Ngô Quốc Anh

January 9, 2010

The maximum principles

Filed under: Nghiên Cứu Khoa Học, PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:02

The maximum principle, in its various forms, is one of the most useful tools for working with elliptic partial differential equations. We gather here a few versions which we use elsewhere in this blog. These versions come from a paper due to James Isenberg published in Class. Quantum Grav. in 1995. In all cases, $\nabla^2$ is the Laplacian (with negative eigenvalues) on a Riemannian manifold $\Sigma^3$ (closed=compact without boundary, or open with boundary, as indicated), and $\psi$ is a real-valued $C^2$ function on $\Sigma^3$.

Version 1. ($\Sigma^3$ closed)

If $\psi$ satisfies either

$\displaystyle\nabla^2 \psi \leq 0, \qquad \nabla^2 \psi=0,$

or

$\displaystyle\nabla^2 \psi \geq 0$

then $\psi$ must be constant.

Hence there is no solution to the equation $\nabla^2 \psi=F(x,\psi)$ with $F(x,\psi) \geq 0$ or $F(x,\psi) \leq 0$ unless in fact $F(x,\psi) =0$.

Version 2. ($\Sigma^3$ closed)

Let $f: \Sigma^3 \to \mathbb R$ be a positive definite ($f(x)>0$) function.

• If $\psi$ satisfies

$\displaystyle\nabla^2\psi+f\psi \geq 0$

then $\psi(x) \geq 0$.

• If $\psi$ satisfies

$\displaystyle\nabla^2\psi+f\psi \geq 0$ (not identically zero)

then $\psi(x) > 0$.

• If $\psi$ satisfies

$\displaystyle\nabla^2\psi+f\psi \leq 0$

then $\psi(x) \leq 0$.

• If $\psi$ satisfies

$\displaystyle\nabla^2\psi+f\psi \leq 0$ (not identically zero)

then $\psi(x) <0$.

Version 3. ($\Sigma^3$ closed)

Let $\mu$ and $\kappa$ be positive constants. If $\psi$ satisfies

$\displaystyle -\nabla^2\psi + \mu\psi \leq \kappa$

then

$\displaystyle \psi(x) \leq \frac{\kappa}{\mu}$.

Version 4. ($\Sigma^3$ open with boundary)

Let $\mu$ be a positive constant, and let $\rho : \partial\Sigma^3 \to \mathbb R$ be a positive definite function. If $\psi$ satisfies either

$\displaystyle -\nabla^2 \psi +\mu\psi=0$ on $\Sigma^3$

and

$\displaystyle\psi = \rho$ on $\partial\Sigma^3$

then $\psi>0$.

We prefer the reader to the book entitled “Maximum Principles in Differential Equations” due to Protter and Weinberger for further discussion and proofs.