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Approximation in the mean by solutions of elliptic equations. (English) Zbl 0538.35029

Let L be a homogeneous polynomial of degree \(m\geq 1\) in \(\underset \tilde{} R^ n\), with complex coefficients, such that \(L(\xi)\neq 0\) if \(\xi\neq 0\), and either \(m<n\) or n is odd. Let L(D) be the associated partial differential operator, let \(1<p<\infty\), and let X be a compact subset of \(\underset \tilde{} R^ n\) with positive Lebesgue measure. The author provides necessary and sufficient conditions for the following statement to hold: If \(f\in L_ p(X)\) and \(L(D)f=0\) on the interior of X, then for each \(\epsilon>0\) there exists a solution u of \(L(D)u=0\) on a neighbourhood of X such that \(\| u-f\|_{p,X}<\epsilon.\) The main result is analogous to one obtained by A. G. Vitushkin [Usp. Mat. Nauk 22, No.6(138), 141-199 (1967; Zbl 0157.394)], and involves certain capacities which are similar to ones defined by V. G. Maz’ya [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 356-385 (1973; Zbl 0256.35065)].
Reviewer: N.A.Watson

MSC:

35J30 Higher-order elliptic equations
35A35 Theoretical approximation in context of PDEs
31C15 Potentials and capacities on other spaces
41A30 Approximation by other special function classes
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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[1] David R. Adams and John C. Polking, The equivalence of two definitions of capacity, Proc. Amer. Math. Soc. 37 (1973), 529 – 534. · Zbl 0251.31005
[2] I. Babuska, Stability of the domain with respect to the fundamental problems in the theory of partial differential equations, mainly in connection with the theory of elasticity. I, II, Czechoslovak Math. J. 11 (86) (1961), 76-105, 165-203. (Russian)
[3] Thomas Bagby, Quasi topologies and rational approximation, J. Functional Analysis 10 (1972), 259 – 268. · Zbl 0266.30024
[4] V. I. Burenkov, The approximation of functions in the space \?_{\?}^{\?}(\Omega ) by compactly supported functions for an arbitrary open set \Omega , Trudy Mat. Inst. Steklov. 131 (1974), 51 – 63, 245 (Russian). Studies in the theory of differentiable functions of several variables and its applications, V.
[5] A. P. Calderón, Lecture notes on pseudo differential operators and elliptic boundary value problems, Cursos de Matemática 1, Instituto Argentino de Matematica, Buenos Aires, 1976.
[6] Lennart Carleson, Mergelyan’s theorem on uniform polynomial approximation, Math. Scand. 15 (1964), 167 – 175. · Zbl 0163.08601 · doi:10.7146/math.scand.a-10741
[7] Claes Fernström and John C. Polking, Bounded point evaluations and approximation in \?^{\?} by solutions of elliptic partial differential equations, J. Functional Analysis 28 (1978), no. 1, 1 – 20. · Zbl 0396.35037
[8] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. · Zbl 0213.40401
[9] Reese Harvey and John C. Polking, A notion of capacity which characterizes removable singularities, Trans. Amer. Math. Soc. 169 (1972), 183 – 195. · Zbl 0249.35012
[10] Reese Harvey and John C. Polking, A Laurent expansion for solutions to elliptic equations, Trans. Amer. Math. Soc. 180 (1973), 407 – 413. · Zbl 0285.35024
[11] V. P. Havin, Approximation by analytic functions in the mean, Dokl. Akad. Nauk SSSR 178 (1968), 1025 – 1028 (Russian).
[12] Lars Inge Hedberg, Approximation in the mean by analytic functions, Trans. Amer. Math. Soc. 163 (1972), 157 – 171. · Zbl 0236.30045
[13] Lars Inge Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299 – 319. · Zbl 0236.31010 · doi:10.1007/BF01181619
[14] Lars Inge Hedberg, Approximation in the mean by solutions of elliptic equations, Duke Math. J. 40 (1973), 9 – 16. · Zbl 0283.35035
[15] Lars Inge Hedberg, Removable singularities and condenser capacities, Ark. Mat. 12 (1974), 181 – 201. · Zbl 0297.30017 · doi:10.1007/BF02384755
[16] Lars Inge Hedberg, Two approximation problems in function spaces, Ark. Mat. 16 (1978), no. 1, 51 – 81. · Zbl 0399.46023 · doi:10.1007/BF02385982
[17] Lars Inge Hedberg, Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981), no. 3-4, 237 – 264. · Zbl 0504.35018 · doi:10.1007/BF02392874
[18] L. I. Hedberg and T. H. Wolff, Thin sets in nonlinear potential theory (to appear). · Zbl 0508.31008
[19] L. Hörmander, Linear partial differential operators, Academic Press, New York, 1963. · Zbl 0108.09301
[20] Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. · Zbl 0067.32101
[21] M. V. Keldych, On the solubility and the stability of Dirichlet’s problem, Uspekhi Matem. Nauk 8 (1941), 171 – 231 (Russian).
[22] P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 747 – 766. · Zbl 0072.33004 · doi:10.1002/cpa.3160090407
[23] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. · Zbl 0253.31001
[24] P. Lindberg, \( L^p\)-approximation by analytic functions in an open region, Uppsala Univ. Dept. of Math. Report No. 1977:7.
[25] Per Lindberg, A constructive method for \?^{\?}-approximation by analytic functions, Ark. Mat. 20 (1982), no. 1, 61 – 68. · Zbl 0495.30031 · doi:10.1007/BF02390498
[26] Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble 6 (1955 – 1956), 271 – 355 (French). · Zbl 0071.09002
[27] V. G. Maz\(^{\prime}\)ja, The (\?,\?)-capacity, imbedding theorems and the spectrum of a selfadjoint elliptic operator, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 356 – 385 (Russian).
[28] S. N. Mergeljan, Uniform approximations to functions of a complex variable, Uspehi Mat. Nauk 7 (1952), 31-122; English transl., Amer. Math. Soc. Transl. (1) 101 (1954). · Zbl 0049.32701
[29] A. G. O’Farrell, Metaharmonic approximation in Lipschitz norms, Proc. Roy. Irish Acad. Sect. A 75 (1975), no. 24, 317 – 330. · Zbl 0324.46026
[30] John C. Polking, Approximation in \?^{\?} by solutions of elliptic partial differential equations, Amer. J. Math. 94 (1972), 1231 – 1244. · Zbl 0266.35026 · doi:10.2307/2373572
[31] John C. Polking, A Leibniz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21 (1971/72), 1019 – 1029. · Zbl 0242.47029 · doi:10.1512/iumj.1972.21.21082
[32] C. Runge, Zur Theorie der Eindeutigen Analytischen Functionen, Acta Math. 6 (1885), no. 1, 229 – 244 (German). · JFM 17.0379.01 · doi:10.1007/BF02400416
[33] È. M. Saak, A capacity criterion for a domain with a stable Dirichlet problem for higher order elliptic equations, Mat. Sb. (N.S.) 100(142) (1976), no. 2, 201 – 209, 335 (Russian).
[34] Georgi E. Shilov, Generalized functions and partial differential equations, Authorized English edition revised by the author. Translated and edited by Bernard D. Seckler, Gordon and Breach, Science Publishers, Inc., New York-London-Paris, 1968.
[35] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[36] A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141 – 199 (Russian).
[37] Barnet M. Weinstock, Uniform approximation by solutions of elliptic equations, Proc. Amer. Math. Soc. 41 (1973), 513 – 517. · Zbl 0273.35032
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