On initial-boundary value problem of the stochastic heat equation in Lipschitz cylinders. (English) Zbl 1286.60059
Summary: We consider the initial boundary value problem of the non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random noises. The space boundary is Lipschitz, and we impose nonzero conditions on the parabolic boundary. We prove a regularity result by finding appropriate spaces for solutions and pre-assigned data in the problem. We use a collection of tools from potential theory, harmonic analysis, and probability. Some lemmas are as important as the main theorem.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic heat equation; Lipschitz cylinders; initial boundary value problem; anisotropic Besov spaceReferences:
| [1] | Brown, R.: The initial-Neumann problem for the heat equation in Lipschitz cylinders. Trans. Am. Math. Soc. 320(1), 1-52 (1990) · Zbl 0714.35037 · doi:10.1090/S0002-9947-1990-1000330-7 |
| [2] | Brown, R.: The method of layer potentials for the heat equation in Lipschitz cylinders. Am. J. Math. 111(2), 339-379 (1989) · Zbl 0696.35065 · doi:10.2307/2374513 |
| [3] | Bergh, J., Lofström, J.: Interpolation Spaces, an Introduction. Springer, Berlin (1976) · Zbl 0344.46071 · doi:10.1007/978-3-642-66451-9 |
| [4] | Chang, T.: Extension and restriction theorems in time variable bounded domains. Commun. Contemp. Math. 12(2), 265-294 (2010) · Zbl 1201.46030 · doi:10.1142/S0219199710003774 |
| [5] | Fabes, E.; Riviere, N., Dirichlet and Neumann problems for the heat equation in C1-cylinders, harmonic analysis in Euclidean spaces. Part 2, No. XXXV, 179-196 (1979), Providence · Zbl 0436.35039 |
| [6] | Jakab, T., Mitrea, M.: Parabolic initial boundary value problems in nonsmooth cylinders with data in anisotropic Besov spaces. Math. Res. Lett. 13(5-6), 825-831 (2006) · Zbl 1115.35057 · doi:10.4310/MRL.2006.v13.n5.a12 |
| [7] | Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161-219 (1995) · Zbl 0832.35034 · doi:10.1006/jfan.1995.1067 |
| [8] | Jones, B.F. Jr.: Lipschitz spaces and the heat equation. J. Math. Mech. 18, 379-409 (1968) · Zbl 0179.42803 |
| [9] | Jonsson, A., Wallin, H.: A Whitney extension theorem in Lp and Besov spaces. Ann. Inst. Fourier 28, 139-192 (1978) · Zbl 0335.46024 · doi:10.5802/aif.684 |
| [10] | Kim, K.: On stochastic partial differential equations with variable coefficients in C1 domains. Stoch. Process. Appl. 112(2) 261-283 (2004) · Zbl 1074.60071 · doi:10.1016/j.spa.2004.02.006 |
| [11] | Kim, K., Krylov, N.V.: On SPDEs with variable coefficients in one space dimension. Potential Anal. 21(3), 203-239 (2004) · Zbl 1059.60075 · doi:10.1023/B:POTA.0000033334.06990.9d |
| [12] | Krylov, N. V., An analytic approach to SPDEs, No. 64, 185-242 (1999), Providence · Zbl 0933.60073 |
| [13] | Krylov, N.V.: A generalization of the Littlewood-Paley inequality and some other results related to stochastic partial differential equations. Ulam Q. 2(4), 16-26 (1994) · Zbl 0870.42005 |
| [14] | Krylov, N.V., Lototsky, S.V.: A Sobolev space theory of SPDEs with constant coefficients on a half line. SIAM J. Math. Anal. 30(2), 298-325 (1999) · Zbl 0928.60042 · doi:10.1137/S0036141097326908 |
| [15] | Krylov, N.V., Lototsky, S.V.: A Sobolev space theory of SPDEs with constant coefficients in a half space. SIAM J. Math. Anal. 31(1), 19-33 (1999) · Zbl 0943.60047 · doi:10.1137/S0036141098338843 |
| [16] | Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Am. Math. Soc., Providence (1967) · Zbl 0164.12302 |
| [17] | Lototsky, S.V.: Dirichlet problem for stochastic parabolic equations in smooth domains. Stoch. Stoch. Rep. 68(1-2), 145-175 (1999) · Zbl 0944.60076 |
| [18] | Peetre, J.: On the trace of potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 2(1), 33-43 (1975) · Zbl 0308.46031 |
| [19] | Stein, E.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton (1970) · Zbl 0207.13501 |
| [20] | Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983) · Zbl 1235.46002 · doi:10.1007/978-3-0346-0416-1 |
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