Appendix C: PDE’s and analytic semigroups

In this appendix we recall some classical results on PDE’s of elliptic and parabolic problems that we use in this book. Namely, we state some wellknown (interior) Schauder estimates, some maximum principle and we see the relations between the theory of PDE’s and the theory of analytic semigroups. Throughout this appendix, by A we denote the elliptic operator defined on

smooth functions by

Au(x) = N∑

qij(x)Diju(x) +

bi(x)Diu(x) + c(x)u(x), (C.0.1)

with real coefficients qij , bi (i, j = 1, . . . , N) and c defined in an open set Ω. We do not assume any smoothness assumption on the domain Ω unless it is explicitly mentioned. The following condition is always assumed in this appendix.