This book aims to propose a unified approach to elliptic and parabolic equations with bounded and smooth coefficients. The book will highlight the connections between these equations and the theory of semigroups of operators. Inhaltsverzeichnis 1. Function spaces. I. Semigroups of bounded operators. 2. Strongly continuous semigroups. 3. Analytic semigroups. II. Parabolic equations. 4. Elliptic and parabolic maximum principles. 5 Prelude to parabolic equations: the heat equation and the Gauss-Weierstrass semigroup in C (Rd). 6. Parabolic equations in Rd. 7. Parabolic equations in Rd + with Dirichlet boundary conditions. 8. Parabolic equations in Rd+ with more general boundary conditions. 9 Parabolic equations in bounded smooth domains . III Elliptic equations. 10. Elliptic equations in R . 11. Elliptic equations in Rd+ with homogeneous Dirichlet boundary conditions. 12. Elliptic equation in Rd+ with general boundary conditions. 13 Elliptic equations on smooth domains . 14 Elliptic operators and analytic semigroups. 15. Kernel estimates. IV. Appendices. A Basic notion of Functional Analysis in Banach spaces. B Smooth domains and extension operators.
