Uniform approximability of functions by polynomial solutions of second-order elliptic equations on compact plane sets

On Dirichlet problem for second-order elliptic equations in the plane and uniform approximation problems for solutions of such equations

“…We present the new proof of the criterion for uniform approximability of functions by L-analytic polynomials on boundaries of Carathéodory domains, see Theorem 3. This result was firstly obtained in [39], but the proof given there is rather involved technically and, moreover, it is not enough complete in a certain place. As a consequence of Theorem 3 one can show that weak maximum modulus principle (i.e.…”

“…where c ∈ C, β ∈ R, |β| 1, and ∂ β = ∂ x + iβ∂ y . This representation was used in several papers and it turned out to be quite useful (see, for instance, [40] and [41]), but we need to modify it a bit more to get a more simple notation system which allows one to distinguish strongly elliptic and not strongly elliptic cases in a more clear way. Note that β −1 if and only if L ∈ SE, while β 1 if and only if L ∈ NSE.…”

“…To prove the second statement let us observe that the function Φ can be chosen in such a way that is has zeros on D, but it has no zeros on T (it is enough to chose a suitable value of Φ(1)). It can be readily verified (see [39], the proof of Corollary 1, for details), that the function 1/F (z) in µ-integrable over ∂ a G and…”

“…We are going to present one criterion in order that the equality A L (∂G, G) = C(∂G) holds for a given Carathéodory domain G (see Theorem 3 below). This result was firstly obtained in [39], but the proof given therein is not enough complete in a certain place. To state Theorem 3 we need to introduce yet another space of functions and give one more definition.…”

“…This theorem was firstly proved in [7] for bianalytic functions, and almost immediately after that the proof was modified for general L in [40]. Theorem G allows us to reduce the problem of L-analytic polynomial approximation on a given compact set X to compact subsets of X having more simple topological structure.…”