Let $v = v(t,y)$ be a strictly convex $C^{1,2}((0,T]\times \mathbb{R})$ function such that
- It solves the heat equation: $v_t = \frac{1}{2} v_{yy}$
- $v_y(t,\cdot)$ has range $\mathbb{R}$.
Define a function $u$ via $$ u(t,x) = \inf_{y\in\mathbb{R}}\{v(t,y)-xy\}$$.
My question: what is the regularity of $u$ in both variable $t$ and $x$?
I was able to get $u(t,\cdot)$ is twice-continuously differentiable, through strict convexity of $v(t,\cdot)$ and the inverse function theorem.