Generalization of the Maximum Principle for elliptic linear PDEs

Question

This question came up when I was trying to solve linear elliptic PDEs. Let $R$ be an open domain and $L$ be a linear elliptic operator such that

$$ L \; u = 0 \; \mathrm{on \; R}, \;\; B \; u = f \; \mathrm{on \; \partial R} $$

where $B$ is an operator that encodes the boundary conditions (they could be Dirichlet, Neumann, Robin, or even mixed).

The maximum principle states that the maximum and the minimum of $u$ both happen on the boundary $\partial R$ in the case that $B$ encodes Dirichlet boundary conditions, i.e. the boundary condition is simply $u =f$. Therefore, if $f \equiv 0$, then $u \equiv 0$ as a consequence.

It seems to me that this could be generalized to any general boundary conditions, i.e.

$$ B \; u = 0 \; \text{on} \; \partial R \;\Rightarrow\; u \equiv 0 \; \text{in} \; R \;\;\; \forall B $$

but I can't seem to find information about such a result.

Answer 1

Yes, it could be true. Actually, The maximum principle does not depend on the boundary conditions. It's really up to the sign of $c$ ($Lu=-\sum a^{ij}u_{x_i x_j}+\sum b^iu_{x_i}+cu$). About the more specific proof, I think you can see Evan's book Partial differential equations or Gilbarg and Trudinger's book or others.

Answer 2

This is not true in general. For instance, if $L$ is the Laplacian and $B$ encodes Neumann boundary conditions, then any constant $u$ satisfies $Lu = 0$ and $Bu = 0$.

See Theorem 6.31 in Gilbarg and Trudinger for some sufficient conditions for your claim to be true with Robin boundary conditions. You might also want to look at Protter and Weinberger's very readable book on maximum principles, which for instance gives techniques for dealing with equations where the coefficient $c$ that chuck mentions has the "wrong sign".