217 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…

Choksi–Peletier conjecture. There exists a unique threshold $\varepsilon^*>0$ such that, for $\varepsilon\leq\varepsilon^*$, the unit ball is the unique minimizer of $G$ under the…

Strict positivity conjecture. For every $b>0$, the second variation at $Y$ is strictly positive definite.

Gurtin's conjecture. In the limit as $\varepsilon\to0$, any family of minimizers of $\{F_\varepsilon\}_{\varepsilon>0}$ should converge to a piecewise constant function that partit…

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…

The Fibonacci set is a point set associated with the Fibonacci permutation, considered as a construction on the two-dimensional torus $\mathbb{T}^2$. Fibonacci-set conjecture. The…

Let $Omegasubset"); mathbb{R}^n$ be a bounded domain, and consider the Aviles–Giga functional … Write $J_{\nabla u}$ for the jump set of $\nabla u$, $[\nabla u]$ for its jump, and…

Aviles–Giga minimizer convergence conjecture. At least when $\Omega$ is convex,

De Giorgi's conjecture. There exists a limit

Kohn–Müller branching conjecture. The minimizers of the Kohn–Müller functional exhibit this branching structure.

Ball's sufficient conditions conjecture. Strict positivity of the second variation, together with suitable versions of quasiconvexity in the interior and at the boundary, should im…

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

Let a system of real-analytic Jordan curves be fixed, and consider minimal surfaces spanning these curves that are relative minima of area. Fix an orientable topological type, mean…

The sufficiency conjecture. The 1-extension property is sufficient for

Capacity conjecture. The existence of minimizers of $E_\kappa$ in $\mathcal{J}$ is completely determined by the value of $\operatorname{cap}(A)$ and the value of $\kappa$.

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…

Let $u$ be the function on the plane defined in polar coordinates by … which is equivalently the imaginary part of $\sqrt{2z/\pi}$ with a cut along the negative real axis. Let…

Let $a$ and $b$ be the “true” rod functions given in the definition of the rod coefficients. The associated operator is of the form … A comparison principle means that, for every i…

Closed elastic rods are subject to a constant twist, and their energy is minimized among admissible rod configurations. A ply is a double helix with a loop on each end. Ply minimiz…

In the de Donder–Weyl formalism, let ${\cal M}^{dDW}_q$ be the submanifold of $\Lambda^nT^*_q{\cal N}$ defined by the constraints $p^{\mu_1\cdots \mu_j}_{i_1\cdots i_j}=0$ for all…

Constant-density conjecture. The space of functions $F$ for which there are two inequivalent solutions consists only of the constant functions.

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

Let $\Omega\subset\mathbb R^2$ be a bounded domain, and consider vector fields $m_\epsilon\colon\Omega\to\mathbb R^2$ with $\operatorname{div}m_\epsilon=0$ and Aviles–Giga energy ……

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…