2 problems

Matching

Ignat–Merlet conjecture on the viscosity solution as a line-energy minimizer
Ignat–Merlet conjecture. Then $(\nabla\varphi_0)^\perp$ is a global minimizer of $\mathcal{I}_f$ .
solvedline-energies lower-semicontinuity variational-problems
05
Ambrosio–De Lellis–Mantegazza relaxation conjecture for line energies
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 $L^1$ .
openline-energies lower-semicontinuity relaxation
05

Ignat–Merlet conjecture on the viscosity solution as a line-energy minimizer