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Degenerate Cases of Uniform Approximation by Solutions of Systems with Surjective Symbols

Published online by Cambridge University Press:  20 November 2018

P. M. Gauthier  and
N. N. Tarkhanov
P. M. Gauthier
Affiliation:
Département de mathématiques et de satistique et Centre de recherches mathématiques Université de Montréal, CP 6128-A Montreal, Québec H3C 3J7 e-mail: [email protected]
N. N. Tarkhanov
Affiliation:
Institute of Physics and Krasnoyarsk University Rossiya
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Abstract

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We prove that each (vector-valued) function in Sobolev space on a compact set K, which in the interior K0 of K satisfies a system of differential equations, can be approximated by solutions in a neighbourhood of K plus sums of potentials of measures supported on the boundary of K. We discuss the particular case where, for all compact sets K, one can dispense with potentials in such approximations

Keywords

35A3531B35elliptic systemssurjective symbolsuniform approximation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 4 , 01 August 1993 , pp. 740 - 757
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Bagby, T., Quasi topologies and rational approximation, J. Funct. Anal. 10(1972), 259–268.Google Scholar
2. Bagby, T. and Gauthier, P. M., Approximation by harmonic functions on closed subsets ofRiemann surfaces, J. Analyse Math. 51(1988), 259–284.Google Scholar
3. Bers, L., An approximation theorem, J. Analyse Math. 14(1965), 1–4.Google Scholar
4. Deny, J., Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier, Grenoble 1(1949), 103–113.Google Scholar
5. Havin, V. P., Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR 178(1968), 1025— 1028, Russian; English transi., Soviet Math. Dokl. 9(1968), 245–252.Google Scholar
6. Hedberg, L. I., Non-linear potentials and approximations in the mean, Math. Z. 129(1972), 299–319.Google Scholar
7. Hedberg, L. I., Two approximation problems in function spaces, Ark. Mat. 16(1978), 51–81.Google Scholar
8. Hedberg, L. I., Spectral synthesis in Sobolev spaces and uniqueness of solutions of the Dirichlet problem, Acta Math. 147(1981), 235–264.Google Scholar
9. Hedberg, L. I., Approximation by harmonic functions, and stability of the Dirichlet problem, Exposition. Math., to appear.Google Scholar
10. Hedberg, L. I. and Wolff, T. H., Thin sets in nonlinear potential theory, Ann. Inst. Fourier 33( 1983), 161–187.Google Scholar
11. Keldysh, M. V., On the solvability and the stability of the Dirichlet problem, Uspekhi Mat. Nauk 8(1941 ), 171-231, Russian; English transi., Amer. Math. Soc. Transi. 51(1966), 1–73.Google Scholar
12. Mateu, J. and Orobitz, J., Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39(1990), 703–736.Google Scholar
13. Mateu, J. and Verdera, J., BMO harmonic approximation in the plane and spectral synthesis for Hardy- Sobolev spaces, Rev. Mat. Iberoamericana4(1988), 291–318.Google Scholar
14. Poiking, J. C., A Leibnitz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21(1972), 1019–1029.Google Scholar
15. Poiking, J. C., Approximation in LP by solutions of elliptic partial differential equations, Amer. J. Math. 94( 1972), 1231–1244.Google Scholar
16. Tarkhanov, N. N., Uniform approximation by solutions of elliptic systems, Mat. Sb. 133(1987), 356–381, Russian; English transi., Math. USSR-Sb 61(1988), 351–377.Google Scholar
17. Tarkhanov, N. N., approximation on compact sets by solutions of systems with surjective symbols, Prepr. Inst. Phys., Krasnoyarsk (48) M(1989), Russian, Uspekhi Mat. Nauk, to appear; English transi., Russ. Math. Surv., to appear.Google Scholar
18. Tarkhanov, N. N., Approximation in Sobolev spaces by solutions of elliptic systems, Dokl. Akad. Nauk SSSR 315 (1990), 1308-1313, Russian; English transi., Soviet Math. Dokl. 42(1991), 902–907 19 , Laurent Series for Solutions of Elliptic Systems, Nauka, Novosibirsk, 1991, Russian.Google Scholar
20. Trent, T. and J. L.-M. Wang, Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc. 81(1981), 62–64.Google Scholar
21. Verdera, J., Approximation by rational modules in Sobolev and Lipschitz norms, J. Funct. Anal. 58(1984), 267–290.Google Scholar
22. Verdera, J., Cm Approximation by solutions of elliptic equations, and Calderón-Zygmund operators, Duke Math. J. 55(1987), 157–187.Google Scholar
23. Verdera, J., On the uniform approximation problem for the square of the Cauchy-Riemann operator, Universitat Autônomade Barcelona, num. 3, 1991, preprint.Google Scholar
24. Wiener, N., The Dirichletproblem, J. Math. Phys. Mass. Inst. Tech. 3(1924), 127–146.Google Scholar