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Smoothness of solutions of parabolic equations in regions with edges

Published online by Cambridge University Press:  22 January 2016

A. Azzam  and
E. Kreyszig
A. Azzam
Affiliation:
Department of Mathematics, University of Windsor, Windsor, Ontario, Canada
E. Kreyszig
Affiliation:
Department of Mathematics, University of Windsor, Windsor, Ontario, Canada
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We consider the mixed initial-boundary value problem for the parabolic equation

in a region Ω × (0, T], where x = (x1, x2) and Ω ⊂ R2 is a simply-connected bounded domain having corners.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 84 , December 1981 , pp. 159 - 168
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[ 1 ] Agmon, S. Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, L, Comm. Pure Appl. Math., 12 (1959), 623–727.Google Scholar
[ 2 ] Azzam, A., On Dirichlet’s problem for elliptic equations in sectionally smooth n-dimensional domains, SIAM J. Math. Anal, 11 (1980), 248–253.CrossRefGoogle Scholar
[ 3 ] Azzam, A. and Kreyszig, E., On parabolic equations in n space variables and their solutions in regions with edges, in press.Google Scholar
[ 4 ] Carslaw, H.S. and Jaeger, J.C., Conduction of heat in solids, Clarendon Press, Oxford (1959).Google Scholar
[ 5 ] Friedman, A., Boundary estimates for second order parabolic equations and their applications, J. Math. Mech., 7 (1958), 771–791.Google Scholar
[ 6 ] Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, Springer, Berlin etc., 1977.Google Scholar
[ 7 ] Grisvard, P., Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain; in: Numerical solutions of partial differential equations III (ed. B. Hubbard), Academic Press, New York (1976).Google Scholar
[ 8 ] Itô, Z., A boundary value problem of partial differential equations of parabolic type, Duke Math. J., 24 (1957), 299–312.Google Scholar
[ 9 ] Kamynin, L.I. and Maslennikova, V.N., Boundary estimates for the solution of the third boundary-value problem for a parabolic equation, Dokl. Akad. Nauk SSSR, 153 (1963), 526–529.Google Scholar
[10] Kondrat’ev, V.A., Singularities of a solution of Dirichlet’s problem for a second order elliptic equation in the neighborhood of an edge, Diff. Uravnen., 13 (1977), 2026–2033.Google Scholar
[11] Laasonen, P., On the degree of convergence of discrete approximations for the solutions of the Dirichlet problem, Ann. Acad. Fenn. Ser. A. I. No. 246 (1957), 19 pp.Google Scholar
[12] Ladyzenskaja, O. A., Solonnikov, V. A. and Ural’ceva, N., Linear and quasi-linear equations of parabolic type, American Mathematical Society, Providence, R. I. 1968.Google Scholar
[13] Protter, M.H. and Weinberger, H.F., Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N. J. (1967).Google Scholar
[14] Volkov, E.A., Differentiability properties of solutions of boundary value problems for the Laplace equation on a polygon, Proc. Steklov Inst. Math., 77 (1965), 127–159.Google Scholar
[15] Zitarasu, N.V., Schauder estimates and solvability of general boundary problems for general parabolic systems with discontinuous coefficients, Soviet Math. Dokl., 7 (1966), 952–956.Google Scholar