38 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.

The framework assumes that the coefficients $\tilde{\omega}$ and $U$ are globally Lipschitz in the state variables, so that strong solutions to backward stochastic differential equ…

Geometric dynamic programming conjecture. The map $w$ should be related to its geometric dynamic programming equation, conjectured to be

The $C^{1,\frac{1}{3}}$ regularity conjecture. Every such viscosity solution belongs to the Hölder space $C^{1,\frac{1}{3}}$.

Let $w$ be the solution of the parabolic equation and let $(\lambda,\phi)$ be a viscosity solution of the ergodic HJB equation. Define … The bounded function $G$ should tend to a c…

Weighted p-Laplace convergence conjecture. The solution of the weighted $p$-Laplace equation converges to the solution of the infinity-Laplace equation as $p\to\infty$.

Classical solvability conjecture. The occupied heat equation always admits a classical solution whenever $g$ has the stated continuity. Unlike the path-dependent heat equation, who…

The $C^{1,\frac{1}{3}}$ regularity conjecture. Every such viscosity solution belongs to the class $C^{1,\frac{1}{3}}$.

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…

Let $\mathcal{V}(t,\mu)$ be the value function and let $\mathcal{V}_1(t,\mu)$ and $\mathcal{V}_2(t,\mu)$ be the expressions appearing in the two control regimes. Let $\Pi_3$ denote…

Consider first-order equations arising from McKean–Vlasov control, where the control $a$ may be a feedback control dependent on the state $X$. Extension conjecture. The uniqueness…

Viscosity free-boundary conjecture. There are examples of viscosity solutions whose topological free boundary is not a set of $\sigma$-finite $\mathcal H^{n-1}$-measure.

Extension conjecture. The sufficient regularity theorem is also valid when $p=2$: every viscosity supersolution of

Universal $\mathcal{C}^{1,\alpha}$ regularity conjecture. Solutions are of class $\mathcal{C}^{1,\alpha}$ for every $\alpha\in(0,1)$.

Discontinuous-viscosity-solution conjecture. The function $V$ is only a discontinuous viscosity solution of

The infinity Laplacian is denoted by $\Delta_\infty$, and an infinity harmonic function is a function satisfying $\Delta_\infty u=0$ in the viscosity sense. For a domain in the pla…

Let $[-1,1]^2\setminus\{0\}$ be the punctured square, and let an infinity harmonic function on this domain be understood in the usual viscosity sense. A ground-state conjecture ass…

Let $u_\lambda$, for any $\lambda>0$, denote the maximal viscosity solution of the discounted problem … that is bounded from below. This conjecture is motivated by the one-dimensio…

Let $U$ be the unmodified optimal cost, let $V$ be the modified optimal cost defined by … and let $f$ be a nonnegative viscosity solution of the Bellman equation. Assume that the s…

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

Let $H(x,p,t)$ be a Hamiltonian and let $g(x)$ be convex. For a path $\gamma\in C^\infty$ and a momentum path $p$, impose … with $\gamma(t)=x$, $\gamma(0)=y$, and $p(t)=v$. General…

Let $H(x,p,t)$ be a Hamiltonian, let $g$ be convex, and let $\varphi$ be the viscosity solution of the Hamilton–Jacobi equation with initial data $g$ in the source. Let the general…

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…

Let $V$ denote the value function associated with conditional value-at-risk and its HJB equation, as in the preceding representation. Separation of variables refers to solving this…