Advertisement

SeMA Journal

, Volume 73, Issue 4, pp 379–426 | Cite as

Regularity of solutions of elliptic or parabolic problems with Dirac measures as data

  • S. Ariche
  • C. De Coster
  • S. NicaiseEmail author
Article

Abstract

In this paper we study the Laplace and heat equations with Dirac right-hand side. We prove some regularity results in a scale of weighted Sobolev spaces, the weight being the distance to the support of the right-hand side. Model situations in dimension three are treated by using Fourier, Laplace or Mellin technique that reduces the problem to a Helmholtz problem in two dimension. Hence the key point stays on estimates for the solution of the Helmholtz problem in standard or weighted Sobolev spaces which are uniform with respect to the parameter.

Keywords

Laplace equation Heat equation Dirac measure  Regularity 

Mathematics Subject Classification

35R06 35B65 

References

  1. 1.
    Agranovich, M.S., Vishik, M.I.: Elliptic problems with a parameter and parabolic problems of general type. Russ. Math. Surveys 19, 53–157 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ali Mehmeti, F.: Transient tunnel effect and Sommerfeld problem. Wave in semi-infinite structures. Mathematical Research, vol. 91. Akademie Verlag, Berlin (1996)Google Scholar
  3. 3.
    Apel, T., Benedix, O., Sirch, D., Vexler, B.: A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 49(3), 992–1005 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Araya, R., Behrens, E., Rodríguez, R.: A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105(2), 193–216 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)Google Scholar
  6. 6.
    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2), 241–273 (1995)Google Scholar
  7. 7.
    Boccardo, L., Dall’Aglio, A., Gallouët, T., Orsina, L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147(1), 237–258 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)zbMATHGoogle Scholar
  9. 9.
    Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. RAIRO Modél. Math. Anal. Numér. 33, 627–649 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Costabel, M., Dauge, M., Nicaise, S.: Mellin analysis of weighted Sobolev spaces with nonhomogeneous norms on cones. In: Around the Research of Vladimir Maz’ya. I, Int. Math. Ser. (N.Y.), vol. 11, pp. 105–136. Springer, New York (2010)Google Scholar
  11. 11.
    Costabel, M., Dauge, M., Nicaise, S.: Corner singularities and analytic regularity for linear elliptic systems (2016, in preparation)Google Scholar
  12. 12.
    Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(4), 741–808 (1999)Google Scholar
  13. 13.
    D’Angelo, C.: Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal. 50(1), 194–215 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D’Angelo, C., Quarteroni, A.: On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18(8), 1481–1504 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dauge, M.: Elliptic boundary value problems on corner domains—smoothness and asymptotics of solutions. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)Google Scholar
  16. 16.
    Duren, P.L.: Theory of \(H^{p}\) spaces. Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)Google Scholar
  17. 17.
    Eskin, G.: Lectures on linear partial differential equations. Graduate Studies in Mathematics, vol. 123. American Mathematical Society, Providence (2011)Google Scholar
  18. 18.
    Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)Google Scholar
  19. 19.
    Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence (1997)Google Scholar
  20. 20.
    Leykekhman, D., Vexler, B.: Optimal a priori error estimates of parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 51(5), 2797–2821 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Maz’ya, V.G., Roßmann, J.: On the Agmon–Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Global Anal. Geom. 9, 253–303 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Maz’ya, V.G., Roßmann, J.: Elliptic equations in polyhedral domains. Mathematical Surveys and Monographs, vol. 162. American Mathematical Society, Providence (2010)Google Scholar
  23. 23.
    Nist Digital Library of Mathematical Functions. http://dlmf.nist.gov/
  24. 24.
    Paley, R.E.A.C., Wiener, N.: Fourier transforms in the complex domain. American Mathematical Society Colloquium Publications, vol. 19. American Mathematical Society, Providence (1987) (reprint of the 1934 original)Google Scholar
  25. 25.
    Petitta, F.: Renormalized solutions of nonlinear parabolic equations with general measure data. Ann. Mat. Pura Appl. (4) 187(4), 563–604 (2008)Google Scholar
  26. 26.
    Petitta, F., Porretta, A.: On the notion of renormalized solution to nonlinear parabolic equations with general measure data. J. Elliptic Parabol. Equ. 1, 201–214 (2015)MathSciNetGoogle Scholar
  27. 27.
    Ponce, A.: Elliptic PDEs, measures and capacities. Tracts in Mathematics, vol. 23. European Mathematical Society, Zürich (2016)Google Scholar
  28. 28.
    Scott, R.: Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973/1974)Google Scholar
  29. 29.
    Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15(fasc. 1), 189–258 (1965)Google Scholar
  30. 30.
    Tanabe, H.: Functional analytic methods for partial differential equations. Monographs and Textbooks in Pure and Applied Mathematics, vol. 204. Marcel Dekker, New York (1997)Google Scholar
  31. 31.
    Tychonov, A.N., Samarski, A.A.: Partial differential equations of mathematical physics, vol. II (translated by S. Radding) Holden-Day, San Francisco (1964)Google Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2016

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et ses Applications de ValenciennesFR CNRS 2956, Université de Valenciennes et du Hainaut-CambrésisValenciennes Cedex 9France

Personalised recommendations