Related papers: Almost everywhere uniqueness of blow-up limits for…
In this article we study a normalised double obstacle problem with polynomial obstacles $ p^1\leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$. In dimension two we give a complete characterisation of blow-up solutions…
For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion…
In this paper, we complete the classification of the blow-up limits of minimizers of the vectorial Bernoulli free boundary problem. Furthermore, we study the vectorial Bernoulli free boundary problem in a bounded box $D$, with a constraint…
We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional…
We study the free-boundary equation \[ \Delta u=\chi_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of \(\partial\{|\nabla u|>0\}\), the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012},…
In this paper we study the blow-ups of the singular points in the boundary of a minimizing cluster lying in the interface of more than two chambers. We establish a sharp lower bound for the perimeter density at those points and we prove…
We prove the uniqueness of $L^1$ blow-down limit at infinity for an entire minimizing solution $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ of a planar Allen-Cahn system with a triple-well potential. Consequently, $u$ can be approximated by a…
In this note, we prove the uniqueness of blowups at singular points of the no-sign obstacle problem $\Delta u=\chi_{_{B_1\backslash \{u=|Du|=0\}}}\ \text{in}\ B_1,$ thus give a positive answer to a problem raised in \cite[Notes of Chaper 7,…
We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to…
In general, solutions $u$ to \[ \Delta u(\mathbf{x})=f(\mathbf{x})\chi_{\{u>\psi\}} \] are not $C^{1,1}$, even for $f$ smooth and $\psi(\mathbf{x})\equiv0$. Points around which $u$ is not $C^{1,1}$ are called singular points, and the set of…
In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of…
We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the…
We study the regularity of free boundaries in the multiple elastic membrane problem in the plane. We prove the uniqueness of blow-ups, and that the free boundaries are $C^{1,\log}$-curves near a regular intersection point.
We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…
Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In…
We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence…
In this work, we consider the singular set in the thin obstacle problem with weight $|x_{n+1}|^a$ for $a\in (-1, 1)$, which arises as the local extension of the obstacle problem for the fractional Laplacian (a non-local problem). We develop…
A simple criterion of the existence of (type-I) blow-up solutions for nonautonomous ODEs is provided. In a previous study [Matsue, SIADS, 24(2025), 415-456], geometric criteria for characterizing blow-up solutions for nonautonomous ODEs are…
We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined…
We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature…