14 problems

Let $\Omega\subset\mathbb{R}^n$ be a bounded uniformly convex domain. Consider the Neumann problem for the $k$-Hessian equation, with $2\leq k\leq n-1$, obtained from … with prescr…

Convergence conjecture.

Ivochkina–Trudinger–Wang conjecture. The condition $f^{\frac{1}{k-1}} \in C^{1,1}(\overline{\Omega})$ should be sufficient for obtaining an existence theorem for the degenerate $k$…

Let $f$ be a smooth strictly convex function defined on a convex domain $\Omega$ in $\mathbb{R}^2$. The Calabi rigidity conjecture. If $f$ satisfies the PDE (1.2) with…

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…

Higher-dimensional constant-data uniqueness conjecture. Then $h$ must be a constant solution.

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

Let $n\geq 1$, let $p\in\mathbb{R}$, and let $h$ be a positive smooth function on the unit sphere $\mathbb{S}^n$. Consider the equation … The theorem establishes that, for…

The concavity conjecture. For every $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n$, the quadratic form

The quadratic-form conjecture. For every $n4$-dimensional vector $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n4$, the quadratic form

Bao–Li exterior Dirichlet conjecture. For every $\varphi\in C^2(\partial D)$, if

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…

Chen's generalized Hessian equation conjecture. The equation

Let $\Omega:=(-0.57,0.57)^2$, let $K>0$ be a constant, and consider the Dirichlet problem … with … The problem concerns convex viscosity solutions, and $K^*>0$ denotes the upper cu…