Interior Schauder estimates with weights

Question

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a bounded function. If we require that $\int_{\Omega_{\rho}}{|u|^2w(x)dx}<\infty$, here $w(x)$ is some weight function, for example, we choose $w(x)=\exp{|x|^{\alpha}}$, $\alpha>0$. Then my question is how to show that $$ \int_{\Omega_{\rho}}{|D^{\beta}u|^2w(x)dx}<\infty $$ is also true for $|\beta|\leq 2$?

I came across this problem when reading a paper, and the reference the author offered was Hormander's The Analysis of Linear Differential operators, the 2nd volume, where I failed to find something similar. However, I found that in the first section of Chapter 17, he gave the interior Shauder estimates by using fundamental solution and fractional integral. But the result is local and without weight. I don't know if this can modified to the case I concerned here. Another resource that maybe helpful is S Agmon, A Douglis, L Nirenberg's classical paper "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions.I" , in thm 15.1, they established the interior estimated for general elliptic operators, but I don't know whether their methods can be applied to the case with weight like $w(x)=\exp{|x|^{\alpha}}$. I'm very appreciated that if someone can point out how this is done. Thanks in advance.

Answer 1

First of all, I would be slightly surprised if you can get higher regularity theory without assuming some smoothness from $V$ (although it might be possible to get the integrability of the second derivatives since that is not too much, yet). This is due to the fact that typically you would like to differentiate the equation, but if $V$ not smooth enough it will cause problems.

Regarding the weighted theory, I think one of the first papers was this one:

Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7(1):77–116, 1982.

Nowadays, there is a quite rich literature regarding interior regularity results with weights. For these I would study the following books and the references therein:

Heinonen, Kilpeläinen and Martio. Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications.

A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics 17, European Mathematical Society, Zürich, 2011, 415 pp, ISBN 978-3-03719-099-9.

I think these books, however, only include Hölder estimates and not the higher regularity theory (although I did not check). In any case, they should have the references there if such results exist.

In order to prove higher regularity theory, you typically would like to differentiate the equation. In rigorous terms this means that you would need to study the difference quotients. Since the Lebesgue measure is translation invariant it does not really play any role when differentiating the equation. With a more general measure, there will be some contribution coming also from the measure, but if the weight is nice enough (say $C^1$) you should be able to estimate it in a rather straightforward manner. Unfortunately, I think the references I had in mind do not include this part of the theory, although you might still want to check.

Answer 2

Since you are only supposed to assume that V is bounded, the assumption on u really is that $|\Delta u| \le C_0 |u|$.

I think the crucial assumption might be that the weight function does not vary too wildly - is tempered in some sense . The refrence Hormander's The Analysis of Linear Differential operators, the 2nd volume does define some tempered weight functions in chapter 10, but they are on the Fourier transform side and probably not relevant here. A reference for a very general treatment of estimates of this kind would be Hormander's The Analysis of Linear Differential operators, the 3rd volume, section 18.5-6 on the Weyl calculus. But the treatment there is quite abstract, the original 1979 paper might contain examples on how to construct the metric and weight function.

A possible strategy along those lines (without using the Weyl calculus machinery) could be the following.

First, I will assume the domain is all of $R^n$. Let $X$ be the unit cube and $Y = [-1,2]^n$. Then we can bound $\int_X|D^\beta u|^2 dx \le (1+C^2_0)C_1 \int_Y|u|^2 dx$ by the assumption on u and interior regulatity. Cover $R^n$ by integer translates of $X$. If the weight function only varies by a fixed factor over any of the translates of $Y$ in the sense that $|w(x)/w(y)| <= C_2$ if $x-y \in Y$, then we get an estimate like $\int_{R^n}|D^\beta u|^2 w(x) dx \le C_3 \int_{R^n}|u|^2 w(x) dx$ summing over the translations. Here it is used that there is a bounded number of overlapping integer translates of $Y$.

For your case, one might be able to cover $\Omega_\rho$ by scaled translates of $X$ and $Y$, using that the $C_1$ above scales in a simple way. The scaling would need to be related to how $w(x)$ varies. I have not carried this through, and the question is, what is the right condition on w that enables it.