Critical Points for Elliptic Equations with Prescribed Boundary Conditions

TL;DR

This paper investigates the existence and behavior of critical points of solutions to elliptic equations with prescribed boundary conditions, revealing differences between two and higher spatial dimensions and implications for inverse problems.

Contribution

It demonstrates that in dimensions three and higher, critical points can be forced to exist for certain coefficients, contrasting with the two-dimensional case, impacting stability in inverse problems.

Findings

Critical points can be enforced in dimensions n≥3 for certain coefficients.

In 2D, the number of critical points relates to boundary oscillations.

High-contrast coefficients affect the topology of the gradient field.

Abstract

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension $n=2$, it is known that the number of critical points (where $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $\sigma$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $\sigma(x)$ such that $\nabla u$ vanishes at least at one point in $X$. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $\sigma(x)$. These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $\sigma(x)$ for which the stability of the reconstructions will inevitably degrade.