The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number, let $\beta\in\mathbb{R}$ be positive, let $W_{N}$ and $H=H_{\beta}$ be as in The Logarithmic Energy of N Ordered Particles on the Weyl Chamber, and let $a_{kj}$ and $S$ be as in The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity; $W_{N}$ is open by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open. Let $V_{1}:\mathbb{R}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}$ (with $n=m=1$; $\mathbb{R}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), write $V_{1}'=\partial_{1}V_{1}$ and $V_{1}''=\partial_{1}\partial_{1}V_{1}$, and let $P:W_{N}\to\mathbb{R}$ be given by

We write $\operatorname{tr}$ for the trace and abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$.

1. (Regularity)¶ $P$ is of class $C^{2}$ on $W_{N}$, and for $x\in W_{N}$ and $k\in[N]$

2. (Penalty)¶ If there are $a_{0},b_{0}\in\mathbb{R}$ with $0<a_{0}$ and $a_{0}|t|-b_{0}\le V_{1}(t)$ for every $t\in\mathbb{R}$, then $P$ is a penalty on $W_{N}$.

3. (Monotone gradient)¶ If $0\le V_{1}''(t)$ for every $t\in\mathbb{R}$, then $0\le\bigl(DP(x)-DP(y)\bigr)\cdot(x-y)$ for all $x,y\in W_{N}$.

4. (Dissipation)¶ Let $\lambda,\kappa\in\mathbb{R}$ with $0<\lambda$ and $0\le\kappa<2\beta$. Assume the hypotheses of claims 2 and 3, and that $V_{1}$ has regular growth: for every positive $\eta\in\mathbb{R}$ there is $C_{\eta}\in\mathbb{R}$ with $V_{1}''(t)\le C_{\eta}+\eta\,V_{1}'(t)^{2}$ for every $t\in\mathbb{R}$. Then there are $\varepsilon,C\in\mathbb{R}$ with $0<\varepsilon\le1$ such that