18 problems

Let $D\subset\mathbb{R}^2$ be a bounded simply connected domain with sufficiently smooth boundary, and let $v\in C^4(D)\cap C^1(\overline D)$. The boundary conditions use the inwar…

Let $w\in H^2_{\operatorname{loc}}(\mathbb{R}^n)$ have constant Laplacian. Constant-Laplacian conjecture. Every such $w$ is a global minimizer of $\mathcal{F}$. The conjecture aris…

Rigidity conjecture. Then $\theta_1(t)\to-i$, $\theta_2(t)\to0$, $\lambda_1(t)\to1$, and

Let $f: [-1,1]^d \to \mathbb{R}$, and let $u$ solve the continuum biharmonic problem … Assume either that $u\in C^5((-1,1)^d)$ with $\|u\|_{C^5((-1,1)^d)}\leq C<\infty$, or that…

Extremal-parameter conjecture. For the problem $Q_{\lambda}$, the extremal parameter is

Extremal-parameter conjecture. The extreme value for $P_{\lambda}$ is

Existence conjecture. The problem has a solution for every $\lambda>0$ under this almost critical assumption.

The space $\mathrm{tr}^{2,\Delta}(H(\Delta,\Omega))$ is considered as a subspace of $H^2(\mathcal{T})'$. In dimension two, an example shows the relevant non-closedness phenomenon;…

Let a domain with corners be given, and consider its domain Green's function for the biharmonic equation. Osher's conjecture. The domain Green's function has infinitely many oscill…

Let $u(r)$) be a radial solution of equation (1.1), and set … The scalar curvature is the scalar curvature of the conformal metric $g=u^{\frac{4}{n-4}}g_0$, where $g_0$ is the Eucl…

Optimal trace estimate. There exists a constant $C$ such that

Let $f$ be a positive continuous function on $(0,\infty)$ and consider the biharmonic inequality … in an exterior domain of $\mathbb{R}^2$. The conjecture. If … then the inequality…

Let $n\ge 3$ and let $f$ be a positive continuous function on $(0,\infty)$ satisfying … and … Consider the biharmonic inequality … in an exterior domain of $\mathbb{R}^n$. The conj…

Consider the biharmonic problem … where $p>1$, and let $p_{JL_4}$ denote the Joseph–Lundgren exponent associated with $\Delta^2$. A solution is smooth stable if it is smooth and sa…

Let $\sigma$ lie in the $L^2$-supercritical and $H^2$-subcritical regime, namely … Consider the radial nonlinear eigenvalue problem for $\tilde S_{\mathrm B}(\rho)$ with $\nu=1$: ……

Let $d$ and $u$ be parameters, and let $frac{4}{d}< u$ when $dleq 4$, while $frac{4}{d}<sigma<frac{4}{d-4}$ when $d4$. A radial self-similar profile $S{mathrm B}(rho)$ solves … wi…

Standing-ring blowup-rate conjecture. By analogy with the nonlinear Schrödinger equation, the blowup rate is conjectured to be $1/4$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, and let $G(x,y)$ be the Green function for the biharmonic equation … with clamped boundary conditions. Hadamard's conjec…