Degenerate Cases of Uniform Approximation by Solutions of Systems with Surjective Symbols
Canadian journal of mathematics, Tome 45 (1993) no. 4, pp. 740-757

Voir la notice de l'article provenant de la source Cambridge University Press

We prove that each (vector-valued) function in Sobolev space on a compact set K, which in the interior K 0 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
DOI : 10.4153/CJM-1993-042-5
Mots-clés : 35A35, 31B35, elliptic systems, surjective symbols, uniform approximation
Gauthier, P. M.; Tarkhanov, N. N. Degenerate Cases of Uniform Approximation by Solutions of Systems with Surjective Symbols. Canadian journal of mathematics, Tome 45 (1993) no. 4, pp. 740-757. doi: 10.4153/CJM-1993-042-5
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[1] 1. Bagby, T., Quasi topologies and rational approximation, J. Funct. Anal. 10(1972), 259–268. Google Scholar

[2] 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] 3. Bers, L., An approximation theorem, J. Analyse Math. 14(1965), 1–4. Google Scholar

[4] 4. Deny, J., Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier, Grenoble 1(1949), 103–113. Google Scholar

[5] 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] 6. Hedberg, L. I., Non-linear potentials and approximations in the mean, Math. Z. 129(1972), 299–319. Google Scholar

[7] 7. Hedberg, L. I., Two approximation problems in function spaces, Ark. Mat. 16(1978), 51–81. Google Scholar

[8] 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] 9. Hedberg, L. I., Approximation by harmonic functions, and stability of the Dirichlet problem, Exposition. Math., to appear. Google Scholar

[10] 10. Hedberg, L. I. and Wolff, T. H., Thin sets in nonlinear potential theory, Ann. Inst. Fourier 33( 1983), 161–187. Google Scholar

[11] 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] 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] 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] 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] 15. Poiking, J. C., Approximation in LP by solutions of elliptic partial differential equations, Amer. J. Math. 94( 1972), 1231–1244. Google Scholar

[16] 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] 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] 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] 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] 21. Verdera, J., Approximation by rational modules in Sobolev and Lipschitz norms, J. Funct. Anal. 58(1984), 267–290. Google Scholar

[22] 22. Verdera, J., Cm Approximation by solutions of elliptic equations, and Calderón-Zygmund operators, Duke Math. J. 55(1987), 157–187. Google Scholar

[23] 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] 24. Wiener, N., The Dirichletproblem, J. Math. Phys. Mass. Inst. Tech. 3(1924), 127–146. Google Scholar

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