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, is the Laplacian (with negative eigenvalues) on a Riemannian manifold (closed=compact without boundary, or open with boundary, as indicated), and is a real-valued function on .

**Version 1**. ( closed)

If satisfies either

or

then must be constant.

Hence there is no solution to the equation with or unless in fact .

**Version 2**. ( closed)

Let be a positive definite () function.

- If satisfies
then .

- If satisfies
(not identically zero)

then .

- If satisfies
then .

- If satisfies
(not identically zero)

then .

**Version 3**. ( closed)

Let and be positive constants. If satisfies

then

.

**Version 4**. ( open with boundary)

Let be a positive constant, and let be a positive definite function. If satisfies either

on

and

on

then .

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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