Critical points of solutions of elliptic equations in divergence form in planar non simply connected domains with smooth or nonsmooth boundary

TL;DR

This paper investigates the critical points of solutions to second-order elliptic equations in divergence form within planar non-simply connected domains, employing complex analysis, quasi-conformal mappings, and numerical experiments to understand their behavior.

Contribution

It introduces a novel approach combining the argument principle and quasi-conformal mappings to analyze critical points in complex elliptic problems, including degenerate coefficients.

Findings

Critical points are characterized using the argument principle.

The method applies to degenerate coefficients and nonsmooth boundaries.

Numerical experiments illustrate the theoretical results.

Abstract

We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on its normal derivatives) along the boundary. The proof combines the argument principle introduced in [1] for elliptic equations with the representation formula (using quasi-conformal mappings) for operators in divergence form in simply connected domains [2]. The case of a degenerate coefficient is also treated where we combine the level lines technique and the maximum principle with the argument principle. Finally, some numerical experiments on illustrative examples are presented. [1] G. Alessandrini and R. Magnanini. The index of isolated critical points and solutions of elliptic equations in the plane. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19(4):567-589, 1992 [2] G. Alessandrini and R. Magnanini. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal., 25(5):1259-1268, 1994