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Vol. 61, no. 1 (2020)
of elliptic equations of even order with Hölder coefficients.
We consider elliptic equations of order $2m$ with Hölder coefficients. We
show local solvability of the Dirichlet problem with $m$ conditions on the
boundary of the upper half space. First we consider local solvability in free
space and then we treat the boundary case. Our method is based on applying the
operator to an approximate solution and iterating in the Hölder spaces. A
priori estimates for the approximate solution is the essential part of the
of subspaces generated by dilations of polynomials. An application to best
We study the convergence of a net of subspaces generated by dilations of
polynomials in a finite dimensional subspace. As a consequence, we extend the
results given by Zó and Cuenya [Advanced Courses of Mathematical Analysis
II (Granada, 2004), 193–213, World Scientific, 2007] on a general
approach to the problems of best vector-valued approximation on small regions
from a finite dimensional subspace of polynomials.