Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains

Journal of Differential Equations

Volumn 251, Issue 4-5, 2011, Pages 860-880

  Nittka, Robin ^a  

^a UNIVERSITY OF ULM   (Germany)

Author keywords

H lder regularity; L coefficients; Lipschitz domains; Parabolic equations; Robin boundary conditions; Second order linear elliptic equations; Strongly continuous semigroups on C( ); Wentzell Robin boundary conditions

Indexed keywords

EID: 79958183659     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2011.05.019     Document Type: Article

References (26)

1. Alibert J.-J., Raymond J.-P. Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 1997, 18(3-4):235-250.
2. Arendt W., ter Elst A. Gaussian estimates for second order elliptic operators with boundary conditions. J. Operator Theory 1997, 38(1):87-130.
3. Arendt W., Warma M. The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 2003, 19(4):341-363.
4. Arendt W., Batty C., Hieber M., Neubrander F. Vector-Valued Laplace Transforms and Cauchy Problems. Monogr. Math. 2001, vol. 96. Birkhäuser Verlag, Basel.
5. Arendt W., Metafune G., Pallara D., Romanelli S. The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions. Semigroup Forum 2003, 67(2):247-261.
6. Bass R., Hsu P. Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 1991, 19(2):486-508.
7. Cazenave T., Haraux A. An Introduction to Semilinear Evolution Equations. Oxford Lecture Ser. Math. Appl. 1998, vol. 13. The Clarendon Press, Oxford University Press, New York, translated from the 1990 French original by Yvan Martel and revised by the authors.
8. Coclite G., Favini A., Goldstein G., Goldstein J., Romanelli S. Continuous dependence on the boundary conditions for the Wentzell Laplacian. Semigroup Forum 2008, 77(1):101-108.
9. Daners D. Heat kernel estimates for operators with boundary conditions. Math. Nachr. 2000, 217:13-41.
10. Daners D. Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 2000, 352(9):4207-4236.
11. Daners D. Inverse positivity for general Robin problems on Lipschitz domains. Arch. Math. (Basel) 2009, 92(1):57-69.
12. Engel K.-J. The Laplacian on C(Ω) with generalized Wentzell boundary conditions. Arch. Math. (Basel) 2003, 81(5):548-558.
13. Engel K.-J., Nagel R. One-Parameter Semigroups for Linear Evolution Equations. Grad. Texts in Math. 2000, vol. 194. Springer-Verlag, New York, with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
14. Evans L., Gariepy R. Measure theory and fine properties of functions. Stud. Adv. Math. 1992, CRC Press, Boca Raton, FL.
15. Favini A., Goldstein G., Goldstein J., Romanelli S. The heat equation with generalized Wentzell boundary condition. J. Evol. Equ. 2002, 2(1):1-19.
16. Favini A., Goldstein G., Goldstein J., Romanelli S. Wentzell boundary conditions in the nonsymmetric case. Math. Model. Nat. Phenom. 2008, 3(7):143-147.
17. Fukushima M., Tomisaki M. Reflecting diffusions on Lipschitz domains with cusps - analytic construction and Skorohod representation. Potential Anal. 1995, 4(4):377-408.
18. Gilbarg D., Trudinger N. Elliptic Partial Differential Equations of Second Order. Classics Math. 2001, Springer-Verlag, Berlin.
19. Griepentrog J., Recke L. Linear elliptic boundary value problems with non-smooth data: normal solvability on Sobolev-Campanato spaces. Math. Nachr. 2001, 225:39-74.
20. Grisvard P. Elliptic problems in nonsmooth domains. Monogr. Stud. Math. 1985, vol. 24. Pitman (Advanced Publishing Program), Boston, MA.
21. Ouhabaz E. Analysis of Heat Equations on Domains. London Math. Soc. Monogr. Ser. 2005, vol. 31. Princeton University Press.
22. Triebel H. Interpolation Theory, Function Spaces, Differential Operators 1995, Johann Ambrosius Barth, Heidelberg. 2nd edition.
23. Troianiello G. Elliptic Differential Equations and Obstacle Problems. Univ. Ser. Math. 1987, Plenum Press, New York.
24. Tröltzsch F. Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen 2005, Vieweg, Wiesbaden.
25. Warma M. The Robin and Wentzell-Robin Laplacians on Lipschitz domains. Semigroup Forum 2006, 73(1):10-30.
26. Ziemer W. Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Grad. Texts in Math. 1989, vol. 120. Springer-Verlag, New York.