In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)$, when $\Omega$ is a bounded domain of $\mathbb{R}^n$ with $\partial \Omega$ regular enough. I think he also takes the Dirichlet boundary conditions equal to 0, but the problem I am dealing with takes into account $\Omega = \mathbb{R}^n$, so it shouldn't be a problem to integrate by parts using sufficently large balls. The thing is, he proves the existence of such a solution by shifting the problem to a minum of a particular functional. He then proves this minumum (that is a weak solution of the equation) belongs to $H^{1}_0 (\Omega)$. Then, later on in the book, he proves that under the hypothesis of some regularity for $\partial \Omega$, the solution is in $H^2(\Omega)\cap H^1 _0 (\Omega)$, and he does that using the so called "method of Nirmberg's translations".

The fact is: I am dealing with the case $\Omega= \mathbb{R}^3$. I think the same results that were true for regular bounded domains should also be true in this case, I think it should be enough to consider the limit of domains $\Omega_N = B(0,N)$ where the proof of Brezis works.

The problem is, I wanted to find the solution explicitly and verify by hand it belongs to $H^{2}(\mathbb{R}^3)$, given that the domain is so simple, not using a known theorem. The solution of the equation is given by:

$$u(x)= \left( \frac{e^{-|y|}}{4 \pi |y|} \ast f \right)(x)$$

But when I try to find a weak second order derivative of $u$ using integration by parts (and I do so by splitting the integration domain in and inside a ball of radius $\epsilon$), I get this very annoying term that doesn't allow me to conclude:

$$\int_{B(0, \epsilon)^c} \partial_{x_1}^2 \left(\frac{1}{4 \pi |x|} \right) e^{-|x|} \phi(x+y) dx=\int_{B(0, \epsilon)^c} \left(\frac{3x_1 ^2 - |x|^2}{4 \pi |x|^5} \right) e^{-|x|} \phi(x+y) dx$$

Where $\phi \in C^{\infty}_0 (\mathbb{R}^3)$. Now, I understand that something like this was forced to appear. Indeed, I know that $\Delta (1/|x|)= \delta(x)$, so this isn't a surprise. However, that expression over there does not converge when $\epsilon \to 0$ (or t least I think so, there should be a singularity like $1/\rho$ if you pass to spherical coordinates) and I would like to obtain a weak derivative for $u$ in which I put all the derivatives on the term $e^{-|x|}/(4 \pi |x|)$ of the convolution.

EDIT: Turns out I can just cut-off a ball of radius $\epsilon$ with a smooth compact supported radial function and then integrating over the angular part first I get either 0 or a constant I don’t remember. Gotta think why I can pass to the limit, but the pointeise convergence is straightforward, and the dominance should follow easiliy, somehow.

  • $\begingroup$ Why wouldn't it converge? The $\exp(-\lvert x\rvert)$ is there to help for large $\lvert x\rvert$. $\endgroup$ – user10354138 Dec 1 '18 at 14:17
  • $\begingroup$ I mean convergence for $\epsilon \to 0$. There I have a singularity of order $1/ \rho$, passing in spherical coordinates. I'll edit the question, though. $\endgroup$ – tommy1996q Dec 1 '18 at 14:26

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