Dirichlet problems on varying domains

Abstract The aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with Dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain. Necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators. Examples are given to illustrate that most results are optimal.

[1]  R. Rannacher.  Zur Stabilität diskreter Teilspektren singulärer Differentialoperatoren zweiter Ordnung gegenüber Störungen der Koeffizienten und der Definitionsgebiete , 1975 .

[2]  Jacques-Louis Lions,et al.  Mathematical Analysis and Numerical Methods for Science and Technology : Volume 4 Integral Equations and Numerical Methods , 2000 .

[3]  Jeffrey Rauch,et al.  Potential and scattering theory on wildly perturbed domains , 1975 .

[4]  Dorin Bucur.  Characterization for the Kuratowski Limits of a Sequence of Sobolev Spaces , 1999 .

[5]  C. Bandle.  Isoperimetric inequalities and applications , 1980 .

[6]  D. Aronson,et al.  Non-negative solutions of linear parabolic equations , 1968 .

[7]  Ura Cnrs,et al.  Continuity with respect to the domain for the Laplacian : a survey , 1994 .

[8]  D. Daners.  A priori estimates for solutions to elliptic equations on non-smooth domains , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  Dorin Bucur,et al.  OPTIMISATION DE FORME SOUS CONTRAINTE CAPACITAIRE , 1994 .

[11]  Friedrich Stummel,et al.  Diskrete Konvergenz linearer Operatoren. II , 1971 .

[12]  Daniel Daners,et al.  Heat Kernel Estimates for Operators with Boundary Conditions , 2000 .

[13]  Giuseppe Buttazzo,et al.  Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions , 1991 .

[14]  Yaping Liu.  Smoothing the domain out for positive solutions , 2000, Bulletin of the Australian Mathematical Society.

[15]  U. Mosco.  Convergence of convex sets and of solutions of variational inequalities , 1969 .

[16]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[17]  V. Sverák,et al.  On optimal shape design , 1992 .

[18]  J. M. Arrieta.  Domain Dependence of Elliptic Operators in Divergence Form , 1997 .

[19]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .

[20]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[21]  On the domain dependence of solutions to the two-phase Stefan problem , 2000 .

[22]  Dorin Bucur,et al.  Uniform Concentration-Compactness for Sobolev Spaces on Variable Domains , 2000 .

[23]  E. N. Dancer.  The effect of domain shape on the number of positive solutions of certain nonlinear equations, II , 1990 .

[24]  Peter Stollmann,et al.  A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains , 1995 .

[25]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[26]  Friedrich Stummel,et al.  Diskrete Konvergenz linearer Operatoren. I , 1970 .

[27]  Friedrich Stummel,et al.  1.—Perturbation Theory for Sobolev Spaces , 1975, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[28]  Gianni Dal Maso,et al.  Wiener's criterion and Γ-convergence , 1987 .

[29]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[30]  Daniel Daners,et al.  Domain Perturbation for Linear and Nonlinear Parabolic Equations , 1996 .

[31]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[32]  W. Arendt,et al.  DOMAIN PERTURBATION FOR THE FIRST EIGENVALUE OF THE DIRICHLET SCHRODINGER OPERATOR , 1995 .

[33]  Limits of nonlinear Dirichlet problems in varying domains , 1988 .

[34]  P. Kunstmann.  Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators , 1999 .

[35]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[36]  Rolf Dieter Grigorieff.  Diskret kompakte Einbettungen in Sobolewschen Räumen , 1972 .

[37]  D. Daners.  Existence and perturbation of principal eigenvalues for a periodic-parabolic problem. , 2000 .

[38]  R. Showalter.  Hilbert Space Methods for Partial Differential Equations , 1979, Electronic Journal of Differential Equations.

[39]  J. Weidmann,et al.  Stetige Abhängigkeit der Eigenwerte und Eigenfunktionen elliptischer Differentialoperatoren vom Gebiet. , 1984 .

[40]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.