On the existence of solutions of the Beltrami equations with conditions on inverse dilatations

Abstract

We have found one of possible conditions under which the degenerate Beltrami equation has a continuous solution of the Sobolev class. This solution is Hölder continuous in the ”weak” (logarithmic) sense with the exponent power 𝛼 = 1/2. Moreover, it belongs to the class \( {W}_{\mathrm{loc}}^{1,2}. \) Under certain additional requirements, it can also be chosen as a homeomorphic solution. We give an appropriate example of the equation that satisfies all the conditions of the main result of the article, but does not have a homeomorphic Sobolev solution.