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.

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