Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains

SIAM Journal on Mathematical Analysis

Volumn 40, Issue 1, 2008, Pages 292-305

  Hieber, Matthias ^a   Rehberg, Joachim ^b  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

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

Author keywords

L coefficients; Mixed dirichlet neumann conditions; Quasilinear parabolic system

Indexed keywords

DIRICHLET-NEUMANN BOUNDARY CONDITIONS; MAXIMAL REGULARITIES; MIXED BOUNDARY CONDITIONS; MIXED DIRICHLET-NEUMANN CONDITIONS; NONSMOOTH DOMAINS; QUASI-LINEAR SYSTEMS; QUASILINEAR PARABOLIC SYSTEM; REACTION-DIFFUSION EQUATIONS;

BOUNDARY CONDITIONS;

EID: 61849116177     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/070683829     Document Type: Article

References (42)

1. H. AMANN, Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel, Boston, Berlin, 1995.
2. H. AMANN, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math. 133, H.-J. Schmeisser et al., eds., Stuttgart, 1993, pp. 9-126.
3. K. ANDREWS, P. SHI, M. SHILLOR, AND S. WRIGHT, Thermoelastic contact with Barber's heat exchange condition, Appl. Math. Optim., 28 (1993), pp. 11-48.
4. W. ARENDT AND A. F. M. TER ELST, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), pp. 87-130.
5. M. CHAPLAIN AND G. LOLAS, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), pp. 1685-1734.
6. S. N. ANTONTSEV AND M. CHIPOT, The thermistor problem: Existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), pp. 1128-1156.
7. N. H. CHANG AND M. CHIPOT, On some mixed boundary value problems with nonlocal diffusion, Adv. Math. Sci. Appl., 14 (2004), pp. 1-24.
8. M. CHIPOT AND B. LOVAT, On the asymptotic behavior of some nonlocal problems, Positivity, 3 (1999), pp. 65-81.
9. P. G. CIARLET, The finite element method for elliptic problems, Stud. Math. Appl., North-Holland, Amsterdam, New York, Oxford, 1979.
10. P. CLÉMENT AND S. LI, Abstract parabolic quasilinear equations and application to a groundwater flow problem, Adv. Math. Sci. Appl., 3 (1994), pp. 17-32.
11. M. DAUGE, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator Theory, 15 (1992), pp. 227-261.
12. R. DENK, M. HIEBER, AND J. PRÜSS, H-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. So.c, 166 (2003), pp. viii+114.
13. X. T. DUONG AND E. M. OUHABAZ, Complex multiplicative perturbations of elliptic operators: Heat kernel bounds and holomorphic functional calculus, Differential Integral Equations, 12 (1999), pp. 395-418.
14. 1,q regularity results for elliptic transmission problems on heterogeneous polyhedra, Math. Models Methods Appl. Sci., 17 (2007), pp. 593-615.
15. 1 interfaces, Interfaces Free Bound., 9 (2007), pp. 233-252.
16. J. ESCHER, On quasilinear fully parabolic boundary value problems, Differential Integral Equations, 7 (1994), pp. 1325-1343.
17. M. FILA AND H. MATANO, Blow up in nonlinear heat equations from the dynamical systems point of view, in Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed., North-Holland, Amsterdam, 2002.
18. J. FUHRMANN AND H. LANGMACH, Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws, Appl. Numer. Math., 37 (2001), pp. 201-230.
19. H. GAJEWSKI, K. GRÖGER, AND K. ZACHARIAS, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
20. H. GAJEWSKI AND K. GRÖGER, Reaction-diffusion processes of electrical charged species, Math. Nachr., 177 (1996), pp. 109-130.
21. E. GIUSTI, Metodi diretti nel calcólo delle variazioni, Unione Matemática Italiana, Bologna, 1994.
22. p, Adv. Math. Sci. Appl., 11 (2001), pp. 87-112.
23. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
24. 1'p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), pp. 679-687.
25. R. HALLER-DINTELMANN, H.-C. KAISER, AND J. REHBERG, Elliptic Model Problems Including Mixed Boundary Conditions and Material Heterogeneities, Preprint 1203, WIAS, Berlin 2007.
26. M. HEINKENSCHLOSS AND F. TROELTZSCH, Analysis of the Langrange-SQP-Newton method for the control of a phase field equation, Control Cybernet., 28 (1999), pp. 177-211.
27. q estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), pp. 1647-1669.
28. D. JERISON AND C. KENIG, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), pp. 161-219.
29. H. C. KAISER, J. REHBERG, AND H. NEIDHARDT, Classical solutions of quasilinear parabolic systems on two dimensional domains, NoDEA Nonlinear Differential Equation Appl., 13 (2006), pp. 287-310.
30. P. KREJCI, E. ROCCA, AND J. SPREKELS, Nonlocal Temperature-Dependent Phase-Field Models for Non-Isothermal Phase Transitions, Preprint 1006, WIAS, Berlin, 2007.
31. M. MARCUS AND V. MIZEL, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), pp. 217-229.
32. V. MAZ'YA, Sobolev Spaces, Springer, Berlin, Heidelberg, New York, Tokyo, 1985.
33. p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 17 (1963), pp. 189-206.
34. p-spaces, Conf. Semin. Math. Univ. Bari, 285 (2002), pp. 1-39.
35. p, in Nonlinear Elliptic and Parabolic Problems. A special tribute to the work of Herbert Amann, Prog. Nonlinear Differential Equations Appl., M. Chipot and J. Escher, eds., 2005, pp. 413-419.
36. S. SELBERHERR, Analysis and Simulation of Semiconductors, Springer, Wien, 1984
37. E. SHAMIR, Regularization of mixed second-order elliptic problems, Israel J. Math. 6, (1968), pp. 150-168.
38. P. SHI, M. SHILLOR, AND X. XU, Existence of a solution to the Stefan problem with Joule's heating, J. Differential Equations, 105 (1993), pp. 239-263.
39. A. SOMMERFELD, Thermodynamics and statistical mechanics, Lectures on Theoretical Physics V, Academic Press, New York, 1956.
40. H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978.
41. A. UNGER AND F. TROELTZSCH, Fast solutions of optimal control problems in the selective cooling of steel, ZAMM Z. Angew. Math. Mech., 81 (2001), pp. 447-456.
42. D. ZANGER, The inhomogeneous Neumann problem in Lipschitz domains, Comm. Partial Differential Equations, 25 (2000), pp. 1771-1808.