A note on boundary value problems for the heat equation in Lipschitz cylinders. (English) Zbl 0806.35055
Summary: We study the initial Dirichlet problem and the initial Neumann problem for the heat equation in Lipschitz cylinders, with boundary data in mixed norm spaces \(L^ q(0,T,L^ p (\partial \Omega))\).
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K05 | Heat equation |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
Keywords:
time-dependent boundary conditionsReferences:
| [1] | Russell M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), no. 2, 339 – 379. · Zbl 0696.35065 · doi:10.2307/2374513 |
| [2] | Russell M. Brown, The initial-Neumann problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 320 (1990), no. 1, 1 – 52. · Zbl 0714.35037 |
| [3] | Russell M. Brown and Zhong Wei Shen, The initial-Dirichlet problem for a fourth-order parabolic equation in Lipschitz cylinders, Indiana Univ. Math. J. 39 (1990), no. 4, 1313 – 1353. · Zbl 0702.35114 · doi:10.1512/iumj.1990.39.39059 |
| [4] | Russell M. Brown and Zhong Wei Shen, Boundary value problems in Lipschitz cylinders for three-dimensional parabolic systems, Rev. Mat. Iberoamericana 8 (1992), no. 3, 271 – 303. · Zbl 0782.35033 |
| [5] | R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur \?² pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361 – 387 (French). · Zbl 0497.42012 · doi:10.2307/2007065 |
| [6] | Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in \?^{\?} for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437 – 465. · Zbl 0658.35027 · doi:10.2307/1971407 |
| [7] | E. B. Fabes and N. M. Rivière, Dirichlet and Neumann problems for the heat equation in \( {C^1}\)-cylinders, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, RI, 1979, pp. 179-196. |
| [8] | Eugene B. Fabes, Nicola Garofalo, and Sandro Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), no. 4, 536 – 565. · Zbl 0625.35006 |
| [9] | 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. · Zbl 0542.35042 |
| [10] | C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107 – 115. · Zbl 0222.26019 · doi:10.2307/2373450 |
| [11] | David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203 – 207. · Zbl 0471.35026 |
| [12] | Carlos E. Kenig, Elliptic boundary value problems on Lipschitz domains, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 131 – 183. |
| [13] | Alexander Nagel and E. M. Stein, Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems, Mathematical Notes, vol. 24, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. · Zbl 0415.47025 |
| [14] | Zhong Wei Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, Amer. J. Math. 113 (1991), no. 2, 293 – 373. · Zbl 0734.35080 · doi:10.2307/2374910 |
| [15] | Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501 |
| [16] | Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572 – 611. · Zbl 0589.31005 · doi:10.1016/0022-1236(84)90066-1 |
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