scholarly journals Second order elliptic boundary value problems in spaces with homogeneous norms

1980 ◽  
Vol 29 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Partial Differential Operator ◽  
Second Order ◽  
Elliptic Boundary Value ◽  
First Order ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

scholarly journals Higher order elliptic boundary value problems in spaces with homogeneous norms

1981 ◽  
Vol 31 (1) ◽  
pp. 92-113 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
A Priori ◽  
Elliptic Boundary Value ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

General edge asymptotics of solutions of second-order elliptic boundary value problems I

1993 ◽  
Vol 123 (1) ◽  
pp. 109-155 ◽  
Author(s):  
Martin Costabel ◽  
Monique Dauge
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Weighted Sobolev Spaces ◽  
Second Order ◽  
Elliptic Boundary Value Problems ◽  
General Situation ◽  
Opening Angle ◽  
Elliptic Boundary Value ◽  
Basic Facts

SynopsisThis is the first of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.

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