J. Rossmann

Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations

Introduction Singularities of solutions to equations of mathematical physics: Prerequisites on operator pencils Angle and conic singularities of harmonic functions The Dirichlet problem for the Lame system Other boundary value problems for the Lame system The Dirichlet problem for the Stokes system Other boundary value problems for the Stokes system in a cone The Dirichlet problem for the biharmonic and polyharmonic equations Singularities of solutions to general elliptic equations and systems: The Dirichlet problem for elliptic equations and systems in an angle Asymptotics of the spectrum of operator pencils generated by general boundary value problems in an angle The Dirichlet problem for strongly elliptic systems in particular cones The Dirichlet problem in a cone The Neumann problem in a cone Bibliography Index List of symbols.

Elliptic equations in polyhedral domains

This is the first monograph which systematically treats elliptic boundary value problems in domains of polyhedral type. The authors mainly describe their own recent results focusing on the Dirichlet problem for linear strongly elliptic systems of arbitrary order, Neumann and mixed boundary value problems for second order systems, and on boundary value problems for the stationary Stokes and Navier-Stokes systems. A feature of the book is the systematic use of Greens matrices. Using estimates for the elements of these matrices, the authors obtain solvability and regularity theorems for the solutions in weighted and non-weighted Sobolev and Holder spaces. Some classical problems of mathematical physics (Laplace and biharmonic equations, Lame system) are considered as examples. Furthermore, the book contains maximum modulus estimates for the solutions and their derivatives. The exposition is self-contained, and an introductory chapter provides background material on the theory of elliptic boundary value problems in domains with smooth boundaries and in domains with conical points. The book is destined for graduate students and researchers working in elliptic partial differential equations and applications.

On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ?n with conical points

In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, Dkvu = gk on ∂G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ℝn (n≥4) that the Agmon-Miranda maximum principle fails in this cone.

Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary

In the first part of the paper, the authors obtain the asymptotics of Green’s function of the first boundary value problem for the heat equation in an m-dimensional cone K. The second part deals with the first boundary value problem for the heat equation in the domain K×Rn−m. Here the right-hand side f of the heat equation is assumed to be an element of a weighted Lp,q-space. The authors describe the behavior of the solution near the (n−m)-dimensional edge of the domain.

ESTIMATES OF GREENS FUNCTION FOR SECOND-ORDER PARABOLIC EQUATIONS NEAR EDGES

We consider the first boundary value problem for a second-order parabolic equation with variable coefficients in the domain K x Rn-m, where K is an m-dimensional cone. The main results of the paper ...

Mixed Boundary Value Problems for Stokes and Navier-Stokes Systems in Polyhedral Domains

The paper deals with a boundary value problem for the stationary Stokes and Navier-Stokes systems, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are prescribed on the faces of a polyhedral domain. Various regularity results in weighted and nonweighted Sobolev and Holder spaces are given here. Furthermore, the paper contains a maximum modulus estimate for the velocity.

Asymptotics of Solutions of the Poisson Equation near Straight Edges

The paper deals with the Dirichlet problem for the Poisson equation ∆u = f in the domain D = K × Rn−m, where K is a cone in Rm. The author describes the singularities of the Green function near the edge of the domain. Using this result, he obtains the asymptotics of the solution of the boundary value problem for a right-hand side f belonging to a weighted Lp Sobolev space. Here, precise formulas for all coefficients in the asymptotics are given.

Contributions of V. Maz’ya to the theory of boundary value problems in nonsmooth domains

A large part of the scientific work of Vladimir Maz′ya is concerned with the theory of boundary value problems in nonsmooth domains. Different variations of this theme are constantly heard in his mathematical symphony. The present article is aimed to give a review of his results in this field.