11 problems

Let $\Sigma$ be the singular set of the anisotropic obstacle problem, and let $A$ be the coefficient matrix with ellipticity constants. $C^{1,\alpha}$-rectifiability conjecture. Th…

Non-trapping obstacle conjecture. The local energy of solutions should decay for any non-trapping obstacle, and consequently such solutions should scatter.

Let $Z\subset\mathbb{R}^2$ be a rectifiable barrier, and define the minimum-time function by … where $R^Z(t)$ is the region reachable from the initial fire $R_0$ by absolutely cont…

Consider the weak Harnack inequality and Hölder continuity results for the solutions of the one-obstacle, two-obstacle and N-membranes problems associated with the nonlocal operato…

Let $s=1$ and consider an obstacle problem of the form of the wave equation with obstacle, with obstacle $g$, and let $u$ be the weak solution obtained through the convex minimizat…

The obstacle problem concerns a weak solution produced by the proposed convex minimization and semi-discrete approximation scheme; in cases where weak solutions are nonunique, comp…

Let $K\subset\mathbb{R}^N$ be a starshaped compact obstacle, and consider the nonlocal obstacle problem and the Liouville property described for compact obstacles in the paper. Lio…

Non-transversal intersection conjecture. The free boundary intersects the fixed boundary non-transversally.

Let $E$ be the limiting functional obtained as the … -convergence, compactness of minimizers under the same boundedness constraint, and a sufficient condition for global optimizati…

Let the free boundary be the boundary separating the contact set from the non-contact set in the fractional obstacle problem, and let $N$ be the ambient dimension. Free-boundary me…

Let $D$ be a convex domain, let $B({\bf x},r)$ be a ball of radius $r$ contained in $D$, and write $\Omega({\bf x})=D\setminus B({\bf x},r)$. Let $\heartsuit(D)$ denote the heart o…