These lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in Holder spaces. Krylov shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations. The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed. This book, with nearly 200 exercises, will provide a good understanding of what kinds of results are available and what kinds of techniques are used to obtain them.
Elliptic equations with constant coefficients in $\mathbb R^d$ Laplace's equation Solvability of elliptic equations with constant coefficients in the Holder spaces Elliptic equations with variable coefficients in $\mathbb R^d$ Second-order elliptic equations in half spaces Second-order elliptic equations in smooth domains Elliptic equations in non-smooth domains Parabolic equations in the whole space Boundary-value problems for parabolic equations in half spaces Parabolic equations in domains Bibliography Index.