51 problems

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 $Omegasubset"); mathbb{R}^n$ 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…

Proposed functional's Gamma-convergence conjecture. The proposed functional converges, in the $\Gamma$-convergence sense, to a total variation type functional on graphs as…

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 ……

Second-order Gamma-limit conjecture. If $g$ has this form, then the limiting functional, with the same scaling $\varepsilon$, is proportional to the size of the jump set $J_g$.

Let $N\geq 1$, $p\geq 1$, and $\gamma\in\mathbb{R}\setminus\{0\}$. Let $Q=(0,1)^N$, let … and define $\kappa_{N,p,\gamma}$ by taking the infimum over measurable families…

Let $\kappa$ be the parameter of the type-I superconductivity model. Define the one-dimensional transition energy by … Let … and let $\sigma_\kappa$ denote the corresponding two-di…

Let $p\in(1,\infty)$ and $t_{\max}>0$. For each natural number $n$, let $(\hat{k}_n,\hat{\mu}_n)\in L^\infty(\Omega;\mathbb{R}_+)^2$ solve the (vp-IMD) problem, respectively the (s…

Let $\{p_n\}_{n=1}^\infty$ be a decreasing sequence with $1<p_n<\infty$ and $p_n\to1$. For each $n$, let $(\hat{k}_n,\hat{\mu}_n)$, $\hat{\boldsymbol{\tau}}_n$, and…

Let $u_0$ be the limit temperature and let $H$ denote the mean curvature of the boundary of the domain. Curvature–temperature dependence conjecture. The design of the shape of the…

Three-dimensional Gamma-limit conjecture. For $n=3$ and $\rho\to0$, if $\zeta\in(\zeta_0,\zeta_1)$, the Gamma-limit of $F'_\rho$ is a quadratic form in the principal curvatures for…

Two-dimensional Gamma-limit conjecture. For $n=2$ and $\rho\to0$, if $\zeta\in(0,\zeta_1)$, the Gamma-limit of $F'_\rho$ is the elastica functional for closed $W^{2,2}$ curves in…

Helfrich-limit conjecture. The Gamma-limit of the rescaled bilayer energy is the Helfrich energy.

Let $G$ be a non-negative integrable kernel, and consider the aggregation-diffusion models and associated free boundary problems discussed in the paper, with energies whose…

Let $\Omega\subset\mathbb R^n$ be a bounded domain, let $u:\Omega\to\mathbb R$ denote the relative concentration of two phases with pure-phase values $u^-<u^+$, and consider the Ca…

Let $n\geq 2$ be an integer, let $E\subset \mathbb{R}^n$ have boundary $\Sigma:=\partial E$ of class $C^2$, and let $\Omega\subset \mathbb{R}^n$ be open. For $\lambda>0$, define ……

Line-singularity conjecture. In the limit, the energy concentrates on line singularities corresponding to interfaces (domain walls) in micromagnetics.

Aviles–Giga conjecture. The $\Gamma$-limit of $AG_{\varepsilon}$ as $\varepsilon\to0$ is a functional finite on such functions and charges jumps of the gradient field…

Aviles–Giga Gamma-limit conjecture. The $\Gamma$-limit of the Aviles–Giga functional is, up to a constant, the total mass of $\mu$.

De Giorgi's conjecture. The functionals $\mathcal{A}_\epsilon$ $\Gamma$-converge in $L^1(\Omega)$ to $\sigma\mathbf{E}_A$:

Consider the diffuse Willmore functional and its Gamma limit for phase-field approximations. In the sharp-interface limit, the relevant one-dimensional configurations are represent…

Let $\mathcal A_\varepsilon$ be the Cahn--Hilliard diffuse-interface energy and let its squared $L^2(\Omega)$-gradient be integrated against the diffuse area measure. De Giorgi's c…

Let $\Omega\subset\mathbb{R}^d$ be a domain and let $MS$ denote the Mumford–Shah functional … where $u\in SBV(\Omega)$, $\nabla u$ is the absolutely continuous part of its gradient…

Let $W$ be the potential defining the nematic-isotropic energy, and let $K$ denote the wall cost obtained from the one-dimensional heteroclinic profile. For a wall, the one-dimensi…

Necessity conjecture. These two additional conditions are not really necessary for the homogenization conclusions of Theorem Homogenization I to remain true. The conjecture concern…