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Year 2022, , 134 - 140, 15.09.2022
https://doi.org/10.33205/cma.1143800

Abstract

References

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  • L. Bers, F. John and M. Schechter: Partial differential equations, Providence R.I.: Wiley (1979).
  • H. Brakhage, P. Werner: Über das Dirichlet’sche Außenraumproblem für die Helmholtz’sche Schwingungschleichung, Arch. Math., 16 (1965), 325–329.
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  • V. G. Maz’ja: Sobolev spaces, Berlin Heidelberg New York Tokyo, Springer Verlag (1985).
  • R. McOwen: The behaviour of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783–795.
  • J. Neˇcas: Les mèthodes directes en thèorie des èquations elliptiques, Prague, Academia (1967).
  • J. Saranen, K. J. Witsch: Exterior boundary value problems for elliptic equations. Ann. Acad. Sci. Fennicae Series A. I. Mathematica, 8 (1983) 3–42.
  • W. I. Smirnow: Lehrgang der höheren Mathematik 4, Berlin: Deutscher Verlag der Wissenschaften (1979).
  • C. G. Simader: On Dirichlet’s boundary value problem, Berlin Heidelberg New York, Springer Lecture Notes 268 (1972).
  • W. Varnhorn: The Poisson equation with weights in exterior domains of $\mathbb{R}^{n}$, Applic. Anal., 43 (1992), 135–145.

On the Poisson equation in exterior domains

Year 2022, , 134 - 140, 15.09.2022
https://doi.org/10.33205/cma.1143800

Abstract

We construct a solution of the Poisson equation in exterior domains $\Omega \subset \mathbb R^n,\;n \ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(\Omega),;1 < q <\infty,$ with methods of potential theory and integral equations. We investigate the corresponding null spaces and prove that its dimensions is equal to $n+1$ independent of $q$.

References

  • R. A. Adams: Sobolev Spaces, New York-San Francisco-London: Academic Press (1975).
  • L. Bers, F. John and M. Schechter: Partial differential equations, Providence R.I.: Wiley (1979).
  • H. Brakhage, P. Werner: Über das Dirichlet’sche Außenraumproblem für die Helmholtz’sche Schwingungschleichung, Arch. Math., 16 (1965), 325–329.
  • A. P. Calderon, A. Zygmund: On singular integrals. Amer. J. Math., 78 (1956) 289–309.
  • J. Deny, J.-L. Lions: Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5 (1953–1954) 305–370.
  • D. Fortunato: On the index of elliptic partial differential operators in $\mathbb{R}^{n}$, Ann. Mat. Pura Appl., 119 (1979), 317–331.
  • J. Giroire: Etude de quelques problemes aux limites exterieurs et resolution par equations integrales. Thèse de doctorat d’etat es sciences mathematiques. Paris 6: Universite Pierre et Marie Curie (1987).
  • N. M. Günter: Die Potentialtheorie und ihre Anwendungen auf Grundaufgaben der mathematischen Physik, Leipzig: Verlagsgessellschaft (1957).
  • G. C. Hsiao, R. Kress: On an integral equation for the two-dimensional exterior Stokes problem. NAM-Bericht 33. Universität Göttingen (1983).
  • R. Leis: Zur Eindeutigkeit der Randwertaufgaben der Helmholtz’schen Schwingungsgleichung, Math. Z., 85 (1964) 141–153.
  • V. G. Maz’ja: Sobolev spaces, Berlin Heidelberg New York Tokyo, Springer Verlag (1985).
  • R. McOwen: The behaviour of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783–795.
  • J. Neˇcas: Les mèthodes directes en thèorie des èquations elliptiques, Prague, Academia (1967).
  • J. Saranen, K. J. Witsch: Exterior boundary value problems for elliptic equations. Ann. Acad. Sci. Fennicae Series A. I. Mathematica, 8 (1983) 3–42.
  • W. I. Smirnow: Lehrgang der höheren Mathematik 4, Berlin: Deutscher Verlag der Wissenschaften (1979).
  • C. G. Simader: On Dirichlet’s boundary value problem, Berlin Heidelberg New York, Springer Lecture Notes 268 (1972).
  • W. Varnhorn: The Poisson equation with weights in exterior domains of $\mathbb{R}^{n}$, Applic. Anal., 43 (1992), 135–145.
There are 17 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Articles
Authors

Werner Varnhorn 0000-0001-9486-1319

Publication Date September 15, 2022
Published in Issue Year 2022

Cite

APA Varnhorn, W. (2022). On the Poisson equation in exterior domains. Constructive Mathematical Analysis, 5(3), 134-140. https://doi.org/10.33205/cma.1143800
AMA Varnhorn W. On the Poisson equation in exterior domains. CMA. September 2022;5(3):134-140. doi:10.33205/cma.1143800
Chicago Varnhorn, Werner. “On the Poisson Equation in Exterior Domains”. Constructive Mathematical Analysis 5, no. 3 (September 2022): 134-40. https://doi.org/10.33205/cma.1143800.
EndNote Varnhorn W (September 1, 2022) On the Poisson equation in exterior domains. Constructive Mathematical Analysis 5 3 134–140.
IEEE W. Varnhorn, “On the Poisson equation in exterior domains”, CMA, vol. 5, no. 3, pp. 134–140, 2022, doi: 10.33205/cma.1143800.
ISNAD Varnhorn, Werner. “On the Poisson Equation in Exterior Domains”. Constructive Mathematical Analysis 5/3 (September 2022), 134-140. https://doi.org/10.33205/cma.1143800.
JAMA Varnhorn W. On the Poisson equation in exterior domains. CMA. 2022;5:134–140.
MLA Varnhorn, Werner. “On the Poisson Equation in Exterior Domains”. Constructive Mathematical Analysis, vol. 5, no. 3, 2022, pp. 134-40, doi:10.33205/cma.1143800.
Vancouver Varnhorn W. On the Poisson equation in exterior domains. CMA. 2022;5(3):134-40.