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

Rolando Magnanini, Serge Nicaise, Madeline Chauvier

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

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

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
Paper Structure (17 sections, 12 theorems, 77 equations, 11 figures)

This paper contains 17 sections, 12 theorems, 77 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem and assumptions
  3. The critical points are isolated
  4. Case of a $C^{1,1}$ exterior boundary
  5. Case of a piecewise $C^{1,1}$ exterior boundary
  6. Number of critical points inside the domain
  7. Degenerate density in case of piecewise $C^{1,1}$ exterior boundary
  8. Critical points are isolated
  9. Number of critical points inside the domain
  10. Checking the Hopf lemma on the exterior/interior boundary
  11. Numerical illustrations
  12. An annulus
  13. The unit disc with three holes
  14. A half-disc with one hole
  15. An half-disc with three holes
  16. ...and 2 more sections

Key Result

Lemma 3.2

Assume that the solution $u$ of EDPinu-bcu belongs to $C^1(\bar{\Omega})$ and satisfies assumderiveenormaleextbdy and assumderiveenormaleintbdy. If$\partial \mathcal{O}$ is $C^{1,1}$, then$u$ has a finite number of critical points in $\Omega$ (and no critical point on $\partial \Omega$).

Figures (11)

  • Figure 1: An illustrative configuration
  • Figure 2: An annulus and a smooth $\rho$: 20 level lines (left) and a gradient map (right)
  • Figure 3: An annulus and a nonsmooth $\rho$: 20 level lines (left) and a gradient map (right)
  • Figure 4: The unit ball with 3 holes and a smooth $\rho$: 20 level lines (left) and a gradient map (right)
  • Figure 5: The unit ball with 3 holes and a degenerate $\rho$: 20 level lines (left) and a gradient map (right)
  • ...and 6 more figures

Theorems & Definitions (30)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 4.1
  • Lemma 4.2
  • ...and 20 more