
This post is about Riesz energy problems, a subject that I explore with Edward B. Saff (Vanderbilt University, USA) and Robert S. Womersley (UNSW Sydney, Australia).
Riesz kernel. For $-2<s<d$, the Riesz $s$-kernel in $\mathbb{R}^d$ is $$
K_s:=\begin{cases}
\displaystyle\frac{1}{s\left|\cdot\right|^{s}} & \text{if } s\neq0\\[1em]
\displaystyle-\log\left|\cdot\right| & \text{if } s=0
\end{cases}.
$$ We recover the Coulomb or Newton kernel when $s=d-2$. This definition of the $s$-kernel allows to pass from $K_s$ to $K_0$ by removing the $1/s$ singularity at $s=0$, in the sense that for $x\neq0$, $$-\log|x|=\lim_{\underset{s\neq0}{s\to0}}\frac{|x|^{-s}-1}{s-0}=\lim_{\underset {s\neq0}{s\to0}}\Bigr(\frac{1}{s|x|^s}-\frac{1}{s}\Bigr).$$
Riesz energy. For $-2<s<d$, the Riesz $s$-energy of a probability measure $\mu$ on $\mathbb{R}^d$ is $$
\mathrm{I}_s(\mu):=\iint K_s(x-y)\mathrm{d}\mu(x)\mathrm{d}\mu(y)
=\int(K_s*\mu)\mathrm{d}\mu.
$$ The Riesz energy is strictly convex and lower semi-continuous with respect to the weak convergence of probability measures with respect to continuous and bounded test functions.
Equilibrium measure. The equilibrium measure on a ball $B_R:=\{x\in\mathbb{R}^d:|x|\leq R\}$ satisfies
$$
\mathrm{I}_s(\mu_{\mathrm{eq}})
=\min_{\substack{\mu\\\mathrm{supp}(\mu)\subset B_R}}\mathrm{I}_s(\mu).
$$
Marcel Riesz original problem (1938). Equilibrium measure on $B_R$ when $d\geq2$ :
$$
\mu_{\mathrm{eq}}
=
\begin{cases}
\sigma_R & \text{if $-2<s\leq d-2$}\\[1em]
\displaystyle\frac{\Gamma(1+\frac{s}{2})}{R^s\pi^{\frac{d}{2}}\Gamma(1+\frac{s-d}{2})}
\frac{\mathbf{1}_{B_R}}{(R^2-|x|^2)^{\frac{d-s}{2}}}\mathrm{d}x &
\text{if $0\leq d-2<s<d$}
\end{cases}
$$ where $\sigma_R$ is the uniform distribution on the sphere $\{x\in\mathbb{R}^d:|x|=R\}$ of radius $R$.
The proof relies on the following integral formula for the variational characterization : $$
\int_{|y|\leq R}
\frac{|x-y|^{-s}}{(R^2-|y|^2)^{\frac{d-s}{2}}}\mathrm{d} y
=\frac{\pi^{\frac{d}{2}+1}}{\Gamma(\frac{d}{2})\sin(\frac{\pi}{2}(d-s))},\,\quad
x\in B_R
$$ The proof of this integral formula involves in turn a Kelvin transform and a reduction to the planar case. It can be found in detail in the Appendix of the book by Landkof (1972), and also with even more details in our 2022 JMAA article. To our knowledge, a simple proof is still lacking!
The Riesz integral formula above reveals a threshold phenomenon : the support condensates on a sphere when $s$ passes the critical value $2$. Our main finding is that this Riesz problem admits a full space extension in which we replace the ball support constraint with an external field. We show that a new threshold phenomenon occurs, related to the strenght of the external field.
External Field Equilibrium Problem. The energy with external field $V$ on $\mathbb{R}^d$ is defined by $$\mathrm{I}(\mu)
:=\iint\left[K_s(x-y)+V(x)+V(y)\right]\mathrm{d}\mu(x)\mathrm{d}\mu(y)$$
and the associated equilibrium measure satisfies $$\mathrm{I}(\mu_{\mathrm{eq}})=\min_{\mu}\mathrm{I}(\mu)$$ The Frostman or Euler-Lagrange variational characterization of $\mu_{\mathrm{eq}}$ reads $$K_s*\mu+V
\begin{cases}
=c& \text{quasi-everywhere on }\mathrm{supp}(\mu)\\
\geq c&\text{quasi-everywhere outside }\mathrm{supp}(\mu)
\end{cases}$$ Quasi-everywhere means except on a set that cannot carry a probability measure of finite energy.
Coulomb case : $s=d-2$. The kernel $K_{d-2}$ is the fundamental solution of the Laplace equation :
$$
\Delta K_{d-2}\overset{\mathcal{D}’}{=}-c_d\delta_0,\quad\text{with}\quad c_d=|\mathbb{S}^{d-1}|.
$$
Also, restricted to the interior of $\mathrm{supp}(\mu_{\mathrm{eq}})$,
$$
\mu_{\mathrm{eq}}\overset{\mathcal{D}’}{=}\frac{\Delta V}{c_d}
$$
If $V=\left|\cdot\right|^\alpha$, $\alpha>0$, then
$$
\mu_{\mathrm{eq}}
=\alpha(\alpha+d-2)\left|\cdot\right|^{\alpha-2}\mathbf{1}_{B_R}\mathrm{d}x
\quad\text{with}\quad R=\bigr(\frac{1}{\alpha}\bigr)^{\frac{1}{d-2+\alpha}}.$$
Findings for the iterated Coulomb case $s=d-2n, n=1,2,3,\ldots$. Then, restricted to the interior of $\mu_{\mathrm{eq}}$, in the sense of distributions,
$$
\mu_{\mathrm{eq}}
\overset{\mathcal{D}’}{=}
\frac{\Delta^{n}V}{c_dC_{d,n}},
\quad\text{where}\quad
C_{d,n}:=(-1)^{n-1}\frac{(d-4)!!(2n-2)!!}{(d-2n-2)!!}.
$$
The proof relies crucially on the local nature of the Laplacian and the fact that
$$\Delta K_u=-c_{d,u}K_{u+2}$, $C_{d,n}=(-1)^{n-1}\prod_{k=0}^{n-2}c_{d,s+dk}.$$ In particular : if $s=d-4$ and $V=\left|\cdot\right|^\alpha$, $\alpha\geq2$, then $C_{d,2}<0$ while $\Delta V=\alpha(\alpha+d-2)\left|\cdot\right|^{\alpha-2}\geq0$ and thus $\mu_{\mathrm{eq}}$ is necessarily singular!
Findings when $s=d-4$. Suppose that $V=\gamma\left|\cdot\right|^\alpha$, $\gamma>0, alpha>0$.
- Let $d\geq4$ and $s=d-4\geq0$.
- If $\alpha\geq2$ then $\mu_{\mathrm{eq}}=\sigma_R$ where $$
R=\Bigr(\frac{2}{(s+4)\alpha\gamma}\Bigr)^{\frac{1}{\alpha+s}}$$ - If $0<\alpha<2$ then $$\mu_{\mathrm{eq}}=\beta fm_d+(1-\beta)\sigma_R$$ where
$$\beta=\frac{2-\alpha}{s+2},\
f=\frac{\alpha+s}{R^{\alpha+s}|\mathbb{S}^{d-1}|}\mathbf{1}_{B_R},\
R=\Bigr(\frac{2}{(\alpha+s+2)\alpha\gamma}\Bigr)^{\frac{1}{\alpha+s}}$$
- If $\alpha\geq2$ then $\mu_{\mathrm{eq}}=\sigma_R$ where $$
- Let $d=3$ and $s=d-4=-1$ (non-singular kernel!).
- If $0<\alpha<1$, then $\mu_{\mathrm{eq}}$ does not exist (blowup)
- If $\alpha=1$ and $\gamma\geq1$, then $\mu_{\mathrm{eq}}=\delta_0$ (collapse).
- If $\alpha>1$, then $\mu_{\mathrm{eq}}$ is as above (mixture).
Findings when $s=d-3$. Suppose that $V=\gamma\left|\cdot\right|^\alpha$, $\gamma>0, \alpha>0$.
- If $s=d-3$ and $\alpha=2$ then $$\mu_{\mathrm{eq}}
=\frac{\Gamma(\frac{s+4}{2})}{\pi^{\frac{s+4}{2}}R^{s+2}}
\frac{\mathbf{1}_{B_R}}{\sqrt{R^2-\left|\cdot\right|^2}}
\mathrm{d}x$$ where $$R=\Bigr(\frac{\sqrt{\pi}}{4\gamma}\frac{\Gamma(\frac{s+4}{2})}{\Gamma(\frac{s+5}{2})}\Bigr)^{\frac{1}{s+2}}$$ - This is also $\mu_{\mathrm{eq}}$ for $s=d-1$ on $B_R$ with this $R$.
Methods of proof.
- Frostman or Euler-Lagrange variational characterization
- Applying Laplacian on support of $\mu_{\mathrm{eq}}$
- Rotational invariance and maximum principle
- Dimensional reduction with Funk-Hecke formula
- Orthogonal polynomials expansions
- Integral formulas and special functions
Challenges.
- Super-harmonic kernel and sub-harmonic external field
- Non-locality of fractional Laplacian
Selected Open Problems.
- Find a simple proof of Riesz formula!
- When $s=d-3$ with $\alpha\neq2$, we conjecture that the support of the equilibrium measure is a ball if $0<\alpha<2$ and a full dimensional shell (annulus) if $\alpha>2$
- When $s=d-6$, it could be that the support of the equilibrium measure is disconnected
- Other norms in kernel and external field
Marcel Riesz (1886 – 1969) is the young brother of Frigyes Riesz (1880 – 1956). I do not known if Naoum Samoilovitch Landkof (1915 – 2004) has ever met in person Marcel Riesz. Landkof was a student of Mikhaïl Alekseïevitch Lavrentiev (1900 – 1980), who gave his name to the Lavrentiev phenomenon in the calcul of variations. Landkof was an expert in potential theory. He advised Vladimir Alexandrovich Marchenko (1922 – ), famous notably for his findings on random operators and matrices with his student Leonid Pastur (1937 – ).
Further reading.
- Marcel Riesz
Intégrales de Riemann–Liouville et potentiels
Acta Sci. Math. Szeged 9 (1938): 1–42, 116–118 - Naoum Samoilovitch Landkof
Foundations of Modern Potential Theory
Grundlehren der mathematischen Wissenschaften 180
Springer 1972 (translated from Russian, Moscow 1966) - Edward B. Saff and Vilmos Totik
Logarithmic potentials with external fields
Grundlehren der Mathematischen Wissenschaften 316
Springer 1997 - Sergiy V. Borodachov, Douglas P. Hardin, and Edward B. Saff
Discrete energy on rectifiable sets
Springer Monographs in Mathematics 2019 - Bartłomiej Dyda, Alexey Kuznetsov, and Mateusz Kwaśnicki
Fractional Laplace operator and Meijer G-function
Constr. Approx. 45, No. 3, 427-448 (2017) - Djalil Chafaï, Edward B. Saff, and Robert S. Womersley
On the solution of a Riesz equilibrium problem and integral identities for special functions
J. Math. Anal. Appl. 515 (2022) - Djalil Chafaï, Edward B. Saff, and Robert S. Womersley
Threshold condensation to singular support for a Riesz equilibrium problem
Preprint arXiv:2206.04956