$\newcommand{\bR}{\mathbb{R}}$ Let $\alpha := e^{-(1 + \sqrt{2})}$. We define the following modulus $\psi : \bR_+ \to \bR_+$ of continuity $$ \psi (x) := \begin{cases} 0 &\text{if} \quad x =0 , \\ x | \ln x|^2 &\text{if} \quad x \in (0, \alpha], \\ x - 2 \alpha \ln (\alpha) &\text{if} \quad x \in (\alpha, \infty). \end{cases} $$
Then $\psi$ is continuous, increasing and concave. The function $\psi$ is taken from a paper about gradient flow by Katy Craig.
Is there a map $V : \bR^d \to \bR$ with the following conditions?
- $V$ is differentiable and not convex.
- $e^{-V}$ is integrable w.r.t. Lebesgue measure.
- There is a constant $C >0$ such that $V(x) \ge -C$ and $|V(x)| + |\nabla V(x)|^2 \le C(1 + |x|^2)$ for $x \in \bR^d$.
- $|\nabla V (x) - \nabla V (y)|^2 \le C \psi (|x-y|^2)$ for $x, y \in \bR^d$.
- $\nabla V$ is not locally Lipschitz.
The difficulty lies mainly in the last two conditions. Thank you for your elaboration.
