We consider the problems of C1 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 C1-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.

Original languageEnglish
Pages (from-to)35-50
Number of pages16
JournalProceedings of the Steklov Institute of Mathematics
Volume298
Issue number1
DOIs
StatePublished - 1 Aug 2017

    Scopus subject areas

  • Mathematics (miscellaneous)

ID: 86669176