10 problems

Let $V^+$ and $V^-$ be the upper and lower value functions of the stochastic differential game of mixed type, defined by … and … Here $\mathcal{A}_1$ and $\mathcal{A}_2$ are the ad…

Generic finite-graph conjecture. For a generic such function $A(x)$, the closure of the singular set of $u$ is a finite graph.

Let $w$ be a Lipschitz subsolution to the generalized $k$-convexity equation for $\beta=1$. A measure-valued Hessian conjecture asserts that there exist signed Borel measures…

Armstrong–Silvestre–Smart's conjecture. The optimal exponent in this estimate is

Let $H(x,p,t)$ be a Hamiltonian satisfying $H(x,p,t)\in C^2$ and convexity with respect to $p$, and let (A5) denote the assumption referenced in the source. Let $\varphi$ be the vi…

Constrained-problem construction conjecture. First solve the unconstrained problem using the formula above. Then, whenever $\min \tilde u\leq 0$, replace $\tilde u$ by $v^2/2$ on t…

Let $(f,\Gamma)$ satisfy conditions (01)--(04), and let $0<v\in C^0_{\mathrm{loc}}(\mathbb{R}^n)$ be a viscosity solution of the equation (B0). Write the family referred to as (vfo…

Let $u^*$ be the convex, $C^{1,1}(\mathbb{R}^n)$ solution associated to the eigenvalue $\lambda^*$, and define … where the infimum is over the relevant $C^2(\mathbb{R}^n)$ supersol…

General comparison with measurable ingredients conjecture. Under these assumptions, the conclusion of the comparison proposition holds with $M^+_{CS}$ in place of $M^+_A$.

Let $\Omega$ be the domain under consideration. Assume that $u,-v\in\operatorname{USC}(\bar\Omega)$ and $f\in C(\Omega)$ satisfy … Suppose that either $u$ has no strict local maxim…