107 problems

Let $F$ be an integrand, and consider conditions involving only $F$ and a finite number of its derivatives. Morrey's conjecture. There is no such condition that is both necessary a…

Mumford–Shah conjecture. If $(K,u)$ is a minimizer of $E_\lambda$, then $K$ has the structure described above.

Lions's conjecture. For every $p>\frac{n}{r}$, an $L^p$ estimate should hold for $D^2u$ whenever $f\in L^p$.

Let $\Omega$ be a bounded open domain in an $n$-dimensional Alexandrov space with curvature bounded from below by $k$, and let $N$ be a compact smooth Riemannian manifold. Suppose…

Finite entropy solutions arise in applications including large deviations, the Aviles–Giga functional, and diffusive-dispersive approximations. The isentropic Euler system with…

Let $(u,K)$ be a local minimizer of the Griffith energy in an open set $\Omega$, and let $\nabla u$ denote the gradient of its displacement field. For $p>2$, write…

Let $\Omega\subset\mathbb{R}^N$ be a domain and consider weak solutions of … Assume that the coefficient matrix satisfies … for $\xi\in\mathbb{R}^N$ and almost every $x\in\Omega$,…

Rivière's conjecture. There exists $\delta=\delta(C,n,N)>0$ such that

Let $p\geq 2$ and let $p'=\frac{p}{p-1}$ be the Hölder conjugate exponent, so that … For functions with bounded $p$-Laplacian, optimal regularity conjecture. The optimal regularity…

Let $k\leq m$, let $n\in\mathbb{N}$, and let $N^k$ be a $k$-dimensional closed submanifold of $\mathbb{R}^m$. For every $C>0$, consider $W^{1,2}$ harmonic maps $u:B^n_2(0)\to N^k$…

Consider the one-dimensional nonlinear conservation law … with bounded initial data and suppose that the non-degeneracy condition of order $\alpha$ holds. Lions–Perthame–Tadmor con…

Let $M$ be a Riemannian manifold, and consider area-minimizing hypersurfaces in $M$. Generic smoothness conjecture for area-minimizing hypersurfaces. Area-minimizing hypersurfaces…

Hildebrandt's conjecture. Stationary points of conformally invariant energies should be of class $C^1$.

Armstrong–Vicol conjecture. Using their approach together with large-scale regularity techniques from quantitative homogenization theory, one can reach $C^0_tC^\frac{1-\beta}{2}_x$…

Let $C$ be a domain with boundary $\partial C$, let $d$ be a thickness function on $\partial C$, and define the radial boundary parametrization by … Let $\Omega$ be the resulting d…

Boundary regularity conjecture. For every $x\in[0,A]$, there exists a version of $s\mapsto p^{![2]}_{r,i,c,k}(x,s,l,j)$ that is continuous at $0$ and at $A$. Moreover, there exist…

Weak-assumption conjecture. The results in the paper can be derived under assumptions on the coefficients that are similarly weak to those used in the cited prior work.

Let $M\subset(X^{n+1},\partial X,g,\sigma)$ be a minimizer of the capillary energy, and let $\operatorname{sing}(M)$ denote its singular set. Critical dimension conjecture. The bou…

Landau equation conditional regularity conjecture. As long as the three density functions $\rho$, $E$, and $H$ are “under control”, solutions of the Landau equation remain smooth,…

Let $\langle 90\rangle_N:=\mathcal{J}(\langle 30\rangle_N^3)$, where $\mathcal{J}$ is the Duhamel operator and $\langle 30\rangle_N$ is the stochastic symbol arising in the decompo…

Let $\Omega$ be the bounded domain in the nonlocal Poisson problem, let $u$ solve that problem, let $f\in L^2(\Omega)$, and let $\beta$ be the kernel exponent. Continuity conjectur…

Local Hölder regularity conjecture. Then $u$ is locally Hölder continuous in $\Omega$.

Let $\Omega_X$ be the domain of a convex function $u:\Omega_X\to\mathbb{R}$, and let $v$ be its convex conjugate. Local detachment conjecture. The gradient $\nabla u$ is $L$-Lipsch…

Let $n,k\ge 1$, let $B\subset\mathbb{R}^n$ be an open ball, and let $g\in L^\infty(B;\mathbb{R}^k)$ satisfy $\|g\|_\infty\leq \sigma$. For $u\in C^1(B;\mathbb{R}^k)$, write…

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