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

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.S
[2] G. Alessandrini and R. Magnanini. Elliptic equations in divergence form, geometric criticalpoints of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal., 25(5):1259–1268, 1994.