2
$\begingroup$

I am interested in what I'll call (perhaps erroneously) multivariate Sturm–Liouville theory, i.e., solutions to equations of the form $$\nabla\cdot(P(x)\nabla Y)+Q(x)Y=-\lambda W(x)Y\tag{1}$$ for $\lambda\in\mathbb R$, $Y:\mathbb R^d\to\mathbb R$, and $P,Q,W:\mathbb R^d\to\mathbb R$.

If $d=1$, then, as shown on the Wikipedia page, there is a very well developed theory for solutions of $(1)$ on an interval $[a,b]$ with fixed boundary conditions. This is called Sturm-Liouville theory.

I suspect that such a theory has been generalized to higher dimensions. However, after googling for a while using keywords such as "multivariate Sturm-Liouville theory", I'm starting to suspect that the study of $(1)$ in higher dimensions has a different name, since I've completely failed to find good comprehensive resources on such problems.

Question: Are there textbooks that treat multivariate problems such as $(1)$ in detail? I'm especially interested in fixed point-type arguments for existence and uniqueness, as well as continuity results for the solution $(Y,\lambda)$ with respect to the "data" $P,Q,W$.

$\endgroup$
3
$\begingroup$

Assuming $P$ has constant sign, this is a well-studied class of problems, with the usual terminology being something like "weighted elliptic eigenvalue problem".

In the case $W=1,$ you have the standard eigenvalue problem for the divergence-form linear elliptic operator $L=\operatorname{div} P\, \nabla + Q,$ which is usually analyzed using Fredholm theory - see, for example, the end of Chapter 8 of Elliptic PDE of Second Order by Gilbarg and Trudinger. There has been a huge amount written about elliptic eigenvalue problems which you should have no problems finding online.

When you add the weight $W,$ you're in to marginally less standard territory; but there's still a lot out there. If $W>0$ then I'm guessing you can say quite a bit by rewriting the problem as a standard eigenvalue problem for the unknown $U = WY.$ In more generality, it seems there have been many articles written on the topic of elliptic eigenvalue problems with an indefinite weight function.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.