The solvability of the first initial-boundary problem for parabolic and degenerate parabolic equations in domains with a conical point

S. P. Degtyarev

Sbornik. Mathematics, Tome 201 (2010) no. 7, pp. 999-1028

Résumé

The first initial-boundary problem for second-order parabolic and degenerate parabolic equations is investigated in a domain with a conical or angular point. The means of attack is already known and uses weighted classes of smooth or integrable functions. Sufficient conditions for a unique solution to exist and for coercive estimates for the solution to be obtained are formulated in terms of the angular measure of the solid angle and the exponent of the weight. It is also shown that if these conditions fail to hold, then the parabolic problem has elliptic properties, that is, it can have a nonzero kernel or can be nonsolvable, and, in the latter case, it is not even a Fredholm problem. A parabolic equation and an equation with some degeneracy or a singularity at a conical point are considered. Bibliography: 49 titles.

Détail

Keywords: irregular domain, coercive estimate, spectral properties.
Mots-clés : parabolic equation

Bibliographie

[1] V. A. Kondrat'ev, “Boundary problems for elliptic equations in domains with conical or angular points”, Trans. Mosc. Math. Soc., 16 (1967), 227–313 | MR | Zbl | Zbl

[2] V. Maz'ya, S. Nazarov, B. Plamenevskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. I, Oper. Theory Adv. Appl., 111, Basel, Birkhäuser, 2000 | MR | Zbl

[3] V. Maz'ya, S. Nazarov, B. Plamenevskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. II, Oper. Theory Adv. Appl., 112, Basel, Birkhäuser, 2000 | MR | Zbl

[4] V. G. Maz'ya, R. Mahnke, “Asymptotics of the solution of a boundary integral equation under a small perturbation of a corner”, Z. Anal. Anwendungen, 11:2 (1992), 173–182 | MR | Zbl

[5] V. G. Mazya, B. A. Plamenevskii, “Otsenki v $L_{p}$ i v klassakh Gëldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 81:1 (1978), 25–82 | DOI | MR | Zbl

[6] V. G. Maz'ya, B. A. Plamenevskii, “Estimates of Green's functions and Schauder estimates for solutions of elliptic boundary problems in a dihedral angle”, Siberian Math. J., 19:5 (1978), 752–764 | DOI | MR | Zbl | Zbl

[7] V. Zaionchkovskii, V. A. Solonnikov, “Neumann problem for second-order elliptic equations in domains with edges on the boundary”, J. Soviet Math., 27:2 (1984), 2561–2586 | DOI | MR | Zbl | Zbl

[8] S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Exp. Math., 13, de Gruyter, Berlin, 1994 | MR | Zbl

[9] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities, Math. Surveys Monogr., 52, Amer. Math. Soc., Providence, RI, 1997 | MR | Zbl

[10] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Math. Surveys Monogr., 85, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[11] V. A. Solonnikov, E. V. Frolova, “Investigation of a problem for the Laplace equation with a boundary condition of a special form in a plane angle”, J. Soviet Math., 62:3 (1991), 2819–2831 | MR | Zbl | Zbl

[12] P. Grisvard, Elliptic problems in nonsmooth domains, Monogr. Stud. Math., 24, Pitman, Boston, MA, 1985 | MR | Zbl

[13] M. Bochniak, M. Borsuk, “Dirichlet problem for linear elliptic equations degenerating at a conical boundary point”, Analysis (Munich), 23:3 (2003), 225–248 | MR | Zbl

[14] M. Borsuk, V. Kondratiev, Elliptic boundary value problems of second order in piecewise smooth domains, North-Holland Math. Library, 69, Elsevier, Amsterdam, 2006 | MR | Zbl

[15] M. Borsuk, “Degenerate elliptic boundary-value problems of second order in nonsmooth domains”, J. Math. Sci. (N. Y.), 146:5 (2007), 6071–6212 | DOI | MR | Zbl

[16] V. A. Solonnikov, “Solvability of the classical initial-boundary-value problems for the heat-conduction equation in a dihedral angle”, J. Soviet Math., 32:5 (1986), 526–546 | DOI | MR | Zbl | Zbl

[17] V. A. Kozlov, V. G. Maz'ya, “Characteristics of solutions of first boundary value problem for thermal conductivity equation in domains with conic points. I”, Soviet Math. (Iz. VUZ), 31:2 (1987), 61–74 | MR | Zbl

[18] V. A. Kozlov, V. G. Maz'ya, “Characteristics of solutions of first boundary value problem for thermal conductivity equation in domains with conic points. II”, Soviet Math. (Iz. VUZ), 31:3 (1987), 49–57 | MR | Zbl

[19] V. A. Kozlov, “Coefficients in the asymptotic solutions of the Cauchy boundary-value parabolic problems in domains with a conical point”, Siberian Math. J., 29:2 (1988), 222–233 | DOI | MR | Zbl | Zbl

[20] V. A. Kozlov, “Asymptotics as $t\to0$ of solutions of the heat equation in a domain with a conical point”, Math. USSR-Sb., 64:2 (1989), 383–395 | DOI | MR | Zbl

[21] V. A. Solonnikov, E. V. Frolova, “On a problem with the third boundary condition for the Laplace equation in a plane angle, and its applications to parabolic problems”, Leningrad Math. J., 2:4 (1991), 891–916 | MR | Zbl | Zbl

[22] M. G. Garroni, V. A. Solonnikov, M. A. Vivaldi, “On the oblique derivative problem in an infinite angle”, Topol. Methods Nonlinear Anal., 7:2 (1996), 299–325 | MR | Zbl

[23] V. A. Solonnikov, E. V. Frolova, “On a certain nonstationary problem in a dihedral angle. II”, J. Math. Sci., 70:3 (1994), 1841–1846 | DOI | MR | Zbl

[24] M. G. Garroni, V. A. Solonnikov, M. A. Vivaldi, “Existence and regularity results for oblique derivative problems for heat equations in an angle”, Proc. Roy. Soc. Edinburgh Sect. A, 128:1 (1998), 47–79 | MR | Zbl

[25] V. A. Solonnikov, “$L_p$-estimates for solutions of the heat equation in a dihedral angle”, Rend. Mat. Appl. (7), 21:1–4 (2001), 1–15 | MR | Zbl

[26] A. I. Nazarov, “$L_{p}$-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”, J. Math. Sci. (New York), 106:3 (2001), 2989–3014 | DOI | MR | Zbl

[27] A. I. Nazarov, “Dirichlet problem for quasilinear parabolic equations in domains with smooth closed edges”, Amer. Math. Soc. Transl. Ser. 2, 209:3 (2003), 115–141 | MR | Zbl

[28] A. Azzam, E. Kreyszig, “On solutions of parabolic equations in regions with edges”, Bull. Austral. Math. Soc., 22:2 (1980), 219–230 | DOI | MR | Zbl

[29] A. Azzam, E. Kreyszig, “On parabolic equations in $n$ space variables and their solutions in regions with edges”, Hokkaido Math. J., 9:2 (1980), 140–154 | MR | Zbl

[30] A. Azzam, E. Kreyszig, “Smoothness of solutions of parabolic equations in regions with edges”, Nagoya Math. J., 84 (1981), 159–168 | MR | Zbl

[31] A. Azzam, “On mixed boundary value problems for parabolic equations in singular domains”, Osaka J. Math., 22:4 (1985), 691–696 | MR | Zbl

[32] A. Azzam, “On the behavior of solutions of elliptic and parabolic equations at a crack”, Hokkaido Math. J., 19:2 (1990), 339–344 | MR | Zbl

[33] V. N. Aref'ev, L. A. Bagirov, “Solutions of the heat equation in domains with singularities”, Math. Notes, 64:2 (1998), 139–153 | DOI | MR | Zbl

[34] D. Guidetti, “The mixed Cauchy–Dirichlet problem for the heat equation in a plane angle in spaces of Hölder-continuous functions”, Adv. Differential Equations, 6:8 (2001), 897–930 | MR | Zbl

[35] B. V. Bazalii, S. P. Degtyarev, “On one boundary-value problem for a strongly degenerate second-order elliptic equation in an angular domain”, Ukrainian Math. J, 59:7 (2007), 955–975 | DOI | MR | Zbl

[36] A. Venni, “Maximal regularity for a singular parabolic problem”, Rend. Sem. Mat. Univ. Politec. Torino, 52:1 (1994), 87–101 | MR | Zbl

[37] Can Zui Ho, G. I. Eskin, “Boundary value problems for parabolic systems of pseudodifferential equations”, Soviet Math. Dokl., 12 (1971), 739–743 | MR | Zbl

[38] S. Shmarev, Differentiability and analyticity of solutions and interfaces in multidimensional reaction-diffusion equation, Universidad de Oviedo, Departamento de Matematicas, 2001, 51 pp.

[39] S. I. Shmarev, “Interfaces in multidimensional diffusion equations with absorption terms”, Nonlinear Anal., 53:6 (2003), 791–828 | DOI | MR | Zbl

[40] I. D. Pukal's'kyi, “Cauchy problem for nonuniformly parabolic equations with degeneracy”, Ukrainian Math. J, 55:11 (2003), 1828–1840 | DOI | MR | Zbl

[41] S. P. Degtyarev, “O razreshimosti pervoi nachalno-kraevoi zadachi dlya vyrozhdayuschikhsya parabolicheskikh uravnenii v oblastyakh s neregulyarnoi granitsei”, Ukr. matem. vest., 5:1 (2008), 59–82 | MR

[42] O. A. Ladyż̌enskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968 | MR | MR | Zbl | Zbl

[43] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[44] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Math. Sci. Engrg., 46, Academic Press, New York–London, 1968 | MR | MR | Zbl | Zbl

[45] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York–London–Toronto, ON, 1980 | MR | MR | Zbl | Zbl

[46] P. Daskalopoulos, R. Hamilton, “Regularity of the free boundary for the porous medium equation”, J. Amer. Math. Soc., 11:4 (1998), 899–965 | DOI | MR | Zbl

[47] B. V. Bazaliy, N. V. Krasnoshchok, “Regularity of a solution to the multi-dimensional free boundary problem for the porous medium equation”, Siberian Adv. Math., 13:3 (2003), 1–53 | MR | MR | Zbl | Zbl

[48] O. A. Oleinik, E. V. Radkevich, Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi, Itogi nauki. Ser. Matematika. Mat. anal. 1969, VINITI, M., 1971 | MR | Zbl

[49] S. M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems, Grundlehren Math. Wiss., 205, Springer-Verlag, New York–Heidelberg, 1975 | MR | MR | Zbl | Zbl