbb{R}}^ n\). Analogous results will be proved in Sobolev classes. But it is known that neither Hölder nor Sobolev classes are adequate to describe all the solutions of elliptic second order equations with non-smooth coefficients. Then we shall consider solutions unendowed with derivatives, that satisfy the equation in a suitable generalized sense (viscosity solutions).”
Reviewer: M.Chicco (Genova)


35J15 Second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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