16 problems

Extremality conjecture. In the class of operators defined by the ellipticity condition in, all critical exponents lie in the same interval; equivalently, the operators…

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…

Critical-curve conjecture. The boundary $\partial\mathcal{C}$ is a critical curve in the $(p,q)$-plane separating the exponents for which regular solutions exist in $\mathbb{R}^N$…

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

Armstrong–Silvestre–Smart conjecture. The dependence of $\epsilon$ on $\Lambda$ in the interior $W^{2,\,\epsilon}$ estimate should be linear in $\Lambda^{-1}$, equivalently…

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

Non-transversal intersection conjecture. The free boundary intersects the fixed boundary non-transversally.

Uniqueness conjecture. The constructed solution $u^0$ is the unique solution of the geodesic equation with fixed boundary datum.

Convexity conjecture for $H_k$. For every $3\leq k\leq n-1$, if $r\in\Gamma^+_k$, then $H_k(r,Y)$ is convex in $(r,Y)$.

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…

Let $F$ be the operator in problem, with principal half-eigenvalues $\lambda_1^+(F)$ and $\lambda_1^-(F)$, and let $f$ be the datum in that problem. Multiplicity conjecture. There…