Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems

M. S. Agranovich

Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 2, pp. 1-22.

Résumé

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space $\mathbb{R}^n$. For such problems, equivalent equations on the boundary in the simplest $L_2$-spaces $H^s$ of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces $H^s_p$ of Bessel potentials and Besov spaces $B^s_p$. Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.

Détail

Keywords: strongly elliptic system, mixed problem, potential type operator, spectral problem, eigenvalue asymptotics.

Bibliographie

[1] M. S. Agranovich, “On a mixed Poincaré–Steklov type spectral problem in a Lipschitz domain”, Russian J. Math. Phys., 13:3 (2006), 239–244 | DOI | MR | Zbl

[2] M. S. Agranovich, “Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary”, Russian J. Math. Phys., 15:2 (2008), 146–155 | DOI | MR | Zbl

[3] M. C. Agranovich, “Operatory tipa potentsiala i zadachi sopryazheniya dlya silno ellipticheskikh sistem $2$-go poryadka v oblastyakh s lipshitsevoi granitsei”, Funkts. analiz i ego pril., 43:3 (2009), 3–25 | DOI | MR

[4] M. C. Agranovich, “Silno ellipticheskie sistemy 2-go poryadka s granichnymi usloviyami na nezamknutoi lipshitsevoi poverkhnosti”, Funkts. analiz i ego pril., 45:1 (2011), 1–15 | DOI | MR

[5] M. C. Agranovich, “Spektralnye zadachi v lipshitsevykh oblastyakh”, Uravneniya v chastnykh proizvodnykh, Sovremennaya matematika. Fundamentalnye napravleniya, 39, RUDN, M., 2011, 11–35

[6] M. C. Agranovich, B. A. Amosov, “Otsenki $s$-chisel i spektralnye asimptotiki dlya integralnykh operatorov tipa potentsiala na negladkikh poverkhnostyakh”, Funkts. analiz i ego pril., 30:2 (1996), 1–18 | DOI | MR | Zbl

[7] M. Sh. Birman, M. Z. Solomyak, “Asimptotika spektra variatsionnykh zadach na resheniyakh ellipticheskikh uravnenii”, Cib. matem. zhurn., 20:1 (1979), 3–22 | MR | Zbl

[8] M. Sh. Birman, M. Z. Solomyak, “Kolichestvennyi analiz v teoremakh vlozheniya Soboleva i prilozheniya k spektralnoi teorii”, «Desyataya matematicheskaya shkola», Inst. matem. AN USSR, Kiev, 1974, 5–189 | MR

[9] M. Sh. Birman, M. C. Solomyak, “Asimptotika spektra psevdodifferentsialnykh operatorov s anizotropno-odnorodnymi simvolami”, Vestnik LGU, No13, matem. mekhan. astron., vyp. 3 (1977), 13–21 | MR | Zbl

[10] R. M. Brown, “The mixed problem for Laplace's equation in a class of Lipschitz domains”, Comm. Partial Differential Equations, 19:7–8 (1994), 1217–1233 | DOI | MR | Zbl

[11] R. M. Brown, I. Mitrea, “The mixed problem for the Lamé system in a class of Lipschitz domains”, J. Differential Equations, 246:7 (2009), 2577–2589 | DOI | MR | Zbl

[12] M. Costabel, “Boundary integral operators in Lipschitz domains: elementary results”, SIAM J. Math. Anal., 19:3 (1988), 613–626 | DOI | MR | Zbl

[13] R. Kurant, D. Gilbert, Metody matematicheskoi fiziki, v. I, GTTI, M., 1934

[14] B. E. J. Dahlberg, C. E. Kenig, G. C. Verchota, “Boundary value problems for the system of elastostatics in Lipschitz domains”, Duke Math. J., 57:3 (1988), 795–818 | DOI | MR | Zbl

[15] G. I. Eskin, Kraevye zadachi dlya ellipticheskikh psevdodifferentsialnykh uravnenii, Nauka, M., 1973 | MR

[16] V. I. Fabrikant, Mixed Boundary Value Problems of Potential Theory and their Applications in Engineering, Kluwer, Dorderecht, 1991 | MR | Zbl

[17] J. A. Griepentrog, K. Gröger, H.-Chr. Kaiser, J. Rehrberg, “Interpolation for function spaces related to mixed boundary value problems”, Math. Nachr., 241 (2002), 110–120 | 3.0.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[18] J. A. Griepentrog, H.-Chr. Kaiser, J. Rehrberg, “Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on $L_p$”, Adv. Math. Sci. Appl., 11 (2001), 87–112 | MR | Zbl

[19] K. Gröger, “A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations”, Math. Ann., 283:4 (1989), 679–687 | DOI | MR

[20] G. C. Hsiao, W. L. Wendland, Boundary Integral Equations, Springer-Verlag, Berlin, 2008 | MR

[21] D. Jerison, C. E. Kenig, “The inhomogeneous Dirichlet problem in Lipschitz domains”, J. Funct. Anal., 130:1 (1995), 161–219 | DOI | MR | Zbl

[22] A. Jonsson, H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$, Harwood Academic Publishers, 1984 | MR

[23] V. I. Lebedev, V. I. Agoshkov, Operatory Puankare–Steklova i ikh prilozheniya v analize, Otdel vychislitelnoi matematiki AN SSSR, M., 1983 | MR

[24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl

[25] G. Métivier, “Valeurs propres de problémes aux limites elliptiques irreguliers”, Bull. Soc. Math. France, Mémoire, 51–52 (1977), 125–219 | DOI | MR | Zbl

[26] S. E. Mikhailov, Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients, arXiv: 0906.3875v1

[27] I. Mitrea, M. Mitrea, “The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains”, Trans. Amer. Math. Soc., 359:9 (2007), 4143–4182 | DOI | MR | Zbl

[28] M. Mitrea, M. Taylor, “Boundary layer methods for Lipschitz domains in Riemannian manifolds”, J. Funct. Anal., 163:2 (1999), 181–251 | DOI | MR | Zbl

[29] M. Mitrea, M. Taylor, “Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev–Besov space results and the Poisson problem”, J. Funct. Anal., 176:1 (2000), 1–79 | DOI | MR | Zbl

[30] D. G. Natroshvili, O. O. Chkadua, E. M. Shargorodskii, “Smeshannye zadachi dlya odnorodnykh anizotropnykh uprugikh sred”, Trudy Instituta prikladnoi matematiki im. Vekua, 39 (1990), 133–178 | MR

[31] J. Nec̆as, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967 | MR | Zbl

[32] O. A. Oleinik, A. S. Shamaev, G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North Holland, Amsterdam, 1992 | MR

[33] K. A. Ott, R. M. Brown, The mixed problem for the Laplacian in Lipschitz domains, arXiv: 0909.0061v2

[34] B. V. Paltsev, “O smeshannoi zadache s neodnorodnymi granichnymi usloviyami dlya ellipticheskikh s parametrom uravnenii vtorogo poryadka v lipshitsevykh oblastyakh”, Matem. sb., 187:4 (1996), 59–116 | DOI | MR

[35] T. von Petersdorff, “Boundary integral equations for mixed Dirichlet, Neumann and transmission problems”, Math. Methods Appl. Sci., 11:2 (1989), 183–213 | MR

[36] G. Rozenblum, G. Tashchiyan, “Eigenvalue asymptotics for potential type operators on Lipschitz surfaces”, Russian J. Math. Phys., 13:3 (2006), 326–339 | DOI | MR | Zbl

[37] V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains”, J. London Math. Soc. (2), 60:1 (1999), 237–257 | DOI | MR | Zbl

[38] G. Savaré, “Regularity and perturbation results for mixed second order elliptic problems”, Comm. Partial Differential Equations, 22:5-6 (1997), 869–899 | DOI | MR | Zbl

[39] E. Shamir, “Regularization of mixed second-order elliptic equations”, Israel J. Math., 6 (1968), 150–168 | DOI | MR | Zbl

[40] I. Ya. Shneiberg, “Spektralnye svoistva lineinykh operatorov v interpolyatsionnykh semeistvakh banakhovykh prostranstv”, Matem. issled., 9:2 (1974), 214–227 | MR

[41] I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory, Elsevier, New York, 1966 | MR

[42] E. Stephan, Boundary integral equations for mixed boundary value problems, screen and transmission problems in $\mathbb{R}^3$, Habilitationsschrift, Darmstadt, THD-preprint 848, 1984 | MR

[43] E. P. Stephan, “Boundary integral equations for mixed boundary value problems in $\mathbb{R}^3$”, Math. Nachr., 134 (1987), 21–53 | DOI | MR | Zbl

[44] T. A. Suslina, “Asimptotika spektra variatsionnykh zadach na resheniyakh odnorodnogo ellipticheskogo uravneniya pri nalichii svyazei na chasti granitsy”, Problemy mat. analiza, 9, LGU, 1984, 84–97 | MR

[45] T. A. Suslina, “Spectral asymptotics of variational problems with elliptic constraints in domains with piecewise smooth boundary”, Russian J. Math. Phys., 6:2 (1999), 214–234 | MR | Zbl

[46] J. D. Sykes, R. M. Brown, “The mixed boundary problem in $L^p$ an Hardy spaces for Laplace's equation on a Lipschitz domain”, Contemporary Mathematics, 227, Amer. Math. Soc., 2001, 1–18 | MR

[47] H. Triebel, “Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers”, Rev. Mat. Complut., 15:2 (2002), 475–524 | DOI | MR | Zbl

[48] G. Uhlmann, “Inverse boundary problems and applications”, Astérisque, 207, 1992, 153–207 | MR | Zbl

[49] G. Verchota, “Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains”, J. Funct. Anal., 59:3 (1984), 572–611 | DOI | MR | Zbl

[50] V. I. Voititskii, N. D. Kopachevskii, P. A. Starkov, “Mnogokomponentnye zadachi sopryazheniya i vspomogatelnye abstraktnye kraevye zadachi”, Sovremennaya matematika. Fundamentalnye napravleniya, 34 (2009), 5–44 | MR