8 problems

Compactness and existence conjecture. If $A_g\in\Gamma_k^+$, then there exists a conformal deformation $\tilde{g}=e^{2u}g$ such that

Let $\Omega$ be the domain and let $f$ be the nonlinearity in the $k$-Hessian problem … has no nontrivial solution for every $\lambda\in(0,\lambda_{*})$ and at least two nontrivial…

Let $\Omega$ be the domain and let $f$ be the nonlinearity in the $k$-Hessian problem … has at least two convex solutions for every $\lambda>\lambda_{*}$, exactly one convex soluti…

Let $u\in C^4(\mathbb{R}^n)$ be an entire 2-convex function, meaning that its Hessian eigenvalues lie in the closure of the $2$-Hessian ellipticity cone, and suppose that … through…

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

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…

Geometric subsolution conjecture. There exists a $k$-positive form $'\text{ in }[]$ satisfying the inequality from the paper with $'$ in place of …

Let $\lambda$ and $\Lambda$ be the ellipticity constants in Proposition 2.1, and let $\varepsilon$ denote the exponent in its $W^{2,\varepsilon}$ estimate. Optimal exponent conject…