Most general conditions for (weak or classical) solutions to Poisson's equation

Question

I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving $$ \left\{ \begin{align} & \Delta \psi = - \rho & & \mbox{in } \mathbb{R}^3, &(1) \\ & \psi(\infty) = 0. & & &(2) \end{align} \right. $$ where $\rho$ is a compactly supported function.
We know the answer should be given (up to constant factors) by \begin{equation} \psi(x) = \int_{\mathbb{R}^3} \frac{\rho(y)dy}{|x-y|}. \qquad (3) \end{equation} My first question is what are the mildest conditions that can be imposed on $\rho$ for (1) and (2) to hold pointwise, and is the solution indeed given by (3)? I think Gilbarg and Trudinger give it to hold for $\rho$ Holder continuous with Holder exponent $\alpha \in (0,1]$.

My second question is what if I relax the condition that (1) hold pointwise, and instead seek weak solutions i.e. $\psi$ satisfying $$ \int_{\mathbb{R}^3}\left[ \nabla\psi\cdot\nabla\varphi - \rho\varphi\right]dx = 0, \qquad \psi(\infty)=0 \qquad (4) $$ should hold for all test functions (i.e. $C_c^\infty$) $\varphi$? Then how does the generality improve? Can it be broadened to allow for measures $\rho$ if we replace $\int\varphi\rho dx$ by $\int \varphi d\rho$ in (4)?

FYI the books I have been consulting include Evan's PDE, Gilbarg and Trudinger's Elliptic PDE, Landkof's Potential Theory, Helms' Potential Theory, Jackson's Electrodynamics. Thank you all in advance.

Answer 1

The name of the subject is Potential Theory (your integral (3) is called the Newtonian potential). Good references are:

N. Landkof, Foundation of modern potential theory, Springer 1972,

M. Brelot, Éléments de la théorie classique du potentiel. 3e édition. Les cours de Sorbonne. 3e cycle. Centre de Documentation Universitaire, Paris 1965,

(these titles are slightly misleading: Brelot's book is more modern and more abstract than Landkof's book. But Landkof contains much more.)

L. Hormander, Notions of convexity, Birkhauser, 1994.

and also many books under the title Subharmonic functions, for example, W. Hayman and P. Kennedy (in 2 vols, first volume is about $R^n$, second about $n=2$.)

In the most general setting $\Delta u$ is considered in the sense of Schwartz distributions, and $\rho$ is a charge (signed measure, difference of two Radon measures, difference of two positive distributions). Distributional solutions of this equation are given by the potential which you wrote, and in the case of positive charge $\Delta u$ (a measure) they exist as honest functions $R^n\to R\cup\{-\infty\}$ (defined pointwise everywhere, not just almost everywhere). These functions are upper semicontinuous and they are called subharmonic.

For a signed charge $\rho$, function $u$ cannot be reasonably defined at every point, but can be defined a. e. and belongs to $L^p_{loc}$ for all $p<\infty$. In fact they are defined quasieverywhere, that is the exceptional set has zero Newton capacity (logarithmic capacity when $n=2$). They are called delta-subharmonic functions.

Pointwise second derivative in the classical sense has little use in this theory and it exists under a suitable conditions on $\rho$ stated in the answer of user111. Potentials which are 2-ce differentiable in the classic sense do not have good properties, expecially with respect to taking limits. Also the important property that pointwise maximum of subharmonic functions is subharmonic is lost if we restrict ourselves to 2-ce differentiable functions. When $n=1$, subharmonic functions are nothing but convex functions, and it is not reasonable to restrict oneself to 2-ce differentiable convex functions.

There is a different setting of "potentials of finite energy" which are elements of the completion of smooth functions with respect to energy norm. This has an advantage that they form a Hilbert space (See Landkof).

Answer 2

1st question : indeed, Hôlder continuity seems to be the weakest possible condition for (1), (2), (3) to hold. Gilbarg and Trudinger give an example of a continuous $\rho$ in the unit ball such that $\psi$ is not second order differentiable at the origin (Problem 4.9 p.71).

2nd question : yes, measures give weak solutions. For a proof see e.g. Ransford's book "Potential theory in the complex plane" (Theorem 3.7.4 p.74).