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.

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