Some simple boundary value problems with corner singularities and boundary layers

Some boundary value problems for a second-order elliptic partial differential equation in a polygonal domain are considered. The highest order terms in the equation are multiplied by a small parameter, leading to a singularly perturbed problem. The singular perturbation causes boundary layers and interior layers in the solution, and the corners of the polygon cause corner singularities in the solution. The paper considers pointwise bounds for derivatives of the solution that show the influence of these layers and corner singularities. Several recent results on this problem are surveyed, and some open problems are stated and discussed.

[1]  R. B. Kellogg,et al.  Differentiability properties of solutions of the equation -ε 2 δ u + ru = f ( x,y ) in a square , 1990 .

[2]  F. G. Leppington.  MATCHING OF ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS , 1994 .

[3]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[4]  K. Stewartson.  High Reynolds-number flows , 1980 .

[5]  R. Bruce Kellogg,et al.  Corner singularities and boundary layers in a simple convection–diffusion problem☆ , 2005 .

[6]  Martin Schechter,et al.  Elliptic Problems in Nonsmooth Domains (P. Grisvard) , 1986 .

[7]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[8]  Martin Stynes,et al.  A singularly perturbed convection–diffusion problem in a half-plane , 2006 .

[9]  R. Bruce Kellogg,et al.  Regularity of Solutions to the Navier-Stokes Equations for Compressible Barotropic Flows on a Polygon , 2002 .

[10]  Douglas N. Arnold,et al.  The boundary layer for the reissner-mindlin plate model , 1990 .

[11]  R. Bruce Kellogg,et al.  Compressible Stokes Problem on Nonconvex Polygonal Domains , 2001 .

[12]  J. Roßmann,et al.  Elliptic Boundary Value Problems in Domains with Point Singularities , 2002 .

[13]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[14]  R. Bruce Kellogg.  Corner singularities and singular perturbations , 2001, ANNALI DELL UNIVERSITA DI FERRARA.

[15]  B. Plamenevskii,et al.  Elliptic Problems in Domains with Piecewise Smooth Boundaries , 1994 .