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

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.

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

Notes

  1. 1.

    By elliptic we mean that its principal part \(L_0(x_0, D_x)\) frozen at any point \(x_0\in \overline{\mathcal {O}}\) satisfies \(L_0(x_0,i\xi )\le -\gamma |\xi |^2\), for all \(\xi \in \mathbb {R}^n\), \(\xi \ne 0\), for some \(\gamma >0\) independent of \(x_0\).

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)

    MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  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)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)

    Google Scholar 

  9. 9.

    Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. RAIRO Modél. Math. Anal. Numér. 33, 627–649 (1999)

    MathSciNet  Article  MATH  Google 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)

  11. 11.

    Costabel, M., Dauge, M., Nicaise, S.: Corner singularities and analytic regularity for linear elliptic systems (2016, in preparation)

  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)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  16. 16.

    Duren, P.L.: Theory of \(H^{p}\) spaces. Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)

  17. 17.

    Eskin, G.: Lectures on linear partial differential equations. Graduate Studies in Mathematics, vol. 123. American Mathematical Society, Providence (2011)

  18. 18.

    Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)

  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)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  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)

  25. 25.

    Petitta, F.: Renormalized solutions of nonlinear parabolic equations with general measure data. Ann. Mat. Pura Appl. (4) 187(4), 563–604 (2008)

  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)

    MathSciNet  Google Scholar 

  27. 27.

    Ponce, A.: Elliptic PDEs, measures and capacities. Tracts in Mathematics, vol. 23. European Mathematical Society, Zürich (2016)

  28. 28.

    Scott, R.: Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973/1974)

  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)

  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)

  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)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Nicaise.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ariche, S., De Coster, C. & Nicaise, S. Regularity of solutions of elliptic or parabolic problems with Dirac measures as data. SeMA 73, 379–426 (2016). https://doi.org/10.1007/s40324-016-0077-x

Download citation

Keywords

  • Laplace equation
  • Heat equation
  • Dirac measure
  • Regularity

Mathematics Subject Classification

  • 35R06
  • 35B65