W1,q regularity results for elliptic transmission problems on heterogeneous polyhedra

Mathematical Models and Methods in Applied Sciences

Volumn 17, Issue 4, 2007, Pages 593-615

  Elschner, J ^a   Kaiser, H C ^a   Rehberg, J ^a   Schmidt, G ^a  

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

Author keywords

Elliptic transmission problems; Polyhedral domains; W1,q regularity

Indexed keywords

EID: 34247326926     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202507002030     Document Type: Article

References (37)

1. H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math., Vol. 133, eds. H.-J. Schmeisser et al. (Teubner-Verlag, 1993), pp. 9-126.
2. P. Auscher and P. Tchamitchian, Square-root problem for divergence operators and related topics, Asterisque 249 (1998) viii + 172 pp.
3. 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl Math. 51 (1998) 1-21.
4. M. Costabel and M. Dauge, Construction of corner singularities for Agmon-DouglisNirenberg elliptic systems, Math. Nachr. 162 (1993) 209-237.
5. M. Costabel, M. Dauge and Y. Lafranche, Fast semi-analytic computation of elastic edge singularities, Comput. Meth. Appl. Mech. Engrg. 190 (2001) 2111-2134.
6. M. Costabel, M. Dauge and S. Nicaise, Singularities of Maxwell interface problems, Math. Model. Numer. And. 33 (1999) 627-649.
7. M. Costabel and E. P. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985) 367-413.
8. M. Dauge, Elliptic Boundary Value Problems in Corner Domain. Smoothness and Asymptotics of Solutions, Lecture Notes in Math., Vol. 1341 (Springer-Verlag, 1988).
9. M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Eqns. Oper. Theory 15 (1992) 227-261
10. H. Gajewski and K. Gröger, Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics, Math. Nachr. 140 (1987) 7-36.
11. H. Gajewski and K. Gröger, Reaction-diffusion processes of electrically charged species, Math. Nachr. 177 (1996) 109-130.
12. 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann. 283 (1989) 679-687.
13. E. M. Il'in, Singularities for solutions of elliptic boundary value problems with discontinuous highest order coefficients, Zap. Nauchn. Semin. LOMI 38 (1973) 33-45 (in Russian).
14. E. M. Il'in, Singularities of the weak solutions of elliptic equations 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).
15. E. Kapon, Semiconductor Lasers I (Academic Press, 1999).
16. R. B. Kellogg, On the Poisson equation with intersecting interfaces, Appl. Anal. 4 (1975) 101-129.
17. D. Knees, On the regularity of weak solutions of quasi-linear elliptic transmission problems on polyhedral domains, Z. Anal. Anwend. 23 (2004) 509-546.
18. V. A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967) 227-313.
19. D. Leguillon and E. Sanchez-Palenzia, Computation of Singular Solutions in Elliptic Problems and Elasticity (Wiley, 1987).
20. Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite materials, Comm. Pure Appl. Math. 56 (2003) 892-925.
21. Y. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rat. Mech. Anal. 153 (2000) 91-151.
22. p, Arch. Rat. Mech. Anal. 171 (2004) 219-262.
23. V. G. Maz'ya and J. Rossmann, Point estimates for Green's matrix to boundary value problems for second order elliptic systems in a polyhedral cone, ZAMM 82 (2002) 291-316.
24. p estimates of solutions to boundary value problems for second order elliptic problems in polyhedral domains, ZAMM 83 (2003) 435-467.
25. D. Mercier, Minimal regularity of the solutions of some transmission problems, Math. Meth. Appl. Sci. 26 (2003) 321-348.
26. p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa 17 (1963) 189-206.
27. S. Nicaise and A.-M. Sandig, General interface problems I, II, Math. Meth. Appl. Sci. 17 (1994) 395-430, 431-450.
28. S. Nicaise and A.-M. Sandig, Transmission problems for the Laplace and elasticity operators: regularity and boundary integral formulation, Math. Mod. Meth. Appl. Sci. 9 (1999) 855-898.
29. M. Petzoldt, Regularity results for Laplace interface problems in two dimensions, Z. Anal, Anwend. 20 (2001) 431-455.
30. G. Savaré, Regularity results for elliptic equations in Lipschitz domains, J. Fund. Anal. 152 (1998) 176-201.
31. C. Schaefer, Einführung in die Theoretische Physik II: Theorie der Wärme, Molekularkinetische Theorie (de Gruyter, 1958).
32. S. Selberherr, Analysis and Simulation of Semiconductors (Springer-Verlag, 1984).
33. A. Sommerfeld, Electrodynamics, Lectures on Theoretical Physics, Vol. III (Academic Press, 1952).
34. A. Sommerfeld, Thermodynamics and Statistical Mechanics, Lectures on Theoretical Physics, Vol. V (Academic Press, 1956).
35. B. A. Szabo and Z. Yosibash, 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.
36. C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, 1991).
37. W. P. Ziemer, Weakly Differentiable Functions (Springer-Verlag, 1989).