Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. (English) Zbl 1225.35077
Summary: For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain \(\Omega \) we show that solutions of the corresponding elliptic problem with Robin and thus in particular with Neumann boundary conditions are Hölder continuous up to the boundary for sufficiently \(L^{p}\)-regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell-Robin boundary conditions is well-posed on \(C \bar {\varOmega}\).
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
second order linear elliptic equations; Lipschitz domains; Robin boundary conditions; Hölder regularity; \(L^{\infty }\)-coefficients; parabolic equations; strongly continuous semigroups on \(C \bar {\varOmega}\); Wentzell-Robin boundary conditionsReferences:
| [1] | Alibert, J.-J.; Raymond, J.-P., Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls, Numer. Funct. Anal. Optim., 18, 3-4, 235-250 (1997) · Zbl 0885.49010 |
| [2] | Arendt, W.; ter Elst, A., Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38, 1, 87-130 (1997) · Zbl 0879.35041 |
| [3] | Arendt, W.; Warma, M., The Laplacian with Robin boundary conditions on arbitrary domains, Potential Anal., 19, 4, 341-363 (2003) · Zbl 1028.31004 |
| [4] | Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems, Monogr. Math., vol. 96 (2001), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0978.34001 |
| [5] | Arendt, W.; Metafune, G.; Pallara, D.; Romanelli, S., The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions, Semigroup Forum, 67, 2, 247-261 (2003) · Zbl 1073.47045 |
| [6] | Bass, R.; Hsu, P., Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab., 19, 2, 486-508 (1991) · Zbl 0732.60090 |
| [7] | Cazenave, T.; Haraux, A., An Introduction to Semilinear Evolution Equations, Oxford Lecture Ser. Math. Appl., vol. 13 (1998), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, translated from the 1990 French original by Yvan Martel and revised by the authors · Zbl 0926.35049 |
| [8] | Coclite, G.; Favini, A.; Goldstein, G.; Goldstein, J.; Romanelli, S., Continuous dependence on the boundary conditions for the Wentzell Laplacian, Semigroup Forum, 77, 1, 101-108 (2008) · Zbl 1149.35313 |
| [9] | Daners, D., Heat kernel estimates for operators with boundary conditions, Math. Nachr., 217, 13-41 (2000) · Zbl 0973.35087 |
| [10] | Daners, D., Robin boundary value problems on arbitrary domains, Trans. Amer. Math. Soc., 352, 9, 4207-4236 (2000) · Zbl 0947.35072 |
| [11] | Daners, D., Inverse positivity for general Robin problems on Lipschitz domains, Arch. Math. (Basel), 92, 1, 57-69 (2009) · Zbl 1163.35011 |
| [12] | Engel, K.-J., The Laplacian on \(C(\overline{\Omega})\) with generalized Wentzell boundary conditions, Arch. Math. (Basel), 81, 5, 548-558 (2003) · Zbl 1064.47043 |
| [13] | Engel, K.-J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194 (2000), Springer-Verlag: 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 · Zbl 0952.47036 |
| [14] | Evans, L.; Gariepy, R., Measure theory and fine properties of functions, Stud. Adv. Math. (1992), CRC Press: CRC Press Boca Raton, FL · Zbl 0804.28001 |
| [15] | Favini, A.; Goldstein, G.; Goldstein, J.; Romanelli, S., The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2, 1, 1-19 (2002) · Zbl 1043.35062 |
| [16] | Favini, A.; Goldstein, G.; Goldstein, J.; Romanelli, S., Wentzell boundary conditions in the nonsymmetric case, Math. Model. Nat. Phenom., 3, 7, 143-147 (2008) · Zbl 1337.35064 |
| [17] | Fukushima, M.; Tomisaki, M., Reflecting diffusions on Lipschitz domains with cusps — analytic construction and Skorohod representation, Potential Anal., 4, 4, 377-408 (1995) · Zbl 0840.60070 |
| [18] | Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order, Classics Math. (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 1042.35002 |
| [19] | Griepentrog, J.; Recke, L., Linear elliptic boundary value problems with non-smooth data: normal solvability on Sobolev-Campanato spaces, Math. Nachr., 225, 39-74 (2001) · Zbl 1009.35019 |
| [20] | Grisvard, P., Elliptic problems in nonsmooth domains, Monogr. Stud. Math., vol. 24 (1985), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, MA · Zbl 0695.35060 |
| [21] | Ouhabaz, E., Analysis of Heat Equations on Domains, London Math. Soc. Monogr. Ser., vol. 31 (2005), Princeton University Press · Zbl 1082.35003 |
| [22] | Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1995), Johann Ambrosius Barth: Johann Ambrosius Barth Heidelberg · Zbl 0830.46028 |
| [23] | Troianiello, G., Elliptic Differential Equations and Obstacle Problems, Univ. Ser. Math. (1987), Plenum Press: Plenum Press New York · Zbl 0655.35002 |
| [24] | Tröltzsch, F., Optimale Steuerung partieller Differentialgleichungen, Theorie, Verfahren und Anwendungen (2005), Vieweg: Vieweg Wiesbaden · Zbl 1142.49001 |
| [25] | Warma, M., The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum, 73, 1, 10-30 (2006) · Zbl 1168.35340 |
| [26] | Ziemer, W., Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math., vol. 120 (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0692.46022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
