Boundary value problems for a class of planar complex vector fields

A Runge theorem for generalized analytic functions

“…We begin by describing the class of its principal part, that is, the degenerate elliptic complex vector fields 𝐿. This class of complex fields was first considered in [3] (see also [7]). Let 𝐿 = 𝑎(𝑥, 𝑦)𝜕∕𝜕𝑥 + 𝑏(𝑥, 𝑦)𝜕∕𝜕𝑦 (2.1) be a locally integrable nonvanishing field defined on the open set  ⊂ ℝ 2 with local first integrals 𝑍(𝑥, 𝑦) of Hölder class 𝐶 1+𝛼 , in particular, 𝐿 has Hölder complex coefficients of class 𝐶 𝛼 , for some 0 < 𝛼 < 1.…”

“…The class of complex vector fields 𝐿 that we deal with in this work has been considered in several recent papers in connection with various topics, including the study of the Riemann-Hilbert problem, the similarity principle, the solvability of the equation 𝐿𝑢 = 𝑓 in the torus, and strong uniqueness results for first-order equations [3][4][5][6][7]13]. It is a class of degenerate elliptic vector fields, which enjoy the following key feature: For each point 𝗉, either 𝐿 is elliptic at 𝗉 or else there exist local coordinates (𝑠, 𝑡) centered at 𝗉 such that, near 𝗉, 𝐿 may be expressed as a nonzero multiple of the finite-type vector field…”

A Lojasiewicz Inequality in Hypocomplex Structures of $\mathbb{R}^2$

“…A generalization of the Cauchy integral operator for vector fields in two variables appeared in papers [16], [17], [18] and then in [10] and [11]. Here we use the generalized Cauchy operator for real analytic hypocomplex structures given by a vector filed (2.…”

“…The questions addressed in this paper are related to those contained in papers: [1], [3], [4], [5], [6], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [23] and others.…”

The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

“…However, the boundary value problems in the above-mentioned references are all scalar and little work has been published for vector systems [18][19][20]. Motivated by the above work, in this article, we discuss the singular perturbations of third-order nonlinear differ In order to study SPBVP (1.1), (1.2), we need to study the following nonlinear unperturbed vector multi-point boundary value problem:…”