Optimal regularity for elliptic transmission problems including C 1 interfaces

Interfaces and Free Boundaries

Volumn 9, Issue 2, 2007, Pages 233-252

  Elschner, Johannes ^a   Rehberg, Joachim ^a   Schmidt, Günther ^a  

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

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547307210     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/163     Document Type: Article

References (60)

1. ADAMS, L., & LI, Z. The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 24 (2002), 463-479. Zbl 1014.65099 MR 1951051
2. AGMON, S., DOUGLIS, A., & NIRENBERG, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12 (1959), 623-727. Zbl 0093.10401 MR 0125307
3. AGRANOVICH, M.S. Spectral properties of potential type operators for a class of strongly elliptic systems on smooth and Lipschitz surfaces. Trudy Moskov. Mat. Obshch. 62 (2001), 3-53 (in Russian);
4. English transi.: Trans. Moscow Math. Soc. 2001, 1-47. Zbl 1056.35128 MR 1907250
5. AGRANOVICH, M. S. Spectral problems for second-order strongly elliptic systems in domains with smooth and nonsmooth boundaries, Uspekhi Mat. Nauk 57 (2002), no. 5, 3-79 (in Russian);
6. English transl.: Russian Math. Surveys 57 (2002), no. 5, 847-920. Zbl 1057.35019 MR 1992082
7. AMANN, H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function Spaces, Differential Operators and Nonlinear Analysis, H.-J. Schmeisser et al. (eds.), Teubner-Texte Math. 133, Stuttgart (1993), 9-126. Zbl 0810.35037 MR 1242579
8. ARENDT, W., & TERELST, A. F. M. Gaussian estimates for second order elliptic operators with boundary conditions. J. Operator Theory 38 (1997), 87-130. Zbl 0879.35041 MR 1462017
9. AUSCHER, P., & TCHAMITCHIAN, P. Square root problem for divergence operators and related topics. Astérisque 249 (1998). Zbl 0909.35001 MR 1651262
10. p theory. Math. Ann. 320 (2001), 577-623. Zbl pre01663012 MR 1846778
11. BANDELOW, U., HÜNLICH, R., & KOPRUCKI, T. Simulation of static and dynamic properties of edge-emitting multiple-quantum-well lasers. IEEE JSTQE 9 (2003), 798-806.
12. BENDANIEL, D., & DUKE, C. Space-charge effects on electron tunneling. Phys. Rev. 152 (1966), 683-692.
13. BERESTYCKI, H., HAMEL, F., & ROQUES, L. Analysis of the periodically fragmented environment model: I-species persistence. J. Math. Biol. 51 (2005), 75-113. Zbl 1066.92047 MR 2214420
14. 1'p estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51 (1998), 1-21. Zbl 0906.35030 MR 1486629
15. CHEN, Z., & ZOU, J. Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998), 175-202. Zbl 0909.65085 MR 1622502
16. CHIPOT, M., KINDERLEHRER, D., & CAFFARELLI, G. V. Smoothness of linear laminates. Arch. Ration. Mech. Anal. 96 (1986), 81-96. Zbl 0617.73062 MR 0853976
17. COSTABEL, M., & DAUGE, M. General edge asymptotics of solutions of second-order elliptic boundary value problems I. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 109-155. Zbl 0791.35032 MR 1204855
18. COSTABEL, M., DAUGE, M., & DUDUCHAVA, R. Asymptotics without logarithmic terms for crack problems. Comm. Partial Differential Equations 28 (2003), 869-926. Zbl pre01924551 MR 1986055
19. COSTABEL, M., DAUGE, M., & LAFRANCHE, Y. Fast semi-analytic computation of elastic edge singularities. Comput. Methods Appl. Mech. Engrg. 190 (2001), 2111-2134. Zbl 1013.74078
20. COULHON, T., & DUONG, X. Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv. Differential Equations S (2000), 343-368. Zbl 1001.34046 MR 1734546
21. DAUGE, M. Neumann and mixed problems on curvilinear polyhedra. Integral Equations Operator Theory 15 (1992), 227-261. Zbl 0767.46026 MR 1147281
22. DUDUCHAVA, R., SANDIG, A. M., & WENDLAND, W. Interface cracks in anisotropic composites. Math. Methods Appl. Sci. 22 (1999), 1413-1446. Zbl 0933.35052 MR 1715761
23. ELSCHNER, J., KAISER, H.-C., REHBERG, J., & SCHMIDT, G. Wli? regularity results for elliptic transmission problems on heterogeneous polyhedra. Math. Models Methods Appl. Sci. 17 (2007), 593-615.
24. ESCAURIAZA, L., FABES, E. B., & VERCHOTA, G. On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries. Proc. Amer. Math. Soc. 115 (1992), 1069-1076. Zbl 0761.35013 MR 1092919
25. FRANZONE, P. C., GUERRI, L., & 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. Zbl 0733.92006
26. GAJEWSKI, H., & GRÖGER, K. Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics. Math. Nachr. 140 (1987), 7-36. Zbl 0681.35081 MR 1015383
27. GAJEWSKI, H., & GRÖGER, K. Reaction-diffusion processes of electrical charged species. Math. Nachr. 177 (1996), 109-130.
28. GRISVARD, P. Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). Zbl 0695.35060 MR 0775683
29. p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989), 679-687. Zbl 0646.35024 MR 0990595
30. p spaces. Comm. Partial Differential Equations 15 (1990), 289-340. Zbl 0723.35091 MR 1044427
31. HENRY, D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840, Springer (1981). Zbl 0456.35001 MR 0610244
32. q estimates for parabolic evolution equations. Comm. Partial Differential Equations 22 (1997), 1647-1669. Zbl 0886.35030 MR 1469585
33. HWANG, T., LIN, W., LIU, J., & WANG, W. Jocobi-Davidson methods for cubic eigenvalue problems. Numer. Linear Algebra Appl. 12 (2005), 605-624. MR 2166875
34. JENTSCH, L., NATROSHVILI, D., & WENDLAND, W. General transmission problems in the theory of elastic oscillations of anisotropic bodies (Mixed interface problems). J. Math. Anal. Appl. 235 (1999), 418-434. Zbl 1066.74546 MR 1703704
35. JERISON, D., & KENIG, C. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1995), 161-219. Zbl 0832.35034 MR 1331981
36. KAPON, E. Semiconductor Lasers I. Academic Press, Boston (1999).
37. KATO, T. Perturbation Theory for Linear Operators. Grundlehren Math. Wiss. 132, Springer, Berlin (1984). Zbl 0531.47014 MR 0407617
38. KENIG, C. Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. Amer. Math. Soc., Providence (1994). Zbl 0812.35001 MR 1282720
39. KNEES, D. On the regularity of weak solutions of quasi-linear elliptic transmission problems on polyhedral domains. Z. Anal. Anwend. 23 (2004), 509-546. MR 2094595
40. LADYZHENSKAYA, O. A., RIVKIND, V. Y., & URAL'TSEVA, N. N. The classical solvability of diffraction problems. Trudy Mat. Inst. Steklova 92 (1966), 116-146 (in Russian).
41. MORREY, C. B. JUN. Multiple Integrals in the Calculus of Variations. Grundlehren Math. Wiss. 130, Springer, Berlin (1966). Zbl 0142.38701 MR 0202511
42. LEGUILLON, D., & SANCHEZ-PALENCIA, E. Computation of Singular Solutions in Elliptic Problems and Elasticity. Wiley, Chichester (1987), Zbl 0647.73010 MR 0995254
43. LI, Y. Y., & VOGELIUS, M. Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal. 153 (2000), 91-151. Zbl 0958.35060 MR 1770682
44. LI, Y., LIU, J., VOSKOBOYNIKOV, O., LEE, C., & SZE, S. Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot. Comput. Phys. Comm. 140 (2001), 399-404. Zbl 1072.82576
45. p. Arch. Ration. Mech. Anal. 171 (2004), 219-262. Zbl 1053.35058 MR 2034754
46. p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963), 189-206. Zbl 0127.31904 MR 0159110
47. OLEǏNIK, O. A. Boundary value problems for linear equations of elliptic and parabolic type with discontinuous coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 3-20 (in Russian).
48. p. Nonlinear Elliptic and Parabolic Problems (Zürich, 2004), Progr. Nonlinear Differential Equations Appl. 64, M. Chipot and J. Escher (eds.), Birkhäuser (2005), 413-419. Zbl pre05005821 MR 2185230
49. ROITBERG, J. A., & SHEFTEL, Z. G. On equations of elliptic type with discontinuous coefficients. Soviet Math. Dokl. 3 (1962), 1491-1494.
50. SAVARE, G. Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998), 176-201. Zbl 0889.35018 MR 1600081
51. SCHECHTER, M. A generalization of the problem of transmission. Ann. Scuola Norm. Sup. Pisa 14(1960), 207-236. Zbl 0094.29603 MR 0131063
52. SOMMERFELD, A. Electrodynamics. Lectures on Theoretical Physics, Vol. Ill, Academic Press, New York (1952). Zbl 0047.43904 MR 0051135
53. SOMMERFELD, A. Thermodynamics and Statistical Mechanics. Lectures on Theoretical Physics, Vol. V, Academic Press, New York (1956). Zbl 0070.43613 MR 0074330
54. TRIEBEL, H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland (1978). Zbl 0387.46032 MR 0500580
55. WANG, W. A jump condition capturing finite difference scheme for elliptic interface problems. SIAM J. Sci. Comput. 25 (2004), 1479-1496. Zbl 1061.65110 MR 2087322
56. WANG, W., HWANG, T., LIN, W., & LIU, J. Numerical methods for semiconductor heterostructures with band nonparabolicity. J. Comput. Phys. 190 (2003), 141-158. Zbl 1027.82040 MR 2046761
57. WEISBUCH, C., & VINTER, B. Quantum Semiconductor Structures: Fundamentals and Applications. Academic Press, Boston (1991).
58. WLOKA, J. Partial Differential Equations. Cambridge Univ. Press, Cambridge (1987). Zbl 0623.35006 MR 0895589
59. ZANGER, D. The inhomogeneous Neumann problem in Lipschitz domains. Comm. Partial Differential Equations 25 (2000), 1771-1808. Zbl 0959.35041 MR 1778780
60. ZIEMER, W. P. Weakly Differentiable Functions. Springer, Berlin (1989). Zbl 0692.46022 MR 1014685