Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains

The book contains a systematic treatment of the qualitative theory of elliptic boundary value problems for linear and quasilinear second order equations in non-smooth domains. The authors concentrate on the following fundamental results: sharp estimates for strong and weak solutions, solvability of the boundary value problems, regularity assertions for solutions near singular points.

Key features:

* New the Hardy – Friedrichs – Wirtinger type inequalities as well as new integral inequalities related to the Cauchy problem for a differential equation.
* Precise exponents of the solution decreasing rate near boundary singular points and best possible conditions for this.
* The question about the influence of the coefficients smoothness on the regularity of solutions.
* New existence theorems for the Dirichlet problem for linear and quasilinear equations in domains with conical points.
* The precise power modulus of continuity at singular boundary point for solutions of the Dirichlet, mixed and the Robin problems.
* The behaviour of weak solutions near conical point for the Dirichlet problem for m – Laplacian.
* The behaviour of weak solutions near a boundary edge for the Dirichlet and mixed problem for elliptic quasilinear equations with triple degeneration.

* Precise exponents of the solution decreasing rate near boundary singular points and best possible conditions for this.
* The question about the influence of the coefficients smoothness on the regularity of solutions.
* New existence theorems for the Dirichlet problem for linear and quasilinear equations in domains with conical points.
* The precise power modulus of continuity at singular boundary point for solutions of the Dirichlet, mixed and the Robin problems.
* The behaviour of weak solutions near conical point for the Dirichlet problem for m - Laplacian.
* The behaviour of weak solutions near a boundary edge for the Dirichlet and mixed problem for elliptic quasilinear equations with triple degeneration.

Introduction.

1. Preliminaries.
2. Integral inequalities.
3. The Laplace operator.
4. Strong solutions of the Dirichlet problem for linear equations.
5. The Dirichlet problem for elliptic linear. divergent equations in a nonsmooth domain.
6. The Dirichlet problem for semilinear equations in a conical domain.
7. Strong solutions of the Dirichlet problem for nondivergence quasilinear equations.
8. Weak solutions of the Dirichlet problem for elliptic divergence form quasilinear equations.
9. The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge.
10. Sharp estimates of solutions to the Robin. boundary value problem for elliptic non divergence second order equations in a neighborhood of the conical point.
Bibliography.
Notation Index.
Index.