Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T21:57:17.105Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATES OF GREEN’S FUNCTION FOR SECOND-ORDER PARABOLIC EQUATIONS NEAR EDGES

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  17 April 2015

V. A. Kozlov  and
J. Rossmann
V. A. Kozlov
Affiliation:
University of Linköping, Institute of Mathematics, S-58183 Linköping, Sweden email vladimir.kozlov@liu.se
J. Rossmann
Affiliation:
University of Rostock, Institute of Mathematics, D-18051 Rostock, Germany email juergen.rossmann@uni-rostock.de
Get access

Abstract

We consider the first boundary value problem for a second-order parabolic equation with variable coefficients in the domain $K\times \mathbb{R}^{n-m}$, where $K$ is an $m$-dimensional cone. The main results of the paper are pointwise estimates of the Green’s function.

MSC classification

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 345 - 369
Copyright
Copyright © University College London 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aronson, D. G., Non-negative solutions of linear parabolic equations. Ann. Sc. Norm. Super. Pisa 22(3) 1968, 607–694.Google Scholar
Besov, O. V., Il’in, V. P. and Nikol’skiĭ, S. M., Integral Representations of Functions and Imbedding Theorems, 2nd edn., Nauka (Moscow, 1996) (in Russian). Engl. transl. of the 1st edn.: Integral Representations of Functions and Imbedding Theorems (Scripta Series in Mathematics 1) (ed. M. H. Taibleson), V. H. Winston/Halsted Press (1978).Google Scholar
Hadamard, J., Mémoire sur le problème d’analyse relatif à l’équilibre des plasques élastiques encastrées. In Œuvres de Jacques Hadamard, Vol. 2, Éditions du Centre National de la Recherche Scientific (Paris, 1968), 515–631.Google Scholar
Kozlov, V. A., Coefficients in the asymptotics of solutions of Cauchy boundary value parabolic problems in domains with a conical point. Sib. Math. J. 29(2) 1988, 75–89.CrossRefGoogle Scholar
Kozlov, V. A., On the asymptotics of Green’s function and Poisson’s kernels for a parabolic problem in a cone I. Z. Anal. Anwend. 8(2) 1989, 131–151 (in Russian).CrossRefGoogle Scholar
Kozlov, V. A., On the asymptotics of Green’s function and Poisson’s kernels for a parabolic problem in a cone II. Z. Anal. Anwend. 10(1) 1991, 27–42 (in Russian).CrossRefGoogle Scholar
Kozlov, V. A. and Maz’ya, V. G., On singularities of solutions of the first boundary value problem for the heat equation in domains with conical points II. Izv. Vyssh. Uchebn. Zaved. Mat. 298(3) 1987, 37–44 (in Russian).Google Scholar
Kozlov, V. A., Maz’ya, V. G. and Rossmann, J., Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations (Mathematical Surveys and Monographs 85), American Mathematical Society (Providence, RI, 2001).Google Scholar
Kozlov, V. A. and Nazarov, A. I., The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients. Math. Nachr. 282(9) 2009, 1220–1241.CrossRefGoogle Scholar
Kozlov, V. A. and Rossmann, J., Asymptotics of solutions of the heat equation in cones and dihedra. Math. Nachr. 285(11 –12) 2012, 1422–1449.CrossRefGoogle Scholar
Kozlov, V. A. and Rossmann, J., Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary. Bound. Value Probl. 142 2012; doi:10.1186/1687-2770-2012-142.Google Scholar
Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural’tseva, N. N., Linear and Quasi-linear Equations of Parabolic Type, Nauka (Moscow, 1967) (in Russian). Engl. transl.: Linear and Quasi-linear Equations of Parabolic Type (Translations of Mathematical Monographs 23), American Mathematical Society (Providence, RI, 1968).Google Scholar
Nazarov, A. I., L p-estimates for a solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension. Probl. Mat. Anal. 22 2001, 126–159 (in Russian). Engl. transl.: J. Math. Sci. 106(3) (2001), 2989–3014.Google Scholar
Solonnikov, V. A., Solvability of classical initial-boundary value problems for the heat equation in a two-sided corner. Zap. Nauchn. Sem. LOMI 138 1984, 146–180. Engl. transl.: J. Sov. Math. 32(5) (1986), 526–546.Google Scholar
Solonnikov, V. A., L p-estimates for solutions of the heat equation in a dihedral angle. Rend. Mat. Appl. (7) 21(1 –4) 2001, 1–15.Google Scholar