Well-Posedness of the Singular-Cost Dyson Hamilton-Jacobi Equation above the Hardy Threshold

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number and let $W_{N}$ be the Weyl chamber, open and nonempty by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open. Let $\lambda,\theta,\omega\in\mathbb{R}$ be positive, let $\kappa\in\mathbb{R}$ be nonnegative, and let $c\in\mathbb{R}$ satisfy the Hardy condition¶ $0<\kappa^{2}+8\theta c$. Let $g:W_{N}\to\mathbb{R}$ be continuous into the real line, let $M\in\mathbb{R}$ satisfy $|g(x)|\le M$ for every $x\in W_{N}$, and let $F$ be the singular-cost Dyson Hamilton-Jacobi operator with pair-cost coefficient $c$, confinement coefficient $\omega$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$.

Let $\sqrt{\cdot}$ denote the nonnegative square root of Existence and Uniqueness of the Nonnegative Square Root, and put

Here $\kappa^{2}+8\theta c$ is positive by the Hardy condition, and $\lambda^{2}+8\theta\omega$ is positive as the sum of the nonnegative $\lambda^{2}$ (claim 2 of Nonnegativity of Squares in an Ordered Field) and the positive $8\theta\omega$; so both square roots exist and are positive, and $\beta>\tfrac{\kappa}{2}\ge0$; and $\lambda<\sqrt{\lambda^{2}+8\theta\omega}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $a>0$. Let $H_{\beta}$ be the logarithmic energy of strength $\beta$ and let $P:W_{N}\to\mathbb{R}$ be given by

Then the following hold.

1. (The profile)¶ $\beta(\beta-\kappa)=2\theta c$ and $2\theta a^{2}+\lambda a=\omega$, and $P$ is a penalty on $W_{N}$.

2. (Comparison)¶ If $u:W_{N}\to\mathbb{R}$ is a viscosity subsolution of $F$ on $W_{N}$ such that $u-P$ has $P$-subordinate growth from above, and $v:W_{N}\to\mathbb{R}$ is a viscosity supersolution of $F$ on $W_{N}$ such that $v-P$ has $P$-subordinate growth from below, then $u(x)\le v(x)$ for every $x\in W_{N}$.

3. (Existence and uniqueness)¶ There is exactly one function $u:W_{N}\to\mathbb{R}$ that is both a viscosity subsolution and a viscosity supersolution of $F$ on $W_{N}$ and such that $u-P$ has $P$-subordinate growth from above and from below. It is continuous on $W_{N}$, as a map into the real line.