Boundary value problems for higher order parabolic equations
HTML articles powered by AMS MathViewer
- by Russell M. Brown and Wei Hu PDF
- Trans. Amer. Math. Soc. 353 (2001), 809-838 Request permission
Abstract:
We consider a constant coefficient parabolic equation of order $2m$ and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order $m-1$ lie in $L^2$ with respect to surface measure. In addition, a regularity result for the solution is obtained if the data has one more derivative. The boundary of the space domain is given by the graph of a Lipschitz function. This provides an extension of the methods of Pipher and Verchota on elliptic equations to parabolic equations.References
- Russell M. Brown, Layer potentials and boundary value problems for the heat equation on Lipschitz cylinders, Ph.D. thesis, University of Minnesota, 1987.
- Russell M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), no. 2, 339–379. MR 987761, DOI 10.2307/2374513
- 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. MR 1087194, DOI 10.1512/iumj.1990.39.39059
- Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
- B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135 (English, with French summary). MR 865663, DOI 10.5802/aif.1062
- Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465. MR 890159, DOI 10.2307/1971407
- Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
- E. B. Fabes and N. M. Rivière, Dirichlet and Neumann problems for the heat equation in $C^{1}$-cylinders, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 179–196. MR 545307
- 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
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Steve Hofmann and John L. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains, Ann. of Math. (2) 144 (1996), no. 2, 349–420. MR 1418902, DOI 10.2307/2118595
- Wei Hu, The initial-boundary value problem for higher order differential operators on Lipschitz cylinders, Ph.D. thesis, University of Kentucky, 1997.
- 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. MR 598688, DOI 10.1090/S0273-0979-1981-14884-9
- John L. Lewis and Judy Silver, Parabolic measure and the Dirichlet problem for the heat equation in two dimensions, Indiana Univ. Math. J. 37 (1988), no. 4, 801–839. MR 982831, DOI 10.1512/iumj.1988.37.37039
- John L. Lewis and Margaret A. M. Murray, The method of layer potentials for the heat equation in time-varying domains, Mem. Amer. Math. Soc. 114 (1995), no. 545, viii+157. MR 1323804, DOI 10.1090/memo/0545
- Changmei Liu, The Helmholtz equation on Lipschitz domains, IMA preprint #1356, 1995.
- Dorina Mitrea, Marius Mitrea, and Jill Pipher, Vector potential theory on nonsmooth domains in $\textbf {R}^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl. 3 (1997), no. 2, 131–192. MR 1438894, DOI 10.1007/s00041-001-4053-0
- Marius Mitrea, Initial boundary-value problems for the parabolic Maxwell system in Lipschitz cylinders, Indiana Univ. Math. J. 44 (1995), no. 3, 797–813. MR 1375350, DOI 10.1512/iumj.1995.44.2009
- Marius Mitrea, The method of layer potentials in electromagnetic scattering theory on nonsmooth domains, Duke Math. J. 77 (1995), no. 1, 111–133. MR 1317629, DOI 10.1215/S0012-7094-95-07705-9
- Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
- Jill Pipher and Gregory Verchota, The Dirichlet problem in $L^p$ for the biharmonic equation on Lipschitz domains, Amer. J. Math. 114 (1992), no. 5, 923–972. MR 1183527, DOI 10.2307/2374885
- Jill Pipher and Gregory C. Verchota, Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators, Ann. of Math. (2) 142 (1995), no. 1, 1–38. MR 1338674, DOI 10.2307/2118610
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- G. C. Verchota and A. L. Vogel, Nonsymmetric systems on nonsmooth planar domains, Trans. Amer. Math. Soc. 349 (1997), no. 11, 4501–4535. MR 1443894, DOI 10.1090/S0002-9947-97-02047-3
- Gregory Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J. 39 (1990), no. 3, 671–702. MR 1078734, DOI 10.1512/iumj.1990.39.39034
- Gregory C. Verchota, Potentials for the Dirichlet problem in Lipschitz domains, Potential theory—ICPT 94 (Kouty, 1994) de Gruyter, Berlin, 1996, pp. 167–187. MR 1404706
Additional Information
- Russell M. Brown
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- MR Author ID: 259097
- Email: rbrown@pop.uky.edu
- Wei Hu
- Affiliation: Department of Mathematics and Computer Science, Houghton College, Houghton, New York 14744
- Email: weih@houghton.edu
- Received by editor(s): June 2, 1998
- Published electronically: October 19, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 809-838
- MSC (2000): Primary 35K35
- DOI: https://doi.org/10.1090/S0002-9947-00-02702-1
- MathSciNet review: 1804519
Dedicated: This paper is dedicated to Gene Fabes