57 problems

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 $M$ 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…

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…

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-…

Liquid-drop minimizer conjecture. The energy $E^{\rm G}(m)$ has a minimizer if and only if $m\leqslant m_*$. Moreover, whenever a minimizer exists, it is a ball. The assertion refl…

Let $\omega\ll 1$ and $\gamma\gg 1$, and let $\mathcal{A}_N^{\omega}$ be the class of periodic step functions considered in the generalized Ohta–Kawasaki model. Let $U_N^{\omega}$…

Let $\Omega$ be a unit circle, and let the upper level set (ULS) be the set where an optimal body's height attains its maximum. Assume that the ULS has nonempty interior. A singula…

Let $B_p\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ be the Burkholder integrand, defined by … where $\lvert A\rvert$ denotes the operator norm. Iwaniec's conjecture. The integrand…

Global minimality conjecture. The solutions of the magnetic Ginzburg–Landau equations provided in Theorem 1.1 and Theorem 1.2 are energy minimizers of the energy functional.

The elliptic-cone higher-integrability conjecture. If

Let $\mu$ be the Onsager interaction parameter, and consider the local and global minimizers of the Onsager free-energy functional. A state is axisymmetric if its density is invari…

Let $p(z)$ denote the function associated with an ESFL reduction by a codimension $1$ partial contact curve, and let the truncated Euler operator and the higher-order Euler operato…

Let $C_{kl}$ be the Lawson cone and let $\Phi$ be an integrand constructed as in Theorem Main. An elliptic extension of $\Phi$ is an extension to $\mathbb{R}^{k+l+3}$ that remains…