Journal of Mathematical Sciences, Vol. 190, No. 4, April, 2013 Dirichlet problem for the general Beltrami equation in Jordan domains Bogdan Bojarski, Vladimir Gutlyanskiˇ ı, and Vladimir Ryazanov Abstract. We study the Dirichlet problem for the general degenerate Beltrami equations ∂f = μ∂f + ν ∂f in a Jordan domain. Some criteria for the existence of regular solutions are given. Keywords. Beltrami equation, Dirichlet problem, regular solution. 1. Introduction Let D be a domain in the complex plane C. Throughout this paper, we use the notation z = x + iy, B(z 0 ,r) := {z ∈ C : |z - z 0 | <r} for z 0 ∈ C and r> 0, B(r) := B(0,r), B := B(1), and C := C ∪∞. The purpose of this paper is to study the Dirichlet problem ⎧ ⎨ ⎩ f z = μ(z ) · f z + ν (z ) · f z , z ∈ D, lim z→ζ Re f (z )= ϕ(ζ ), ∀ ζ ∈ ∂D, (1.1) in a Jordan domain D of the complex plane C with continuous boundary data ϕ(ζ ) ≡ const. Here, μ(z ) and ν (z ) stand for measurable coefficients satisfying the inequality |μ(z )| + |ν (z )| < 1 a.e. in D. The degeneracy of the ellipticity for the Beltrami equations f z = μ(z ) · f z + ν (z ) · f z (1.2) is controlled by the dilatation coefficient K μ,ν (z ) := 1+ |μ(z )| + |ν (z )| 1 -|μ(z )|-|ν (z )| ∈ L 1 loc . (1.3) Note that the Beltrami equations of the second type f z = ν (z ) · f z take a key part in many problems of mathematical physics (see, e.g., [19]). We will look for a solution of the Dirichlet problem (1.1) as a continuous, discrete, and open mapping f : D → C of the Sobolev class W 1,1 loc and such that the Jacobian J f (z ) =0 a.e. in D. Such a solution is called a regular solution of the Dirichlet problem (1.1) in a domain D. Recall that a mapping f : D → C is called discrete, if the preimage f -1 (y) consists of isolated points for every y ∈ C, and open, if f maps every open set U ⊆ D onto an open set in C. In the uniformly elliptic case, i.e., where K μ,ν (z ) ≤ K< ∞ a.e. in D, the Dirichlet problem was studied in [2] and [31]. The solvability of the Dirichlet problem in the partial case where ν (z )=0, and the degeneracy of the ellipticity for the Beltrami equations of the first type f z = μ(z ) · f z (1.4) Translated from Ukrains’ki˘ ı Matematychny˘ ı Visnyk, Vol. 9, No. 4, pp. 460–476, October–November, 2012. Original article submitted October 10, 2012 1072 – 3374/13/1904–0525 c  2013 Springer Science+Business Media New York 525