Elliptic Boundary Value Problems in Domains with Piecewise Smooth Boundary

  • B. A. Plamenevskij
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 79)

Abstract

This paper is a sketch of the theory of general elliptic boundary value problems in domains with edges of various dimensions on the boundary. In particular, the class of admissible domains contains polygons, cones, lenses and polyhedrons. Discontinuities in the coefficients of the operators along edges are allowed. We discuss solvability of the problems and obtain asymptotic formulas for solutions near singularities of the boundary and of the coefficients.

Keywords

Manifold Stratification Convolution Cylin Zine 

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References

  1. Agmon, S., Nirenberg, L. (1963): Properties of solutions of ordinary differential equations in Banach space. Commun. Pure Appl. Math. 16, 121–239. Zbl. 117,100MathSciNetMATHCrossRefGoogle Scholar
  2. Agranovich, M.S., Vishik, M.I. (1964): Elliptic problems with a parameter and parabolic problems of general type. Usp. Mat. Nauk 19(3), 53–161. English transl.: Russ. Math. Surv. 19(3) (1964), 53–157. Zbl. 137,296MATHGoogle Scholar
  3. Birman, M.S., Solomyak, M.Z. (1987): L2-theory of the Maxwell operator in arbitrary domains. Usp. Mat. Nauk 42(6), 61–76. English transl.: Russ. Math. Surv. 42(6) (1987), 75–96. Zbl. 635.35075MathSciNetGoogle Scholar
  4. Bueckner, H. F. (1970) : A novel principle for the computation of stress intensity factors. Z. Angew. Math. Mech. 50, 529–546. Zbl. 213,266MathSciNetMATHGoogle Scholar
  5. Dauge, M. (1988) : Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. Lecture Notes in Mathematics 1341, Springer. Zbl. 638.35001MATHGoogle Scholar
  6. Eskin, G. (1985) : Boundary-value problems for second-order elliptic equations in domains with corners. Proc. Sympos. Pure Math. 43, 105–131. Zbl. 574.35029MathSciNetCrossRefGoogle Scholar
  7. Fichera, G. (1975) : Asymptotic behaviour of the electric field and density of the electric charge in the neighborhood of singular points of a conducting surface. Usp. Mat. Nauk 30(3), 105–124. English transl.: Russ. Math. Surv. 30(3) (1975), 107–127. Zbl. 328.31008MathSciNetMATHGoogle Scholar
  8. Gohberg, I.C., Sigal, E.I. (1971): An operator generalization of the logarithmic residue theorem and the theorem of Rouche. Mat. Sb., Nov. Ser. 84, 607–629. English transl.: Math. USSR, Sb. 13 (1971), 603–625. Zbl. 254.47046MathSciNetGoogle Scholar
  9. Grisvard, P. (1985) : Elliptic Problems in Nonsmooth Domains. Pitman. Zbl. 695.35060MATHGoogle Scholar
  10. Komech, A.I. (1973) : Elliptic boundary value problems on manifolds with a piecewise smooth boundary. Mat. Sb., Nov. Ser. 92, 89–134. English transl.: Math. USSR, Sb. 31 (1973), 91–135. Zbl. 286.35027Google Scholar
  11. Kondrat’ev, V.A. (1967): Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. O.-va 16, 209–292. English transl.: Trans. Moscow Math. Soc. 16 (1967), 227–313. Zbl. 162,163MATHGoogle Scholar
  12. Kondrat’ev, V.A. (1970): The smoothness of solution of Dirichlet’s problem for second-order elliptic equations in a region with piecewise-smooth boundary. Differ. Uravn. 6, 1831–1843. English transl.: Differ. Equations 6 (1970), 1392–1401. Zbl. 209,411MATHGoogle Scholar
  13. Kondrat’ev, V.A. (1977): Singularities of a solution of Dirichlet’s problem for a second-order elliptic equation in a neighborhood of an edge. Differ. Uravn. 13, 2026–2032. English transl.: Differ. Equations 13 (1977), 1411–1415. Zbl. 379.35020MATHGoogle Scholar
  14. Kondrat’ev, V.A., Oleinik, O.A. (1983): Boundary value problems for partial differential equations in non-smooth domains. Usp. Mat. Nauk 38(2), 3–76. English transl.: Russ. Math. Surv. 38(2) (1983), 1–86. Zbl. 523.35010MathSciNetMATHGoogle Scholar
  15. Koplienko, L.S., Plamenevskij, B.A. (1983): A radiation principle for periodic problems. Differ. Uravn. 19, 1713–1723. English transl.: Differ. Equations 19 (1983), 1273–1281. Zbl. 543.35027Google Scholar
  16. Kozlov, V.A. (1989) : The strong zero theorem for an elliptic boundary value problem in an angle. Mat. Sb. 180, 831–849. English transl.: Math. USSR, Sb. 67 (1990), 283–302. Zbl. 695.35005Google Scholar
  17. Kozlov, V.A., Maz’ya, V.G. (1988): Spectral properties of the operator bundles generated by elliptic boundary value problems in a cone. Funkts. Anal. Prilozh. 22(2), 38–46. English transl.: Funct. Anal. Appl. 22 (1988), 114–121. Zbl. 672.35050MathSciNetGoogle Scholar
  18. Kufner, A., Sändig, A.-M. (1987): Some Applications of Weighted Sobolev Spaces. Teubner, Leipzig. Zbl. 662.46034MATHGoogle Scholar
  19. Maz’ya, V.G. (1973): On the oblique derivative problem in a domain with edges of different dimensions. Vestn. Leningr. Univ., Mat. Mekh. Astron. 7, 34–39. English transl.: Vestn. Leningr. Univ. Math. 6 (1979), 148–154. Zbl. 257.35032Google Scholar
  20. Maz’ya, V.G., Morozov, N.F., Plamenevskij, B.A. (1979): On nonlinear bending of a plate with a crack. Differential and integral equations. Boundary value problems (I.N. Vekua Memorial Collection), Tbilisi, 145–163. English transl.: Am. Math. Soc. Transl. (Ser. 2) 123 (1984), 125–139. Zbl. 451.73030Google Scholar
  21. Maz’ya, V.G., Nazarov, S.A. (1986): The vertex of a cone can be nonregular in the Wiener sense for a fourth-order elliptic equation. Mat. Zametki 39, 24–28. English transl.: Math. Notes 39 (1986), 14–16. Zbl. 604.35016MathSciNetGoogle Scholar
  22. Maz’ya, V.G., Nazarov, S. A.(1989): Singularities of solutions of the Neumann problem at a conical point. Sib. Mat. Zh. 30(3), 52–63. English transl.: Sib. Math. J. 30 (1989) , 387–396. Zbl. 701.35021MathSciNetGoogle Scholar
  23. Maz’ya, V.G., Nazarov, S.A., Plamenevskij, B.A. (1983): On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone. Mat. Sb., Nov. Ser. 122, 435–456. English transl.: Math. USSR, Sb. 50 (1985), 415–437Google Scholar
  24. Maz’ya, V.G., Plamenevskij, B.A. (1971): Problems with oblique derivatives in regions with piecewise smooth boundaries. Funkts. Anal. Prilozh. 5(3), 102–103. English transl.: Funct. Anal. Appl. 5 (1971), 256–258. Zbl. 232.35027Google Scholar
  25. Maz’ya, V.G., Plamenevskij, B.A. (1973a): Elliptic boundary value problems in a domain with piecewise smooth boundary. Proc. Sympos. Continuum Mechanics and Related Problems of Analysis. Tbilisi, Mecnieraba 1, 171–181 (Russian) . Zbl. 283.35037Google Scholar
  26. Maz’ya, V.G., Plamenevskij, B.A. (1973b): The asymptotic behavior of solutions of the Navier-Stokes equations near edges. Dokl. Akad. Nauk SSSR 210, 803–806. English transl.: Sov. Phys., Dokl. 18 (1973/1974), 379–381. Zbl. 295.35006MathSciNetGoogle Scholar
  27. Maz’ya, V.G., Plamenevskij, B.A. (1975a): On boundary value problems for a second order elliptic equation in a domain with edges. Vestn. Leningr. Univ., Mat. Mekh. Astron. 1, 102–108. English transl.: Vestn. Leningr. Univ. Math. 8 (1980), 99–106. Zbl. 296.35029Google Scholar
  28. Maz’ya, V.G., Plamenevskij, B.A. (1975b): On the coefficients in the asymptotic expansion of the solution of elliptic boundary value problems in a cone. Zap. Nauchn. Semin. LOMI 52, 110–127. English transl.: J. Sov. Math. 9 (1978), 750–764. Zbl. 351.35010MATHGoogle Scholar
  29. Maz’ya, V.G., Plamenevskij, B.A. (1976): On the coefficients in the asymptotics of solutions of elliptic boundary value problems near the edge. Dokl. Akad. Nauk SSSR 229, 33–36. English transl.: Sov. Math., Dokl. 17 (1976), 970–974. Zbl. 355.35032MathSciNetGoogle Scholar
  30. Maz’ya, V.G., Plamenevskij, B.A. (1977a): The coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr. 76, 29–60 English transl.: Am. Math. Soc., Transl. (Ser. 2) 123 (1984), 57–88. Zbl. 359.35024MathSciNetMATHCrossRefGoogle Scholar
  31. Maz’ya, V.G., Plamenevskij, B.A. (1977b): Elliptic boundary value problems on manifolds with singularities. Probl. Mat. Anal. 6, 85–142 (Russian) . Zbl. 453.58022MATHGoogle Scholar
  32. Maz’ya, V.G., Plamenevskij, B.A. (1978a): Estimates in Lp and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr. 81, 25–82. English transl.: Am. Math. Soc., Transl. (Ser. 2) 123 (1984), 1–56. Zbl. 371.35018MathSciNetMATHCrossRefGoogle Scholar
  33. Maz’ya, V.G., Plamenevskij, B.A. (1978b): Lp-estimates of solutions of elliptic boundary value problems in domains with edges. Tr. Mosk. Mat. O.-va 37, 49–93. English transl.: Trans. Moscow Math. Soc. 37 (1980), 49–97. Zbl. 441.35028MATHGoogle Scholar
  34. Maz’ya, V.G., Plamenevskij, B.A. (1978c): Schauder estimates of solutions of elliptic boundary value problems in domains with edges of the boundary. Proc. Semin. S.L. Sobolev 2, Novosibirsk, 69–102. English transl.: Am. Math. Soc., Transl. (Ser. 2) 123 (1984), 141–169. Zbl. 423.35021Google Scholar
  35. Maz’ya, V.G., Plamenevskij, B.A. (1978d): Estimates of Green’s functions and Schauder estimates for solutions of elliptic boundary value problems in a dihedral angle. Sib. Mat. Zh. 19, 1065–1082. English transl.: Sib. Math. J. 19 (1978), 752–764. Zbl. 408.35014Google Scholar
  36. Maz’ya, V.G., Plamenevskij, B.A. (1978e): Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points. Elliptische Differentialgleichungen (Meeting, Rostock, 1977), Wilhelm-Pieck-Univ., Rostock, 161–190. English transl.: Am. Math. Soc., Transl. (Ser. 2) 123 (1984), 89–107. Zbl. 429.35031Google Scholar
  37. Maz’ya, V.G., Plamenevskij, B.A. (1979): Asymptotic behavior of the fundamental solutions of elliptic boundary value problems in domains with conical points. Probl. Mat. Anal. 7, 100–145 English transl.: Sel. Math. Sov. 4 (1985), 363–397. Zbl. 417.35014MATHGoogle Scholar
  38. Maz’ya, V.G., Plamenevskij, B.A. (1980): A problem on the motion of a fluid with a free surface in a container with piecewise smooth walls. Dokl. Akad. Nauk SSSR 250, 1315–1318. English transl.: Sov. Math., Dokl. 21 (1980), 317–319. Zbl. 444.35067MathSciNetGoogle Scholar
  39. Maz’ya, V.G., Plamenevskij, B.A. (1981): On the properties of solutions of threedimensional problems of elasticity theory and hydrodynamics in domains with isolated singular points. Din. Sploshnoj. Sredy 50, 99–121. English transl.: Am. Math. Soc., Transl. (Ser. 2) 123 (1984), 109–123. Zbl. 561.73020MATHGoogle Scholar
  40. Maz’ya, V.G., Plamenevskij, B.A. (1983): The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries. Z. Anal. Anwend. 2, 335–359. Zbl. 532.35065MATHGoogle Scholar
  41. Maz’ya, V.G., Plamenevskij, B.A., Stupyalis, L.I. (1979): The three-dimensional problem of steady-state motion of a fluid with a free surface. Differ. Uravn. Primen. 23 1–155. English transl.: Am. Math. Soc., Transl. (Ser. 2) 123 (1984), 171–286. Zbl. 431.76027Google Scholar
  42. Maz’ya, V.G., Rossman, J. (1988): Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138, 27–53. Zbl. 672.35020MathSciNetMATHCrossRefGoogle Scholar
  43. Maz’ya, V.G., Rossman, J. (1984): Uber die Lösbarkeit und die Asymptotik der Lösungen elliptischer Randwertaufgaben in Gebieten mit Kanten. Preprint Akadem. Wiss. DDR, P-MATH.-31/84, 1–44. Zbl. 547.35042Google Scholar
  44. Mo F (1988) : Mechanics of Fracture and Strength of Materials (in 4 volumes) . Vol. 2, Naukova Dumka, Kiev (Russian)Google Scholar
  45. Nazarov, S.A. (1988): Estimates near an edge for the solution of the Neumann problem for an elliptic system. Vestn. Leningr. Univ., Mat. Mekh. Astron. 21, 37–42. English transl.: Vestn. Leningr. Univ. 21 (1988), 52–59. Zbl. 684.35021Google Scholar
  46. Nazarov, S.A., Plamenevskij, B.A. (1991a): Neumann problem for self-adjoint systems in a domain with piecewise smooth boundary. Tr. Leningr. Mat. O.-va 1, 174–211. English transl.: Am. Math. Soc., Transl. (Ser. 2) 155 (1993), 169–206. Zbl. 778.35033Google Scholar
  47. Nazarov, S.A., Plamenevskij, B.A. (1991b): Radiation principles for self-adjoint elliptic systems. Probl. Mat. Fiz. 13, 192–245 (Russian)Google Scholar
  48. Nazarov, S.A., Plamenevskij, B.A. (1990): On radiation conditions for self-adjoint elliptic problems. Dokl. Akad. Nauk SSSR 311, 523–535. English transl.: Sov. Math., Dokl. 41 (1990), 274–277. Zbl. 725.35027Google Scholar
  49. Nazarov, S.A., Plamenevskij, B.A. (1994): Elliptic Problems in Domains with Piecewise Smooth Boundary. W. de Gruyter&Co, Berlin New YorkCrossRefGoogle Scholar
  50. Nikishkin, V.A. (1979): Singularities of the solutions to the Dirichlet problem for a second-order equation in a neighborhood of an edge. Vestn. Mosk. Univ., Ser. I Mat. Mekh. 1979, No. 2, 51–62. English transl.: Moscow Univ. Math. Bull. 34(2) (1979), 53–64. Zbl. 399.35049MathSciNetMATHGoogle Scholar
  51. Parton, V.Z., Perlin, P.I. (1981): Methods of Mathematical Elasticity Theory. Nauka (Russian) . Zbl. 506.73005MATHGoogle Scholar
  52. Pazy, A. (1967) : Asymptotic expansions of solutions of ordinary differential equations in Hilbert space. Arch. Rat. Mech. Anal. 24(2), 193–218. Zbl. 147,123MathSciNetMATHCrossRefGoogle Scholar
  53. Plamenevskij, B.A., Tashchiyan, G.M. (1990): Convolution operator in weighted spaces. Probl. Mat. Anal. 11, 208–237. English transl.: J. Sov. Math. 64(6) (1993), 1363–1381Google Scholar
  54. Rice, J.R. (1972): Some remarks on elastic crack-tip stress fields. Int. J. Solids Struct. 8(6), 751–758. Zbl. 245.73003MATHCrossRefGoogle Scholar
  55. Solonnikov, V.A. (1979a): Solvability of a problem on the plane motion of a heavy viscous incompressible capillary liquid partially filling a container. Izv. Akad. Nauk SSSR, Ser. Mat. 43, 203–236. English transl.: Math. USSR, Izv. 14 (1980), 193–221. Zbl. 411.76019MathSciNetMATHGoogle Scholar
  56. Solonnikov, V.A. (1979b): Solvability of the three-dimensional problem with a free boundary for the stationary Navier-Stokes system. Zap. Nauchn. Semin. LOMI 84, 252–285. English transl.: J. Sov. Math. 21 (1983), 427–450. Zbl. 414.35062MathSciNetMATHGoogle Scholar
  57. Zajaczkowski, W., Solonnikov, V.A. (1983): The Neumann problem for second-order elliptic equations in domains with ribs on the boundary. Boundary value problems of mathematical physics and related questions in the theory of functions. Zap. Nauchn. Semin. LOMI 127, 7–48. English transl.: J. Sov. Math. 27 (1984), 2561–2586MathSciNetMATHGoogle Scholar

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