Skip to main content
SpringerLink
Log in

Elliptic Boundary Value Problems in Domains with Piecewise Smooth Boundary

  • Chapter
Partial Differential Equations IX

Part of the book series: Encyclopaedia of Mathematical Sciences ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Agmon, S., Nirenberg, L. (1963): Properties of solutions of ordinary differential equations in Banach space. Commun. Pure Appl. Math. 16, 121–239. Zbl. 117,100

    Article  MathSciNet  MATH  Google Scholar 

  • 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. 137,296

    MATH  Google Scholar 

  • 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. 635.35075

    MathSciNet  Google Scholar 

  • Bueckner, H. F. (1970) : A novel principle for the computation of stress intensity factors. Z. Angew. Math. Mech. 50, 529–546. Zbl. 213,266

    MathSciNet  MATH  Google Scholar 

  • Dauge, M. (1988) : Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. Lecture Notes in Mathematics 1341, Springer. Zbl. 638.35001

    MATH  Google Scholar 

  • Eskin, G. (1985) : Boundary-value problems for second-order elliptic equations in domains with corners. Proc. Sympos. Pure Math. 43, 105–131. Zbl. 574.35029

    Article  MathSciNet  Google Scholar 

  • 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. 328.31008

    MathSciNet  MATH  Google Scholar 

  • 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. 254.47046

    MathSciNet  Google Scholar 

  • Grisvard, P. (1985) : Elliptic Problems in Nonsmooth Domains. Pitman. Zbl. 695.35060

    MATH  Google Scholar 

  • 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. 286.35027

    Google Scholar 

  • 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. 162,163

    MATH  Google Scholar 

  • 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. 209,411

    MATH  Google Scholar 

  • 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. 379.35020

    MATH  Google Scholar 

  • 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. 523.35010

    MathSciNet  MATH  Google Scholar 

  • 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. 543.35027

    Google Scholar 

  • 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. 695.35005

    Google Scholar 

  • 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. 672.35050

    MathSciNet  Google Scholar 

  • Kufner, A., Sändig, A.-M. (1987): Some Applications of Weighted Sobolev Spaces. Teubner, Leipzig. Zbl. 662.46034

    MATH  Google Scholar 

  • 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. 257.35032

    Google Scholar 

  • 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. 451.73030

    Google Scholar 

  • 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. 604.35016

    MathSciNet  Google Scholar 

  • 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. 701.35021

    MathSciNet  Google Scholar 

  • 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), 415–437

    Google Scholar 

  • 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. 232.35027

    Google Scholar 

  • 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. 283.35037

    Google Scholar 

  • 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. 295.35006

    MathSciNet  Google Scholar 

  • 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. 296.35029

    Google Scholar 

  • 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. 351.35010

    MATH  Google Scholar 

  • 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. 355.35032

    MathSciNet  Google Scholar 

  • 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. 359.35024

    Article  MathSciNet  MATH  Google Scholar 

  • Maz’ya, V.G., Plamenevskij, B.A. (1977b): Elliptic boundary value problems on manifolds with singularities. Probl. Mat. Anal. 6, 85–142 (Russian) . Zbl. 453.58022

    MATH  Google Scholar 

  • 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. 371.35018

    Article  MathSciNet  MATH  Google Scholar 

  • 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. 441.35028

    MATH  Google Scholar 

  • 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. 423.35021

    Google Scholar 

  • 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. 408.35014

    Google Scholar 

  • 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. 429.35031

    Google Scholar 

  • 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. 417.35014

    MATH  Google Scholar 

  • 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. 444.35067

    MathSciNet  Google Scholar 

  • 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. 561.73020

    MATH  Google Scholar 

  • 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. 532.35065

    MATH  Google Scholar 

  • 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. 431.76027

    Google Scholar 

  • Maz’ya, V.G., Rossman, J. (1988): Ãœber die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138, 27–53. Zbl. 672.35020

    Article  MathSciNet  MATH  Google Scholar 

  • 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. 547.35042

    Google Scholar 

  • Mo F (1988) : Mechanics of Fracture and Strength of Materials (in 4 volumes) . Vol. 2, Naukova Dumka, Kiev (Russian)

    Google Scholar 

  • 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. 684.35021

    Google Scholar 

  • 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. 778.35033

    Google Scholar 

  • Nazarov, S.A., Plamenevskij, B.A. (1991b): Radiation principles for self-adjoint elliptic systems. Probl. Mat. Fiz. 13, 192–245 (Russian)

    Google Scholar 

  • 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. 725.35027

    Google Scholar 

  • Nazarov, S.A., Plamenevskij, B.A. (1994): Elliptic Problems in Domains with Piecewise Smooth Boundary. W. de Gruyter&Co, Berlin New York

    Book  Google Scholar 

  • 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. 399.35049

    MathSciNet  MATH  Google Scholar 

  • Parton, V.Z., Perlin, P.I. (1981): Methods of Mathematical Elasticity Theory. Nauka (Russian) . Zbl. 506.73005

    MATH  Google Scholar 

  • Pazy, A. (1967) : Asymptotic expansions of solutions of ordinary differential equations in Hilbert space. Arch. Rat. Mech. Anal. 24(2), 193–218. Zbl. 147,123

    Article  MathSciNet  MATH  Google Scholar 

  • 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), 1363–1381

    Google Scholar 

  • Rice, J.R. (1972): Some remarks on elastic crack-tip stress fields. Int. J. Solids Struct. 8(6), 751–758. Zbl. 245.73003

    Article  MATH  Google Scholar 

  • 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. 411.76019

    MathSciNet  MATH  Google Scholar 

  • 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. 414.35062

    MathSciNet  MATH  Google Scholar 

  • 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), 2561–2586

    MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. B. A. Plamenevskij
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

M. S. Agranovich Yu. V. Egorov M. A. Shubin

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Plamenevskij, B.A. (1997). Elliptic Boundary Value Problems in Domains with Piecewise Smooth Boundary. In: Agranovich, M.S., Egorov, Y.V., Shubin, M.A. (eds) Partial Differential Equations IX. Encyclopaedia of Mathematical Sciences, vol 79. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06721-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-06721-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08169-9

  • Online ISBN: 978-3-662-06721-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation