4 problems

Ignat–Merlet conjecture. Then $(\nabla\varphi_0)^\perp$ is a global minimizer of $\mathcal{I}_f$.

Ambrosio–De Lellis–Mantegazza conjecture. If $f(t)=t^p$ with $1\leq p\leq3$, then $\overline{\mathcal{I}_f}$ is lower semicontinuous for the strong topology in $L^1$.

Micromagnetic line-energy conjecture. Some line energies $\mathcal{I}_f$ arising from micromagnetics are conjectured to be the $\Gamma$-limit of the corresponding functionals in th…

Aviles–Giga conjecture. If $f(t)=\frac{1}{3}t^3$, then $\mathcal{I}_f$ is the $\Gamma$-limit of $AG_\varepsilon$.