Elliptic model problems including mixed boundary conditions and material heterogeneities

Journal des Mathematiques Pures et Appliquees

Volumn 89, Issue 1, 2008, Pages 25-48

  Haller Dintelmann, Robert ^a   Kaiser, Hans Christoph ^a   Rehberg, Joachim ^a  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Elliptic transmission problems; Mixed boundary problems; W1, p regularity

Indexed keywords

EID: 37349054168     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2007.09.001     Document Type: Article

References (55)

1. Amann H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser H.-J., et al. (Ed). Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math. vol. 133 (1993) 9-126
2. Adams L., and Li Z. The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 24 (2002) 463-479
3. Auscher P., and Tchamitchian P. Square root problem for divergence operators and related topics. Astérisque 249 (1998)
4. Bandelow U., Kaiser H.-C., Koprucki T., and Rehberg J. Modeling and simulation of strained quantum wells in semiconductor lasers. In: Jäger W., and Krebs H.-J. (Eds). Mathematics-Key Technology for the Future. Joint Projects Between Universities and Industry (2003), Springer, Berlin/Heidelberg/New York 377-390
5. 2, p-regularity for solutions of mixed problems. J. Math. Pures Appl. 53 (1974) 279-290
6. Berestycki H., Hamel F., and Roques L. Analysis of the periodically fragmented environment model: I-species persistence. J. Math. Biol. 51 (2005) 75-113
7. Bey R., Loheac J.-P., and Moussaoui M. Singularities of the solution of a mixed boundary problem for a general second order elliptic equation and boundary stabilization of the wave equation. J. Math. Pures Appl. 78 (1999) 1043-1067
8. Brown R. The mixed problem for Laplace's equation in a class of Lipschitz domains. Commun. Partial Differential Equations 19 (1994) 1217-1223
9. Chen Z., and Zou J. Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175-202
10. Costabel M., Dauge M., and Lafranche Y. Fast semi-analytic computation of elastic edge singularities. Comput. Meth. Appl. Mech. Engrg. 190 (2001) 2111-2134
11. Dauge M. Neumann and mixed problems on curvilinear polyhedra. Integral Equations Oper. Theory 15 (1992) 227-261
12. Ebmeyer C., and Frehse J. Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains. Math. Nachr. 203 (1999) 47-74
13. 1, q regularity results for elliptic transmission problems on heterogeneous polyhedra. Math. Models Methods Appl. Sci. 17 (2007) 593-615
14. 1 interfaces. Interfaces Free Bound. 9 (2007) 233-252
15. Evans L.C., and Gariepy R.F. Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (1992), CRC Press, Boca Raton, New York, London, Tokyo
16. Franzone P.C., Guerri L., and Rovida S. Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulation. J. Math. Biol. 28 (1990) 121-176
17. Gajewski H. Analysis und Numerik von Ladungstransport in Halbleitern (Analysis and numerics of carrier transport in semiconductors). Mitt. Ges. Angew. Math. Mech. 16 (1993) 35-57
18. Giusti E. Metodi diretti nel calcolo delle variazioni (1994), Unione Matematica Italiana, Bologna
19. Glitzky A., and Hünlich R. Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures. Appl. Anal. 66 (1997) 205-226
20. Griepentrog J.A., and Recke L. Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces. Math. Nachr. 225 (2001) 39-74
21. Griepentrog J.A. Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals. Math. Nachr. 243 (2002) 19-42
22. 1, p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679-687
23. Griepentrog J.A., Gröger K., Kaiser H.-C., and Rehberg J. Interpolation for function spaces related to mixed boundary value problems. Math. Nachr. 241 (2002) 110-120
24. Grisvard P. Elliptic Problems in Nonsmooth Domains (1985), Pitman, Boston
25. Harutjunjan G., and Schulze B.-W. Parametrices of mixed elliptic problems. Math. Nachr. 277 (2004) 56-82
26. M. Hieber, J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions, WIAS-Preprint 1194, 2006
27. Hwang T., Lin W., Liu J., and Wang W. Jacobi-Davidson methods for cubic eigenvalue problems. Numer. Linear Algebra Appl. 12 (2005) 605-624
28. Il'yin E.M. Singularities for solutions of elliptic boundary value problems with discontinuous highest order coefficients. Zap. Nauchn. Semin. LOMI 38 (1973) 33-45 (in Russian)
29. Il'yin E.M. Singularities of the weak solutions of elliptic equations with discontinuous higher coefficients. II. Corner points of the lines of discontinuity. Zap. Nauchn. Semin. LOMI 47 (1974) 166-169 (in Russian)
30. Jerison D., and Kenig C. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219
31. Kato T. Perturbation Theory for Linear Operators. Reprint of the corr. print of the second ed. Classics in Mathematics (1980), Springer, Berlin/Heidelberg/New York
32. Knees D. On the regularity of weak solutions of quasi-linear elliptic transmission problems on polyhedral domains. Zeitschr. Anal. Anwend. 23 (2004) 509-546
33. Koprucki T., Kaiser H.-C., and Fuhrmann J. Electronic states in semiconductor nanostructures and upscaling to semi-classical models. In: Mielke A. (Ed). Analysis, Modeling and Simulation of Multiscale Problems (2006), Springer, Berlin/Heidelberg/New York 367-396
34. Ladyzhenskaya O.A., and Ural'tseva N.N. Linear and Quasilinear Elliptic Equations. Mathematics in Science and Engineering (1968), Academic Press, New York/London
35. Leguillon D., and Sanchez-Palenzia E. Computation of Singular Solutions in Elliptic Problems and Elasticity (1987), Wiley, Chichester
36. Li Y., Liu J., Voskoboynikov O., Lee C., and Sze S. Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot. Comp. Phys. Comm. 140 (2001) 399-404
37. Maz'ya V. Sobolev Spaces (1985), Springer, Berlin/Heidelberg/New York
38. p estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. Z. Angew. Math. Mech. 83 (2003) 435-467
39. p. Arch. Rat. Mech. Anal. 171 (2004) 219-262
40. Mercier D. Minimal regularity of the solutions of some transmission problems. Math. Meth. Appl. Sci. 26 (2003) 321-348
41. Mitrea I., and Mitrea M. The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains. Trans. Amer. Math. Soc. 359 (2007) 4143-4182
42. p. In: Chipot M., et al. (Ed). Nonlinear elliptic and parabolic problems. A special tribute to the work of Herbert Amann. Progress in Nonlinear Differential Equations and their Applications vol. 64 (2005), Birkhäuser, Basel 413-419
43. Rempel S., and Schulze B.-W. Asymptotics for Elliptic Mixed Boundary Problems (Pseudo-Differential and Mellin Operators in Spaces with Conormal Singularity). Math. Res. vol. 50 (1989), Akademie-Verlag, Berlin
44. Savaré G. Regularity and perturbation results for mixed second order elliptic problems. Commun. Partial Differential Equations 22 (1997) 869-899
45. Savaré G. Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998) 176-201
46. Selberherr S. Analysis and Simulation of Semiconductors (1984), Springer, Wien
47. p theory. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17 (1963) 117-139
48. Shamir E. Regularization of mixed second-order elliptic problems. Isr. J. Math. 6 (1968) 150-168
49. Sommerfeld A. Electrodynamics. Lectures on Theoretical Physics vol. III (1952), Academic Press, New York
50. Sommerfeld A. Thermodynamics and Statistical Mechanics. Lectures on Theoretical Physics vol. V (1956), Academic Press, New York
51. Szabo B.A., and Yosibash Z. Numerical analysis of singularities in two dimensions. Part II: Computation of generalized flux/stress intensity factors. Int. J. Numer. Meth. Engrg. 39 (1996) 409-434
52. Wang W., Hwang T., Lin W., and Liu J. Numerical methods for semiconductor heterostructures with band nonparabolicity. J. Comp. Phys. 190 (2003) 141-158
53. Wang W. A jump condition capturing finite difference scheme for elliptic interface problems. SIAM J. Sci. Comp. 25 (2004) 1479-1496
54. Weisbuch C., and Vinter B. Quantum Semiconductor Structures: Fundamentals and Applications (1991), Academic Press, Boston
55. Zanger D. The inhomogeneous Neumann problem in Lipschitz domains. Commun. Partial Differential Equations 25 (2000) 1771-1808