Abstract
Second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions are studied. Existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder are proved using methods from the theory of integrated semigroups, showing in particular the well-posedness of the abstract Cauchy problem in spaces of continuous functions. Under natural assumptions on the coefficients and the inhomogeneity the solutions are shown to converge to an equilibrium or to be asymptotically almost periodic.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
W. Arendt: Resolvent positive operators and inhomogeneous boundary conditions. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29 (2000), 639–670.
W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics 96, Birkhäuser, Basel, 2001.
W. Arendt, M. Chovanec: Dirichlet regularity and degenerate diffusion. Trans. Am. Math. Soc. 362 (2010), 5861–5878.
W. Arendt, R. Nittka: Equivalent complete norms and positivity. Arch. Math. 92 (2009), 414–427.
W. Arendt, R. Schätzle: Semigroups generated by elliptic operators in non-divergence form on C0(Ω). Ann. Sc. Norm. Super. Pisa, Cl. Sci. 13 (2014), 417–434.
H. Bohr: Almost Periodic Functions. Chelsea Publishing Company, New York, 1947.
D. M. Bošković, M. Krstić, W. Liu: Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE Trans. Autom. Control 46 (2001), 2022–2028.
P. Cannarsa, F. Gozzi, H. M. Soner: A dynamic programming approach to nonlinear boundary control problems of parabolic type. J. Funct. Anal. 117 (1993), 25–61.
R. Chapko, R. Kress, J. -R. Yoon: An inverse boundary value problem for the heat equation: the Neumann condition. Inverse Probl. 15 (1999), 1033–1046.
D. Daners: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217 (2000), 13–41.
R. Dautray, J. -L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I. Springer, Berlin, 1992.
R. Denk, M. Hieber, J. Prüss: Optimal L p-L q-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257 (2007), 193–224.
G. Dore: L p regularity for abstract differential equations. Functional Analysis and Related Topics (H. Komatsu, ed.). Proc. Int. Conference, Kyoto University, 1991. Lect. Notes Math. 1540, Springer, Berlin, 1993, pp. 25–38.
K. -J. Engel: The Laplacian on C(\(\overline \Omega \)) with generalized Wentzell boundary conditions. Arch. Math. 81 (2003), 548–558.
K. -J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194, Springer, Berlin, 2000.
D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 ed. Classics in Mathematics, Springer, Berlin, 2001.
J. A. Griepentrog: Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Adv. Differ. Equ. 12 (2007), 1031–1078.
R. Haller-Dintelmann, J. Rehberg: Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Equations 247 (2009), 1354–1396.
O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural’ceva: Linear and Quasi-Linear Equations of Parabolic Type. Translations of Mathematical Monographs 23, American Mathematical Society, Providence, 1968.
G. M. Lieberman: Second Order Parabolic Differential Equations. World Scientific, Singapore, 1996.
J. -L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems and Applications. Vol. II. Die Grundlehren der mathematischenWissenschaften, Band 182, Springer, Berlin, 1972.
R. Nittka: Quasilinear elliptic and parabolic Robin problems on Lipschitz domains. NoDEA, Nonlinear Differ. Equ. Appl. 20 (2013), 1125–1155.
R. Nittka: Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equations 251 (2011), 860–880.
R. Nittka: Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains. PhD Thesis, University of Ulm, 2010.
W. Stepanoff: Über einige Verallgemeinerungen der fast periodischen Funktionen. Math. Ann. 95 (1926), 473–498. (In German.)
M. Warma: The Robin andWentzell-Robin Laplacians on Lipschitz domains. Semigroup Forum 73 (2006), 10–30.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nittka, R. Inhomogeneous parabolic Neumann problems. Czech Math J 64, 703–742 (2014). https://doi.org/10.1007/s10587-014-0127-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-014-0127-4
Keywords
- parabolic initial-boundary value problem
- inhomogeneous Robin boundary conditions
- existence of weak solution
- continuity up to the boundary
- asymptotic behavior
- asymptotically almost periodic solution