Reinforced Maximum Principle

Question

Let $U\subset{\mathbb R}^n$ be a bounded open domain with smooth boundary. I assume that $U$ is diffeomorphic to a ball. You may think of $L=\Delta$ and $U$ is the unit ball.

Let $L=\operatorname{div}(A\nabla\cdot)$ be a uniformly elliptic operator over $U$ with smooth coefficients. I consider solutions of the Dirichlet boundary-value problem $$Lu=0\quad\hbox{in }\,U,\qquad u=g\quad\hbox{over }\,\partial U.$$

Let me assume that $g$ is a $C^1$-function, with only two critical points $a_\pm$, at which it reaches its minimum and maximum.

From Hopf's Maximum Principle, we know that on the one hand $u$ has no non local minimum/maximum in $U$, and on the other hand $\pm\nabla u(a_\pm)\cdot\vec n>0$, where $\vec n$ is the unit outer normal to $\partial U$. In particular $u$ has no critical point on the boundary.

Is it true that $u$ has no critical point at all in $U$ ?

Observation : suppose $n=2$, and that $u$ is a Morse function. Then the claim seems to be a consequence of the Morse theory, plus the fact that all critical points have index $1$.

Answer 1

As soon as $n=2$, and $A$ is a symmetric matrix, the answer is surely yes: $u$ has no critical points in $U$. Indeed, we can have a look at what is possibly (to my knowledge) the most recent paper on these matters, i.e. [1], by Giovanni Alessandrini. He considers the more general elliptic equation (not necessarily in divergence form) $$ \begin{cases} \displaystyle\sum_{i,j=1}^2 a_{ij} u_{x_ix_j} + \sum_{i=1}^2 b_i u_{x_i} =0 & \text{on }\Omega \\ \displaystyle u= g & \text{on }\partial{\Omega} \end{cases}\label{1}\tag{1} $$ where $\Omega$ is a bounded domain, $a_{ij}=a_{ji}\in C^1(\Omega)$, $b_i\in C^0(\Omega)$, and $g\in C^0(\bar\Omega)$ and proves the following theorem.

Theorem 1.1 ([1], §1, p. 232). Let $u \in W^2_\text{loc}(\Omega)\cap C^0(\bar\Omega)$ be a solution of the Dirichlet problem \eqref{1}. If $g|_{\partial\Omega}$ has $N$ maxima (and $N$ minima), then the interior critical points of $u$ are finite in number and, denoting by $m_l,\ldots,m_K$ their multiplicities, the following estimate holds $$ \sum_{i=1}^K m_i\le N-1\label{2}\tag{2} $$ And as we see, if $N=1$ i.e. $g|_{\partial\Omega}$ has only two critical points $a_\pm\in\partial\Omega$ where it respectively reach its maximum value and its minimum value, then as a consequence of \eqref{2} it must necessarily be $K=0$ i.e. $u$ has no critical points on the interior of $\Omega$.

Addendum: some further notes.

• Magnanini in his survey [A2] seems to confirm my statement at the begin of the begin of the answer, i.e. that despite being published in 1987, [1] is still the state of the art study on the analysis of critical points of solutions to the Dirichlet problem for general linear elliptic operators.

• On the other and, for the equation $$ \operatorname{div}(\sigma(x)\nabla u)=0, $$ in their report [A1] Alberti, Bal and Di Cristo show that if the space dimension is bigger than $3$ the knowledge of the maximum at the boundary alone is not sufficient to conclude something about on the structure of the critical points at the interior. In their own words

  We show that the situation is different in dimension $n \ge 3$. 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 $X$.

Reference

[1] Giovanni Alessandrini, "Critical points of solutions of elliptic equations in two variables", Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 2, 229-256 (1987), MR939628, Zbl 0649.35026.

Addendum references

[A1] Giovanni S. Alberti and Guillaume Bal and Michele Di Cristo, "Critical Points for Elliptic Equations with Prescribed Boundary Conditions", Seminar for Applied Mathematics, ETH Zürich, Research Report No. 2016-50}, Switzerland, (November 2016), published also on the Archive for Rational Mechanics and Analysis (MR3686001, Zbl 1378.35099).

[A2] Rolando Magnanini, "An introduction to the study of critical points of solutions of elliptic and parabolic equations", Rendiconti dell’Istituto di Matematica dell’Università di Trieste 48, 121-166 (2016), MR3592440, Zbl 1408.35006.