Work

Critical Points of Solutions of Elliptic Equations in Divergence Form in Planar Non-simply-connected Domains with Smooth or Nonsmooth Boundary

Year: 2026

Type: article

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 [G. Alessandrini and R. Magnanini, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), pp. 567–589] 4 for elliptic equations with the representation formula (using quasi-conformal mappings) for operators in divergence form in simply connected domains [G. Alessandrini and R. Magnanini, SIAM J. Math. Anal., 25 (1994), pp. 1259–1268] 5 . 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.

Source: SIAM Journal on Mathematical Analysis

DOI: 10.1137/25m1765705

Authors: Rolando Magnanini, Serge Nicaise, Madeline Chauvier

Institutions: University of Florence, Centre National de la Recherche Scientifique, Université de Valenciennes et du Hainaut-Cambrésis, Université Polytechnique Hauts-de-France, University of Mons

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni

Award: G1962231944