Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains

M. S. Agranovich

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 5, pp. 847-920

Résumé

Spectral boundary-value problems with discrete spectrum are considered for second-order strongly elliptic systems of partial differential equations in a domain $\Omega\subset\mathbb R^n$ whose boundary $\Gamma$ is compact and may be $C^\infty$, $C^{1,1}$, or Lipschitz. The principal part of the system is assumed to be Hermitian and to satisfy an additional condition ensuring that the Neumann problem is coercive. The spectral parameter occurs either in the system (then $\Omega$ is assumed to be bounded) or in a first-order boundary condition. Also considered are transmission problems in $\mathbb R^n\setminus\Gamma$ with spectral parameter in the transmission condition on $\Gamma$. The corresponding operators in $L_2(\Omega)$ or $L_2(\Gamma)$ are self-adjoint operators or weak perturbations of self-adjoint ones. Under some additional conditions a discussion is given of the smoothness, completeness, and basis properties of eigenfunctions or root functions in the Sobolev $L_2$-spaces $H^t(\Omega)$ or $H^t(\Gamma)$ of non-zero order $t$ as well as of localization and the asymptotic behaviour of the eigenvalues. The case of Coulomb singularities in the zero-order term of the system is also covered.

Bibliographie

[1] M. S. Agranovich, “Spektralnye svoistva zadach difraktsii”, Dobavlenie k knige [126], 288–416

[2] M. S. Agranovich, “O skhodimosti ryadov po kornevym vektoram operatorov, ochen blizkikh k samosopryazhennym”, Trudy MMO, 41 (1980), 163–180 | MR | Zbl

[3] M. S. Agranovich, “Ob ellipticheskikh psevdodifferentsialnykh operatorakh na zamknutoi krivoi”, Trudy MMO, 47 (1984), 22–67 | MR | Zbl

[4] M. S. Agranovich, “Ellipticheskie operatory na zamknutykh mnogoobraziyakh”, Dif. uravneniya s chastnymi proizvodnymi VI, Itogi nauki i tekhniki. Sovr. problemy matem. Fund. napravl., 63, VINITI, M., 1990, 5–129 | MR

[5] M. S. Agranovich, “O nesamosopryazhennykh zadachakh s parametrom, ellipticheskikh po Agmonu–Duglisu–Nirenbergu”, Funkts. analiz i ego pril., 24:1 (1990), 59–61 | MR | Zbl

[6] M. S. Agranovich, “O modulyakh sobstvennykh znachenii nesamosopryazhennykh zadach s parametrom, ellipticheskikh po Agmonu–Duglisu–Nirenbergu”, Funkts. analiz i ego pril., 26:2 (1992), 51–55 | MR | Zbl

[7] M. S. Agranovich, “O ryadakh po kornevym vektoram operatorov, opredelyaemykh formami s samosopryazhennoi glavnoi chastyu”, Funkts. analiz i ego pril., 28:3 (1994), 1–21 | MR | Zbl

[8] M. S. Agranovich, “Elliptic boundary problems”, Partial Differential Equations. IX: Elliptic Boundary Value Problems, Encyclopaedia Math. Sci., 79, Springer-Verlag, Berlin, 1997, 1–144 | MR | Zbl

[9] M. S. Agranovich, “Spektralnye svoistva operatorov tipa potentsiala dlya nekotorogo klassa silno ellipticheskikh sistem na gladkikh i lipshitsevykh poverkhnostyakh”, Trudy MMO, 62 (2001), 3–53 | Zbl

[10] M. S. Agranovich, “Spektralnye zadachi dlya sistemy Diraka so spektralnym parametrom v lokalnykh granichnykh usloviyakh”, Funkts. analiz i ego pril., 35:3 (2001), 1–18 | MR | Zbl

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

[12] M. S. Agranovich, B. A. Amosov, M. Levitin, “Spectral problems for the Lamé system with spectral parameter in boundary conditions on smooth or nonsmooth boundary”, Russian J. Math. Phys., 6:3 (1999), 247–281 | MR | Zbl

[13] M. S. Agranovich, Z. N. Golubeva, “O nekotorykh zadachakh dlya sistemy Maksvella so spektralnym parametrom v granichnom uslovii”, Dokl. AN SSSR, 231:4 (1976), 777–780 | MR | Zbl

[14] M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH, Berlin, 1999 | MR | Zbl

[15] M. S. Agranovich, R. Menniken, “Spektralnye zadachi dlya uravneniya Gelmgoltsa so spektralnym parametrom v granichnykh usloviyakh na negladkoi poverkhnosti”, Matem. sb., 190:1 (1999), 29–68 | MR | Zbl

[16] M. S. Agranovich, M. I. Vishik, “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, 19:3 (1964), 53–161 | MR | Zbl

[17] I. Berg, I. Lëfstrëm, Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[18] O. V. Besov, V. P. Ilin, S. M. Nikolskii, Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[19] M. Sh. Birman, M. Z. Solomyak, “Spektralnaya asimptotika negladkikh ellipticheskikh operatorov. I, II”, Trudy MMO, 27 (1972), 3–52 ; 28 (1973), 3–24 | MR | Zbl | MR

[20] M. Sh. Birman, M. Z. Solomyak, “Otsenki singulyarnykh chisel integralnykh operatorov”, UMN, 32:1 (1977), 17–84 | MR | Zbl

[21] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya operatorov v gilbertovom prostranstve, Izd-vo LGU, L., 1980

[22] M. Bramanti, “Potential theory for stationary Schrödinger operators: a survey of results obtained with non-probabilistic methods”, Matematiche (Catania), 47:1 (1992), 25–61 | MR | Zbl

[23] P. G. Burke, “$R$-matrix theory: some recent applications”, Many-Body Atomic Physics, eds. J. J. Boyle, M. S. Pindzola, Cambridge Univ. Press, Cambridge, 1996, 305–348

[24] A. P. Calderón, “Cauchy integrals on Lipschitz curves and related operators”, Proc. Natl. Acad. Sci. USA, 74 (1977), 1324–1327 | DOI | MR | Zbl

[25] A. P. Calderón, “Boundary value problems for the Laplace equation in Lipschitzian domains”, Recent Progress in Fourier Analysis, Proc. semin. (El Escorial, 1983), North-Holland Math. Stud., 111, North-Holland, Amsterdam, 1985, 33–48 | MR | Zbl

[26] R. R. Coifman, A. McIntosh, Y. Meyer, “L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes”, Ann. of Math. (2), 116 (1982), 361–387 | DOI | MR | Zbl

[27] D. Kolton, R. Kress, Metody integralnykh uravnenii v teorii rasseyaniya, Mir, M., 1987 | MR

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

[29] M. Costabel, W. L. Wendland, “Strong ellipticity of boundary integral operators”, J. Reine Angew. Math., 372 (1986), 34–63 | MR | Zbl

[30] R. Kurant, D. Gilbert, Metody matematicheskoi fiziki, 1, Gostekhteorizdat, M.–L., 1933

[31] B. Dahlberg, G. Verchota, “Galerkin methods for the boundary integral equations of elliptic equations in nonsmooth domains”, Contemp. Math., 107 (1990), 39–60 | MR | Zbl

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

[33] B. Dahlberg, C. Kenig, J. Pipher, G. Verchota, “Area integral estimates for higher order elliptic equations and systems”, Ann. Inst. Fourier (Grenoble), 47:5 (1997), 1425–1461 | MR | Zbl

[34] G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math., 1465, Springer-Verlag, Berlin, 1991 | MR | Zbl

[35] Z. Ding, “A proof of the trace theorem of Sobolev spaces on Lipschitz domains”, Proc. Amer. Math. Soc., 124:2 (1996), 591–600 | DOI | MR | Zbl

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

[37] E. Fabes, “Layer potential methods for boundary value problems on Lipschitz domains”, Potential Theory: Surveys and Problems, Lecture Notes in Math., 1344, Springer-Verlag, Berlin, 1988, 55–80 | MR

[38] E. Fabes, M. Jodeit, N. Rivière, “Potential techniques for boundary value problems on $C^1$ domains”, Acta Math., 141 (1978), 165–186 | DOI | MR | Zbl

[39] E. Fabes, O. Mendez, M. Mitrea, “Boundary layers on Sobolev–Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains”, J. Funct. Anal., 159:2 (1998), 323–368 | DOI | MR | Zbl

[40] W. Gao, “Layer potentials and boundary value problems for elliptic systems in Lipschitz domains”, J. Funct. Anal., 95:2 (1991), 377–399 | DOI | MR | Zbl

[41] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965

[42] Z. N. Golubeva, “Nekotorye skalyarnye zadachi difraktsii i svyazannye s nimi nesamosopryazhennye operatory”, Radiotekhnika i elektronika, 21:2 (1976), 219–227 | MR

[43] P. Grisvard, “Caractérisation de quelques espaces d'interpolation”, Arch. Ration. Mech. Anal., 25 (1967), 40–63 | DOI | MR | Zbl

[44] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985 | MR | Zbl

[45] V. V. Grushin, “O fundamentalnykh resheniyakh gipoellipticheskikh uravnenii”, UMN, 16:4 (1961), 147–153 | MR | Zbl

[46] L. Khërmander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, t. 3, 4, Mir, M., 1987; 1988

[47] V. Ya. Ivrii, “O tochnykh spektralnykh asimptotikakh dlya ellipticheskikh operatorov, deistvuyuschikh v rassloeniyakh”, Funkts. analiz i ego pril., 16:2 (1982), 30–38 | MR | Zbl

[48] V. Ivrii, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Math., 1100, Springer-Verlag, Berlin, 1984 | MR | Zbl

[49] V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, Berlin, 1998 | MR | Zbl

[50] V. Ivrii, “Sharp spectral asymptotics for operators with irregular coefficients”, Internat. Math. Res. Notes, 22 (2000), 1155–1166 | DOI | MR | Zbl

[51] D. Jerison, C. Kenig, “The Dirichlet problem in non-smooth domains”, Ann. of Math. (2), 113 (1981), 367–382 | DOI | MR | Zbl

[52] D. Jerison, C. Kenig, “Boundary value problems on Lipschitz domains”, MAA Studies in Math., 23 (1982), 1–68 | MR | Zbl

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

[54] H.-C. Kaiser, H. Neidhardt, J. Rehberg, Macroscopic current induced boundary conditions for Schrödinger-type operators, Preprint 630, Weierstrass-Institute for Applied Analysis and Stochastics, Berlin, 2001 | MR

[55] H. Kalf, U.-W. Schmincke, J. Walter, R. Wüst, “On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials”, Spectral Theory and Differential Equations, Lecture Notes in Math., 448, Springer-Verlag, Berlin, 1975, 182–226 | MR

[56] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl

[57] V. E. Katsnelson, O skhodimosti i summiruemosti ryadov po kornevym vektoram nekotorykh klassov nesamosopryazhennykh operatorov, Kandidatskaya diss., Kharkov, 1967

[58] V. E. Katsnelson, “Ob usloviyakh bazisnosti sistemy kornevykh vektorov nekotorykh klassov operatorov”, Funkts. analiz i ego pril., 1:2 (1967), 39–51 | MR | Zbl

[59] C. Kenig, “Recent progress on boundary-value problems on Lipschitz domains”, Pseudodifferential Operators and Applications, Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, 1985, 175–205 | MR

[60] C. Kenig, “Elliptic boundary value problems on Lipschitz domains”, Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., 112, Princeton Univ. Press, Princeton, NJ, 1986, 131–183 | MR

[61] C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Regional Conf. Series in Math., 83, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[62] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy MMO, 16 (1967), 209–292 | MR | Zbl

[63] V. A. Kondratev, S. D. Eidelman, “Ob usloviyakh na granichnuyu poverkhnost v teorii ellipticheskikh granichnykh zadach”, Dokl. AN SSSR, 246:4 (1979), 812–815 | MR | Zbl

[64] V. A. Kondratev, O. A. Oleinik, “Kraevye zadachi dlya sistemy teorii uprugosti v neogranichennykh oblastyakh. Neravenstva Korna”, UMN, 43:5 (1988), 55–98 | MR | Zbl

[65] G. P. Kostometov, “Ob asimptotike spektra integralnykh operatorov s polyarnymi yadrami”, Vestnik LGU, 1977, no. 13, 166–167 | MR | Zbl

[66] V. A. Kozlov, “Traction boundary value problem for anizotropic elasticity in polyhedral domains”, Russian J. Math. Phys., 8:3 (2001), 275–286 | MR | Zbl

[67] 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

[68] 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

[69] H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge, MA, 1974

[70] V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, T. V. Burchuladze, Trekhmernye zadachi matematicheskoi teorii uprugosti i termouprugosti, Nauka, M., 1976 | MR

[71] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, 1-e izd., Nauka, M., 1964 ; 2-Рμ РёР·Рґ., 1973 | MR

[72] V. B. Lidskii, “O summiruemosti ryadov po glavnym vektoram nesamosopryazhennykh operatorov”, Trudy MMO, 11 (1962), 3–35 | MR

[73] J.-L. Lions, “Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs”, J. Math. Soc. Japan, 14:2 (1962), 233–241 | MR | Zbl

[74] Zh.-L. Lions, E. Madzhenes, Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[75] A. S. Markus, “O razlozhenii po kornevym vektoram slabo vozmuschennogo samosopryazhennogo operatora”, Dokl. AN SSSR, 142:3 (1962), 538–541 | MR | Zbl

[76] A. S. Markus, Vvedenie v spektralnuyu teoriyu polinomialnykh operatornykh puchkov, Shtiintsa, Kishinev, 1986 | MR | Zbl

[77] A. S. Markus, V. I. Matsaev, “Operatory, porozhdennye polutoralineinymi formami, i ikh spektralnye asimptotiki”, Matem. issled., 61 (1981), 86–103 | MR | Zbl

[78] A. S. Markus, V. I. Matsaev, “O skhodimosti razlozhenii po sobstvennym vektoram operatora, blizkogo k samosopryazhennomu”, Matem. issled., 61 (1981), 104–129 | MR | Zbl

[79] A. S. Markus, V. I. Matsaev, “Teoremy sravneniya spektrov lineinykh operatorov i spektralnye asimptotiki”, Trudy MMO, 45 (1982), 133–181 | MR | Zbl

[80] V. G. Mazya, T. O. Shaposhnikova, Multiplikatory v prostranstvakh differentsiruemykh funktsii, Izd-vo LGU, L., 1986 | MR | Zbl

[81] R. Mazzeo, R. McOwen, “Singular Sturm–Liouville theory on manifolds”, J. Differential Equations, 176:2 (2001), 387–444 | DOI | MR | Zbl

[82] A. McIntosh, “On representing closed accretive sesquilinear forms as ($A^{1/2}u,A^{*1/2}v$)”, Nonlinear Partial Differential Equations and Their Applications, College de France Sem. V. III, Res. Notes Math., 70, 1982, 252–267 | MR | Zbl

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

[84] G. Métivier, “Valeurs propres de problèmes aux limites elliptiques irrégulieres”, Bull. Soc. Math. France. Suppl. Mém., 1977, no. 51–52, 125–219 | MR | Zbl

[85] S. G. Mikhlin, Kurs matematicheskoi fiziki, Nauka, M., 1968 | MR

[86] D. Mitrea, “The method of layer potentials for non-smooth domains with arbitrary topology”, Integral Equations Operator Theory, 29:3 (1997), 320–338 | DOI | MR | Zbl

[87] D. Mitrea, M. Mitrea, “General second order, strongly elliptic systems in low dimensional nonsmooth manifolds”, Contemp. Math., 277 (2001), 61–86 | MR | Zbl

[88] D. Mitrea, M. Mitrea, J. Pipher, “Vector potential theory on nonsmooth domains in $\mathbb R^3$ and applications to electromagnetic scattering”, J. Fourier Anal. Appl., 3:2 (1997), 131–192 | DOI | MR | Zbl

[89] D. Mitrea, M. Mitrea, M. Taylor, Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds, Mem. Amer. Math. Soc., 713, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[90] M. Mitrea, “Dirichlet integrals and Gaffney–Friedrichs inequalities in convex domains”, Forum Math., 13:4 (2001), 531–567 | DOI | MR | Zbl

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

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

[93] Y. Miyazaki, “The eigenvalue distribution of elliptic operators with Hölder continuous coefficients. II”, Osaka Math. J., 30:2 (1993), 267–301 | MR | Zbl

[94] C. B. Morrey, Jr, “Second-order elliptic systems of differential equations”, Contributions to the Theory of Partial Differential Equations, Ann. Math. Stud., 33, eds. L. Bers et al., Princeton Univ. Press, Princeton, 1954, 101–159 | MR

[95] T. Muramatu, “On Besov spaces and Sobolev spaces of generalized functions defined on a general region”, Publ. Res. Inst. Math. Sci., 9 (1974), 325–396 | DOI | MR | Zbl

[96] F. J. Narcovich, “Mathematical theory of the $R$ matrix. I. The eigenvalue problem”, J. Math. Phys., 15:10 (1974), 1626–1634 ; “II. The $R$ matrix and its properties”, 1635–1642 | DOI | MR

[97] D. G. Natroshvili, Issledovanie kraevykh i nachalno-kraevykh zadach matematicheskoi teorii uprugosti i termouprugosti dlya odnorodnykh anizotropnykh sred metodom potentsiala, Doktorskaya diss., Tbilisi, 1984

[98] D. G. Natroshvili, “Boundary integral equation method in the steady state oscillation problems for anisotropic bodies”, Math. Methods Appl. Sci., 20:2 (1997), 95–119 | 3.0.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[99] 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

[100] J. Nec̆as, “Sur les domaines du type $\mathscr N$”, Czechoslovak Math. J., 12 (1962), 274–287 | MR

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

[102] M. I. Neiman-Zade, A. A. Shkalikov, “Operatory Shrëdingera s singulyarnymi potentsialami iz prostranstv multiplikatorov”, Matem. zametki, 66:5 (1999), 723–733 | MR | Zbl

[103] L. Nirenberg, “Remarks on strongly elliptic partial differential equations”, Comm. Pure Appl. Math., 8 (1955), 649–675 | DOI | MR | Zbl

[104] O. A. Oleinik, A. S. Shamaev, G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Stud. Math. Appl., 26, North-Holland, Amsterdam, 1992 | MR

[105] V. I. Paraska, “Ob asimptotike sobstvennykh i singulyarnykh chisel lineinykh operatorov, povyshayuschikh gladkost”, Matem. sb., 68:4 (1965), 621–631 | MR

[106] L. Payne, H. Weinberger, “New bounds for solutions of second order elliptic partial differential equations”, Pacific J. Math., 8 (1958), 551–573 | MR | Zbl

[107] Sh. Rempel, B.-V. Shultse, Teoriya indeksa ellipticheskikh kraevykh zadach, Mir, M., 1986 | MR | Zbl

[108] R. Rokafellar, Vypuklyi analiz, Mir, M., 1973

[109] G. V. Rozenblyum, M. Z. Solomyak, M. A. Shubin, “Spektralnaya teoriya differentsialnykh operatorov”, Dif. uravneniya s chastnymi proizvodnymi VII, Itogi nauki i tekhniki. Sovr. problemy matem. Fund. napr., 64, VINITI, M., 1989, 5–247 | MR

[110] Yu. Safarov, D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Transl. Math. Monogr., 155, Amer. Math. Soc., Providence, RI, 1991 | MR

[111] G. Savaré, “Regularity results for elliptic equations in Lipschitz domains”, J. Funct. Anal., 152:1 (1988), 176–201 | DOI | MR

[112] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 66:6 (1999), 897–912 | MR | Zbl

[113] M. Schechter, “A generalization of the problem of transmission”, Ann. Scuola Norm. Sup. Pisa. Ser. III, Sci. Fis. Mat., 14 (1960), 207–236 | MR | Zbl

[114] R. T. Seeley, “Refinement of the functional calculus of Calderón and Zygmund”, Konink. Nederl. Acad. Wetensch. Proc. Ser. A, 68:3 (1965), 521–531 | MR | Zbl

[115] R. T. Seeley, “Singular integrals and boundary value problems”, Amer. J. Math., 88 (1966), 781–809 | DOI | MR | Zbl

[116] R. T. Seeley, “Interpolation in $L^p$ with boundary conditions”, Studia Math., 44 (1972), 47–60 | MR | Zbl

[117] V. S. Serov, “O skhodimosti ryadov Fure po sobstvennym funktsiyam operatora Shrëdingera s potentsialom Kato”, Matem. zametki, 67:5 (2000), 755–763 | MR | Zbl

[118] I. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[119] R. Szmytkowski, Metoda $R$-macierzy dla równań Schrödingera i Diraca, Politechnika Gdańska, Gdańsk, 1999

[120] R. H. Torres, G. V. Welland, “The Helmholtz equation and transmission problems with Lipschitz interfaces”, Indiana Univ. Math. J., 42:4 (1993), 1457–1485 | DOI | MR | Zbl

[121] B. R. Vainberg, “Printsipy izlucheniya, predelnogo pogloscheniya i predelnoi amplitudy v obschei teorii uravnenii s chastnymi proizvodnymi”, UMN, 21:3 (1966), 115–194 | MR | Zbl

[122] B. R. Vainberg, Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki, Izd-vo MGU, M., 1982 | MR | Zbl

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

[124] G. Verchota, “Remarks on 2nd order elliptic systems in Lipschitz domains”, Operator Theory and Partial Differential Equations, Miniconf. Ryde/Aust., 1986. Proc. Centre Math. Anal. Austral. Nat. Univ, 14, 1986, 303–325 | MR | Zbl

[125] M. I. Vishik, “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Dokl. AN SSSR, 74:5 (1950), 881–884 | MR | Zbl

[126] N. N. Voitovich, B. Z. Katsenelenbaum, A. N. Sivov, Obobschennyi metod sobstvennykh kolebanii v teorii difraktsii, Nauka, M., 1977 | MR | Zbl

[127] Kh. Uitni, Geometricheskaya teoriya integrirovaniya, IL, M., 1960

[128] L. Zielinski, “Asymptotic distribution of eigenvalues for some elliptic operators with simple remainder estimates”, J. Operator Theory, 39:2 (1998), 249–282 | MR | Zbl

[129] A. Zigmund, Trigonometricheskie ryady, 2, Mir, M., 1965 | MR