Interior regularity of solutions of differential equations

• Lars Hörmander
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 116)

Abstract

The simplest case of the results proved in this chapter is the fact that every u Є C 2 satisfying the Laplace equation
$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaadwhacaGGVaGa % eyOaIyRaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaey4kaSIaey % OaIy7damaaCaaaleqabaWdbiaaikdaaaGccaWG1bGaai4laiabgkGi % 2kaadMhapaWaaWbaaSqabeaapeGaaGOmaaaakiabg2da9iaaicdaaa % a!47EB! {\partial ^2}u/\partial {x^2} + {\partial ^2}u/\partial {y^2} = 0$$
is actually in C and can even be expanded in a convergent power series in x and y. The literature devoted to results of this kind is very extensive, so we shall only mention here a few papers which are particularly closely related to the results and methods of this chapter.

Keywords

Convolution Topo