Geometric Properties of the Zeros of the Derivatives of Solutions to Chaplygin Quasilinear Equations on Ellipticity Domains

摘要

We consider the zeros of the derivatives of solutions to Chaplygin quasilinear elliptic equations and lines going from these zeros on which the derivatives take zero values. We show that each zero line reaches the boundary of the domain under examination not passing through the other zeros of derivatives, which makes it possible to estimate the number of zeros (with orders takes into account) in terms of the number of zeros of derivatives on the boundary. The zeros of derivatives are interesting because, in bounded domains, the points at which all derivatives vanish coincide with the branch points of the level lines of the dependent variables (the infinite point must be considered separately). In turn, the presence of branch points and their orders determine, to a large extent, the topology of the level lines of the functions under examination and other properties of the solution under consideration.