65 problems

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let $1<p<\infty$, and let $u\in C(\Omega)\cap W^{1,p}(\Omega)$ be continuous and Lebesgue monotone. Determine whether $u$ can…

Let $(u,K)$ be a generalized global minimizer of the planar Mumford–Shah functional $\mathcal{E}(u,K;B)=\int_{B\setminus K}|\nabla u|^2\,dx+\mathcal{H}^1(K\cap B)$ for every bounde…

Let $V_n$ be a sequence of divergence-free matrix fields with $\sup_n\|V_n\|_{L^1}<\infty$, each supported in an open set $A_n$ satisfying $|A_n|\to 0$. The conjecture asserts a co…

For every spatial dimension $n\ge 2$ and every admissible elasticity tensor (isotropic or anisotropic) defining a Hooke-law energy, consider the minimum-compliance shape-optimizati…

For every integer $n\ge 2$ and every smooth uniformly elliptic parametric integrand $\Phi$ on $\mathbb{R}^{n+1}$, there exists an exponent $\alpha=\alpha(n,\Phi)>0$ such that every…

For every dimension $n\ge 2$ and every $p>2$, let $\Omega\subset\mathbb{R}^n$ be open and let $u\in W^{1,p}_{\mathrm{loc}}(\Omega)$ be a weak solution of the inhomogeneous $p$-Lapl…

For every pair of positive integers $n,m$, every bounded open set $\Omega\subset\mathbb{R}^n$, and admissible data $g:\Omega\to\mathbb{R}^m$, consider minimizers…

Let $3\le n\le 6$ and $0<a<1$. Consider the variational problem…

Let $N\ge 1$, $p\in[1,\infty)$, $\gamma>0$, and let $\Omega\subset\mathbb{R}^N$ be a bounded open interval if $N=1$ or a bounded Lipschitz domain if $N\ge 2$. For $\lambda>0$, defi…

Let $B_1\subset\mathbb{R}^2$ be the unit disk, let $\varepsilon>0$, and define the Ginzburg--Landau energy…

Let $m_{\operatorname{ball}}$ be the critical mass below which balls minimize the liquid drop energy, let $m_\mathrm{ex}$ be the critical mass up to which minimizers exist, and let…

Morrey's conjecture. Every rank-one convex function $\mathbf E:\mathbb R^{2\times2}\rightarrow\mathbb R$ is quasiconvex.

Let $M$ be a manifold, let $nl^{0}(M)$ denote the space of null Lagrangians of order zero on $M$, and let $J_{c}(M)$ denote the relevant compactly supported jet space. A classical…

Let … where the $U_j(x)$ are fixed non-random potentials of class $C^1$ satisfying condition, and $(a_j^\omega(t),1\leq j\leq N)$ is a realization of a stationary vector-valued ran…

In the terminology of Constantin Carathéodory, two problems are Carathéodory-equivalent when their respective Lagrangians differ by a total derivative; the corresponding extremals…

Quasiconvexity conjecture. The displayed function is quasiconvex at some matrix $A_0$. By homogeneity, quasiconvexity at $A_0$ implies quasiconvexity at $0$. This proposed strength…

Multiple minimizers via symmetry. The constrained problem admits at least three distinct minimizers related by rotations through $2\pi/3$.

Let $d$ be the dimension, let $cmathcal H(d)$ denote the Hankel subspace, and let $p^*=\max\{p,p/(p-1)\}$ be the conjugate exponent. Write $C(\mathcal H(d),p)$ for the optimal…

Let $n\geq 2$ and $0<\alpha<n$, and let $m_{\operatorname{ball}}$, $m_\mathrm{ex}$, and $m_\mathrm{crit}$ denote respectively the critical mass below which balls minimize the liqui…

Suppose the functional $I$ has the integral form … for a continuous integrand … Let $I_i$ be its invariant hull with respect to an invariant class $\mathcal D_0$, and define … wher…

Choksi–Peletier conjecture. For $V\leq V_*$, the round ball of volume $V$ uniquely minimizes $\mathcal E(\cdot)$ among measurable sets $\Omega\subset\mathbb{R}^3$ with $|\Omega|=V$…

Let $d$ and $q$ be integers with $1\leq q\leq d-1$, and consider the exterior derivative acting on $(d-q)$-forms. The operator is $k$-balanceable when it satisfies the balanceabili…

Uniqueness conjecture. The pair $(K,u)$ is either an elementary global minimizer or a cracktip.

Let $X\subset[0,1]\cap\mathbb Q$ be such that … for all $z$, and let $\theta^\ast\in X$ be an energy well satisfying … for some $z$. Non differentiability at the energy wells. The…

Let $n\geq 2$, let $\Omega=(0,1)^n$, and consider functions with bounded Hessian--Schatten variation, together with the class $\operatorname{CPWL}(\Omega)$ of continuous piecewise-…