The oblique derivative problem for the heat equation in Lipschitz cylinders
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- by Russell M. Brown PDF
- Proc. Amer. Math. Soc. 107 (1989), 237-250 Request permission
Abstract:
We consider a class of initial-boundary value problems for the heat equation on $(0.T) \times \Omega$ with $\Omega$ a bounded Lipschitz domain in ${{\mathbf {R}}^n}$. On the lateral boundary, $(0,T) \times \partial \Omega = {\Sigma _T}$, we specify $\left \langle {\alpha ,\nabla u} \right \rangle$ where $\nabla u$ denotes the spatial gradient of the solution and $\alpha :{\Sigma _T} \to \{ x:|x| = 1\}$ is a continuous vector field satisfying $\left \langle {\alpha ,\nu } \right \rangle \geq \mu > 0$ with $\nu$ the unit normal to $\partial \Omega$. On the initial surface, $\{ 0\} \times \Omega$, we require that the solution vanish. The lateral data is taken from ${L^p}({\Sigma _T})$. For $p \in (2 - ,\infty )$, we show existence and uniqueness of solutions to this problem with estimates for the parabolic maximal function of the spatial gradient of the solution.References
-
R. M. Brown, Layer potentials and boundary value problems for the heat equation on Lipschitz cylinders, Thesis, University of Minnesota, 1987.
- Russell M. Brown, Area integral estimates for caloric functions, Trans. Amer. Math. Soc. 315 (1989), no. 2, 565–589. MR 994163, DOI 10.1090/S0002-9947-1989-0994163-7
- Russell M. Brown, The initial-Neumann problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 320 (1990), no. 1, 1–52. MR 1000330, DOI 10.1090/S0002-9947-1990-1000330-7 A. P. Calderón, Boundary value problems in Lipschitzian domains, Recent Progress in Fourier Analysis, Elsevier Science Publishers, 1985, 33-48.
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- Eugene Fabes and Sandro Salsa, Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 279 (1983), no. 2, 635–650. MR 709573, DOI 10.1090/S0002-9947-1983-0709573-7
- David S. Jerison and Carlos E. Kenig, Boundary value problems on Lipschitz domains, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 1–68. MR 716504
- John T. Kemper, Temperatures in several variables: Kernel functions, representations, and parabolic boundary values, Trans. Amer. Math. Soc. 167 (1972), 243–262. MR 294903, DOI 10.1090/S0002-9947-1972-0294903-6
- Carlos E. Kenig and Jill Pipher, The oblique derivative problem on Lipschitz domains with $L^p$ data, Amer. J. Math. 110 (1988), no. 4, 715–737. MR 955294, DOI 10.2307/2374647
- Gary M. Lieberman, Intermediate Schauder theory for second order parabolic equations. I. Estimates, J. Differential Equations 63 (1986), no. 1, 1–31. MR 840590, DOI 10.1016/0022-0396(86)90052-5 —, Intermediate Schauder theory for second order parabolic equations. III, The tusk condition. preprint.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
- Neil S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. MR 226168, DOI 10.1002/cpa.3160210302
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 237-250
- MSC: Primary 35K05
- DOI: https://doi.org/10.1090/S0002-9939-1989-0987608-5
- MathSciNet review: 987608