Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T06:01:43.056Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounded Pointwise Approximation of Solutions of Elliptic Equations

Published online by Cambridge University Press:  20 November 2018

A. Bonilla  and
R. Trujillo-González
A. Bonilla
Affiliation:
Departamento de Análisis Matemático Universidad de La Laguna 38271 La Laguna, Tenerife Spain
R. Trujillo-González
Affiliation:
Departamento de Análisis Matemático Universidad de La Laguna 38271 La Laguna, Tenerife Spain
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize open subsets U of ℝN in which the bounded solutions of certain elliptic equations can be approximated pointwise by uniformly bounded solutions that are continuous in Ū. This result is established in terms of certain capacities. For closed subsets X, this characterization allows us to approximate bounded solutions in X° uniformly on relatively closed subsets of X° by solutions continuous on certain subsets of the boundary of X.

Keywords

35A3541A3035J3031B15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 496 - 511
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Bagby, T., Approximation in the mean by solutions of elliptic equations, Trans. Amer. Math. Soc. 281 (1984), 761–784.Google Scholar
2. Boivin, A. and Verdera, J., Approximation parfonctions holomorphes dans les espaces LP, Lip α et BMO, Indiana Univ. Math. J. 40 (1991), 393–418.Google Scholar
3. Bourbaki, N.,Variétés Differentielles et Analytiques, Fascicule de résultats, Actualités Sci. Indust. 1333, 1347. Hermann, Paris, 1967. 1971.Google Scholar
4. Davie, A., Analytic capacity and approximation problems, Trans. Amer. Math. Soc. 173( 1972), 409–444.Google Scholar
5. Gamelin, T.W.,Uniform Algebras, Chelsea Publishing Company, New York, 1984.Google Scholar
6. Gamelin, T.W. and Garnett, J., Constructive techniques in rational approximation, Trans. Amer. Math. Soc. 143 (1969), 187–200.Google Scholar
7. Hadjiiski, N.H.,Vitushkin s type theorems for meromorphic approximation on unbounded sets, Proc. Conf. Complex Analysis and Applications 1981, Varna, 229–238. Bulgarian Acad. Sci., Sofia, 1984.Google Scholar
8. John, F.,Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955.Google Scholar
9. O'Farrell, A.G., Rational approximation in Lipschitz norm II, Proc. Roy. Irish Acad. Sect. A 79 (1979), 104–114.Google Scholar
10. Stein, E.M.,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970.Google Scholar
11. Stein, E.M. and Weiss, G.,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, New Jersey, 1975.Google Scholar
12. Stray, A., Approximation and interpolation, Pacific J. Math. 40 (1972), 463–475.Google Scholar
13. Tarkhanov, N.N., Uniform approximation by solutions of elliptic systems, Math. USSR-Sb. 61 (1988), 351–377.Google Scholar
14. Tarkhanov, N.N., Laurent expansion and local properties of solutions of elliptic systems, Siberian Math. J. 29 (1988), 970–979.Google Scholar
15. Verdera, J., C approximation by solutions of elliptic equations and Calderon-Zygmund operators, Duke Math. J. 55 (1987), 157–187.Google Scholar
16. Vitushkin, A.G., Analytic capacity of sets in problems of approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199. English transl, Russian Math. Surveys 22 (1967), 139–200.Google Scholar