11 problems

Schaeffer's conjecture. Generically, the weak solution of the obstacle problem is also a strong solution, in the sense that the free boundary is a $C^\infty$ manifold. Equivalently…

Let $u$ be a solution of the no-sign obstacle problem … and let $\Sigma_{n-1}$ denote the top stratum of the free boundary, with $\Sigma_{n-1,a}$ its anomalous part. Here…

Let $u$ solve the obstacle problem in a domain $d4aa$ with boundary datum $g$, and let the singular part of its free boundary denote the set of free boundary points where the conta…

Let the initial free surface lie in a strip of width $2C$, and let $\kappa$ be the geometric constant such that the corresponding Hele-Shaw semi-flow becomes analytic after time…

Classification conjecture. The coincidence set of $u$ is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.

Paraboloid blow-up conjecture. For $C^\infty$ coefficients, paraboloids do occur as coincidence sets of blow-up limits with moving centers approaching singularities.

Let $\Lambda^*$ denote the parameter governing the classification of global solutions in the obstacle problem. The paper considers the description of the regular free boundary near…

Let $w$ be the function arising in the two-dimensional analysis of the obstacle problem, and write points in polar coordinates $(r,\theta)$. The $3/2$-profile conjecture. For $N=2$…

Let $u$ be a global solution of the obstacle problem … The coincidence-set classification conjecture. The coincidence set of each global solution is a half-space, an ellipsoid, a p…

Consider a global solution $u_0$ of the $p$-parabolic equation for $t<0$ such that $u_0\geq 0$, $u_0(0,0)=0$, and $u_0$ has growth of order $R^q$ for large $R$, where … Under the a…

Let $\mathcal{D}subseteq \mathbb{R}^3$ be an open set with smooth boundary and bounded complement, and consider the wave equation on $\mathbb{R}timesmathcal{D}$ with Dirichlet boun…