A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations

M. Ya. Mazalov

Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 13-44

Résumé

Let $X$ be an arbitrary compact subset of the plane. It is proved that if $L$ is a homogeneous elliptic operator with constant coefficients and locally bounded fundamental solution, then each function $f$ that is continuous on $X$ and satisfies the equation $Lf=0$ at all interior points of $X$ can be uniformly approximated on $X$ by solutions of the same equation having singularities outside $X$. A theorem on uniform piecemeal approximation of a function is also established under weaker constraints than in the standard Vitushkin scheme. Bibliography: 24 titles.

Bibliographie

[1] L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren Math. Wiss., 256, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[2] E. P. Dolženko, “Approximation on closed regions and zero-sets”, Soviet Math. Dokl., 3 (1962), 472–475 | MR | Zbl

[3] N. N. Tarkhanov, The analysis of solutions of elliptic equations, Math. Appl., 406, Kluwer Acad. Publ., Dordrecht, 1997 | MR | MR | Zbl | Zbl

[4] L. I. Hedberg, “Two approximation problems in function spaces”, Ark. Mat., 16:1 (1978), 51–81 | DOI | MR | Zbl

[5] A. G. Vitushkin, “The analytic capacity of sets in problems of approximation theory”, Russian Math. Surveys, 22:6, 139–200 | DOI | MR | Zbl | Zbl

[6] M. V. Keldych, “On the solvability and stability of the Dirichlet problem”, Amer. Math. Soc. Transl. Ser. 2, 51 (1966), 1–73 | MR | Zbl

[7] J. Deny, “Systèmes totaux de fonctions harmoniques”, Ann. Inst. Fourier (Grenoble), 1 (1949), 103–113 | MR

[8] J. Mateu, Y. Netrusov, J. Orobitg, J. Verdera, “BMO and Lipschitz approximation by solutions of elliptic equations”, Ann. Inst. Fourier (Grenoble), 46:4 (1996), 1057–1081 | MR | Zbl

[9] P. V. Paramonov, K. Yu. Fedorovskii, “Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations”, Sb. Math., 190:2 (1999), 285–307 | DOI | MR | Zbl

[10] J. Verdera, “On the uniform approximation problem for the square of the Cauchy–Riemann operator”, Pacific J. Math., 159:2 (1993), 379–396 | MR | Zbl

[11] J. Verdera, “The uniform approximation problem for the square of the Cauchy–Riemann operator”, Linear and complex analysis. Problem book 3, Lecture Notes in Math., 1574, Springer-Verlag, Berlin–Heidelberg, 1994, 122–123 | DOI | MR | Zbl

[12] M. Ya. Mazalov, “Uniform approximation of functions continuous on a compact subset of $\mathbb C$ and analytic in its interior by functions bianalytic in its neighborhoods”, Math. Notes, 69:1–2 (2001), 216–231 | DOI | MR | Zbl

[13] M. Ya. Mazalov, “Uniform approximations by bianalytic functions on arbitrary compact subsets of $\mathbb C$”, Sb. Math., 195:5–6 (2004), 687–709 | DOI | MR | Zbl

[14] R. Harvey, J. Polking, “Removable singularities of solutions of linear partial differential equations”, Acta Math., 125:1 (1970), 39–56 | DOI | MR | Zbl

[15] J. Verdera, “$C^m$ approximation by solutions of elliptic equations, and Calderón–Zygmund operators”, Duke Math. J., 55:1 (1987), 157–187 | DOI | MR | Zbl

[16] P. V. Paramonov, “On harmonic approximation in the $C^1$-norm”, Math. USSR-Sb., 71:1 (1992), 183–207 | DOI | MR | Zbl | Zbl

[17] P. V. Paramonov, “Some new criteria for uniform approximability of functions by rational fractions”, Sb. Math., 186:9 (1995), 1325–1340 | DOI | MR | Zbl

[18] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | MR | Zbl | Zbl

[19] W. Rudin, Functional analysis, McGraw-Hill, New York–Düsseldorf–Johannesburg, 1973 | MR | MR | Zbl

[20] K. Yosida, Functional analysis, Grundlehren Math. Wiss., 123, Springer-Verlag, Berlin–Göttingen–Heidelberg; Academic Press, New York, 1965 | MR | MR | Zbl | Zbl

[21] S. V. Kolesnikov, “On a theorem of M. V. Keldysh concerning pointwise convergence of a sequence of polynomials”, Math. USSR-Sb., 52:2 (1985), 553–555 | DOI | MR | Zbl

[22] G. E. Shilov, Mathematical analysis. A special cours, Pergamon Press, Oxford–New York–Paris, 1965 | MR | MR | Zbl | Zbl

[23] P. W. Jones, “Square functions, Cauchy integrals, analytic capacity, and harmonic measure”, Harmonic analysis and partial differential equations (El Escorial, Spain, 1987), Lecture Notes in Math., 1384, Springer-Verlag, Berlin–Heidelberg, 1989, 24–68 | DOI | MR | Zbl

[24] G. David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Math., 1465, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl