Ngô Quốc Anh

July 21, 2014

A strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:08

In this note, we consider a strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation. In the context of closed and compact Riemannian manifolds $(M,g)$ of dimension $n$, it can be stated as follows

Theorem (Maximum Principle). Let $(M,g)$ be a compact Riemannian manifold of dimension $n$. Let $u \in C^1(M)$ be such that

$\displaystyle -\Delta_g u \geqslant f(\cdot, u)$

on $M$ where $f$ is a function such that

$\displaystyle \partial_u f \leqslant 0$

(i.e. $f$ is monotone increasing w.r.t the variable $u$) and that

$\displaystyle |f(x,r)| \leqslant C(K+|r|)|r|$

for all $(x,r) \in M \times \mathbb R$ and for some constant $C>0$. If $u \geqslant 0$ in $M$ and $u$ does not vanish identically, then $u>0$ in $M$.

To prove this theorem, we need two auxiliary results. The first result is the weak comparison principle which can be stated as the following.

Lemma (Weak Comparison Principle). Suppose that $\partial_u f \leqslant 0$ in $B$. Let $u_1, u_2 \in H^1(B)$ satisfy

$\displaystyle \int_B \nabla u_1 \nabla \varphi \leqslant \int_B f(x,u_1) \varphi$

and

$\displaystyle \int_B \nabla u_2 \nabla \varphi \geqslant \int_B f(x,u_2) \varphi$

for all non-negative $\varphi \in H_0^1 (M)$. Then the inequality

$u_1 \leqslant u_2$

on $\partial B$ implies

$u_1 \leqslant u_2$

in $B$.

A proof for the weak comparison principle is rather standard as follows: Thanks to the condition $u_1-u_2 \leqslant 0$ on the boundary $\partial B$, we can use $\varphi = (u_1 - u_2)^+$ as a test function. Clearly,

$\displaystyle \int_{\{u_1 \geqslant u_2\}} \nabla u_1 \nabla (u_1 - u_2) \leqslant \int_{\{u_1 \geqslant u_2\}} f(x,u_1) (u_1 - u_2)$

and

$\displaystyle -\int_{\{u_1 \geqslant u_2\}} \nabla u_2 \nabla (u_1 - u_2) \leqslant -\int_{\{u_1 \geqslant u_2\}} f(x,u_2) (u_1 - u_2).$

Hence,

$\displaystyle \int_{\{u_1 \geqslant u_2\}} |\nabla (u_1 - u_2)|^2 \leqslant \int_{\{u_1 \geqslant u_2\}} \big( f(x,u_1) - f(x,u_2) \big) (u_1 - u_2) \leqslant 0$

due to the monotonicity of $f$. Hence, the proof follows.

The second auxilary result is the so-called Hopf Maximum Principle as the following

Lemma (Hopf’s Maximum Principle). Let $B$ is a geodesic ball contained in $M$ and assume that $u>0$ in $B$ and $u(x_0)=0$ for some point $x_0 \in \partial B$. Furthermore, we assume that all assumptions in the Maximum Principle are fulfilled. Then

$\displaystyle \nabla u(x_0) \ne 0.$

To prove, W.L.O.G, we may assume that $B$ is the unit geodesic ball $B_1(y_0)$ centered in $y_0$ in $M$. Then we define

$\displaystyle b(x)=k \big( \exp (-\alpha \text{dis}(x,y_0)^2) - \exp(-\alpha) \big)$

for some $k>0$ and $\alpha>0$.

An elementary calculation shows that there is an $\alpha>0$ satisfying

$\displaystyle-\Delta b \leqslant -c(K+|b|)|b| \leqslant f(b)$

in $B_1(y_0) - B_{1/2}(y_0)$ independent of $k>0$. Now, we choose $k$ in such a way that

$b \leqslant u$

on $\partial \big( B_1(y_0) - B_{1/2}(y_0) \big)$. The Weak Comparison Principle then implies that

$b \leqslant u$

in $B_1(y_0) - B_{1/2}(y_0)$. Then the Lemma follows.

We are now in a position to prove our main theorem.

Proof of Theorem. We prove the theorem by contradiction. Indeed, suppose that there is a point $y \in M$  such that $u(y)=0$. Then, we can find a small ball $B_r(y_0)$ in such a way that $y \in \partial B_r(y_0)$ and $u>0$ in $B_r(y_0)$.

Hence, $y$ is a minimum point of $u$ in $B_r(y_0)$, this implies that $\nabla u(y) = 0$. However, by the Hopf Maximum Principle above, it should be $\nabla u(y) \ne0$, a contradiction.

We now consider a simple example arising from the prescribed scalar curvature equation given by

$-\Delta_g u + hu=fu^\frac{n+2}{n-2}$

for smooth functions $h$ and $f$. Suppose that $u$ is a non-negative of the equation, we show that either $u \equiv 0$ or $u>0$. To do so, we simply write

$-\Delta_g u + Ku=fu^\frac{n+2}{n-2} - hu +Ku$

where $K>0$ is a constant chosen in such a way that

$\displaystyle fr^\frac{n+2}{n-2} - hr + Kr \geqslant 0$

for all $|r| \leqslant \|u\|_\infty$. Hence, there holds

$-\Delta_g u \geqslant -Ku$

pointwise in $M$. It is immediately to see that our function $-Ku$ fulfills all conditions of the Maximum Principle (monotonicity + growth), hence either $u \equiv 0$ or $u>0$ as claimed.

It is important to note that the advantage of the above result is that $f$ has no fix sign which could be sign-changing.