We consider the problems of C 1 approximation of functions by polynomial solutions and by solutions with localized singularities of homogeneous elliptic second-order systems of partial differential equations on compact subsets of the plane ℝ2. We obtain a criterion of C 1-weak polynomial approximation which is analogous to Mergelyan’s criterion of uniform approximability of functions by polynomials in the complex variable. We also discuss the problem of uniform approximation of functions by solutions of the above-mentioned systems. Moreover, we consider the Dirichlet problem for systems that are not strongly elliptic and prove a result on the lack of solvability of such problems for any continuous boundary data in domains whose boundaries contain analytic arcs.
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J. J. Carmona, “Mergelyan’s approximation theorem for rational modules,” J. Approx. Theory 44 (2), 113–126 (1985).
J. J. Carmona, P. V. Paramonov, and K. Yu. Fedorovskiy, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions,” Mat. Sb. 193 (10), 75–98 (2002) [Sb. Math. 193, 1469–1492 (2002)].
L. K. Hua, W. Lin, and C.-Q. Wu, Second-Order Systems of Partial Differential Equations in the Plane (Pitman Adv. Publ. Program, Boston, 1985), Res. Notes Math. 128.
M. Ya. Mazalov, “A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations,” Mat. Sb. 199 (1), 15–46 (2008) [Sb. Math. 199, 13–44 (2008)].
M. Ya. Mazalov, “The Dirichlet problem for polyanalytic functions,” Mat. Sb. 200 (10), 59–80 (2009) [Sb. Math. 200, 1473–1493 (2009)].
M. Ya. Mazalov, P. V. Paramonov, and K. Yu. Fedorovskiy, “Conditions for C m-approximability of functions by solutions of elliptic equations,” Usp. Mat. Nauk 67 (6), 53–100 (2012) [Russ. Math. Surv. 67, 1023–1068 (2012)].
P. V. Paramonov, “On harmonic approximation in the C1-norm,” Mat. Sb. 181 (10), 1341–1365 (1990) [Math. USSR, Sb. 71 (1), 183–207 (1992)].
P. V. Paramonov, “On approximation by harmonic polynomials in the C 1-norm on compact sets in R2,” Izv. Ross. Akad. Nauk, Ser. Mat. 57 (2), 113–124 (1993) [Russ. Acad. Sci. Izv. Math. 42 (2), 321–331 (1994)].
P. V. Paramonov, “Cm-approximations by harmonic polynomials on compact sets in Rn,” Mat. Sb. 184 (2), 105–128 (1993) [Russ. Acad. Sci. Sb. Math. 78 (1), 231–251 (1994)].
P. V. Paramonov and K. Yu. Fedorovskiy, “Uniform and C1-approximability of functions on compact subsets of R2 by solutions of second-order elliptic equations,” Mat. Sb. 190 (2), 123–144 (1999) [Sb. Math. 190, 285–307 (1999)].
P. V. Paramonov and J. Verdera, “Approximation by solutions of elliptic equations on closed subsets of Euclidean space,” Math. Scand. 74 (2), 249–259 (1994).
I. G. Petrovsky, Lectures on Partial Differential Equations (Nauka, Moscow, 1961; Dover Publ., New York, 1991).
G. C. Verchota and A. L. Vogel, “Nonsymmetric systems on nonsmooth planar domains,” Trans. Am. Math. Soc. 349 (11), 4501–4535 (1997).
J. Verdera, “Cm approximation by solutions of elliptic equations, and Calderón–Zygmund operators,” Duke Math. J. 55 (1), 157–187 (1987).
J. Verdera, “On the uniform approximation problem for the square of the Cauchy–Riemann operator,” Pac. J. Math. 159 (2), 379–396 (1993).
A. B. Zaitsev, “Uniform approximation by polynomial solutions of second-order elliptic equations, and the corresponding Dirichlet problem,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 253, 67–80 (2006) [Proc. Steklov Inst. Math. 253, 57–70 (2006)].
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 298, pp. 42–57.
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Bagapsh, A.O., Fedorovskiy, K.Y. C 1 approximation of functions by solutions of second-order elliptic systems on compact sets in ℝ2 . Proc. Steklov Inst. Math. 298, 35–50 (2017). https://doi.org/10.1134/S0081543817060037