On the theory of convergence and compactness for Beltrami equations

Abstract

Theorems on convergence and compactness are proved for the classes of regular solutions of degenerate Beltrami equations with restrictions of integral type imposed on the dilatation.

To the theory of variational method for Beltrami equations

“…Then it is obvious that µ τ (z) = µ(z 0 + τ z), K µτ (z) = K µ (z 0 + τ z), and the approximative continuity of µ(z) at the point z 0 is equivalent to the convergence in measure µ τ → µ 0 = µ(z 0 ) as τ → 0. Thus, h τ → h 0 as τ → 0 by virtue of Theorem 2 and Lemma 2 in [24] with regard for (6.10), (6.5), and Corollary 6.1, cf. the estimate of the integral I τ in the proof of Theorem 6.1.…”

“…We note also that f τ do not take values of 1 and ∞ inside D. Therefore, f τ , τ ∈ (0, τ 0 ], form a normal family (see Theorem 2 in [24]). Thus, f τ , τ ∈ (0, τ 0 ], is the equicontinuous family by Proposition 7.1 in [26].…”

“…The mentioned approach to the construction of variations uses the convexity of the set of complex coefficients, which turns out to be the common property of compact classes of regular solutions of the Beltrami equation. The criteria of existence and compactness of classes of regular solutions and other references can be found in [4,5,7,18,19,[24][25][26][30][31][32][33]. Everywhere below, D(z 0 , r) = {z ∈ C : |z − z 0 | < r}, D(r) = D(0, r), D = D(0, 1), dist (E, F ) = sup x∈E, y∈F |x−y| is the Euclidean distance between the sets E and F in C, mes E is the Lebesgue measure of the set E ⊂ C, dm(z) corresponds to the Lebesgue measure in C, and dS(z) = 1 + |z| 2 −2 dm(z) stands for the element of a spherical area in C.…”

On global behavior of mappings with integral constraints

“…However, let us note that it is in the form given below that the indicated statement seems to be the most interesting from the point of view of applications to the problem of compactness of solutions of the Beltrami equations and the Dirichlet problem (see, for example, [Dyb,Theorem 2]) and [L,Theorem 1]).…”

On compact classes of solutions of Dirichlet problem in simply connected domains

“…Note that the relation (1.5) holds for the corresponding function K µ = K µ f (see e.g. [L,Lemma 1]). Therefore, f ∈ F M ϕ,Φ,z 0 (D).…”