Related papers: The singular set in the Stefan problem
Given $k\in \mathbb{R},$ $v,$ $D>0,$ and $n\in \mathbb{N},$ let $\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\geq k,$ volume $>v,$ and…
In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ \Delta u = - \chi_{\{u>0\}} \qquad \hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite…
We prove that for the Navier Stokes equation with dissipation $(-\Delta)^{\alpha}$, where $1<\alpha<{5/4}$, and smooth initial data, the Hausdorff dimension of the singular set at time of first blow up is at most $5-4\alpha$. This unifies…
We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild…
A fundamental problem in the dimension theory of self-affine sets is the construction of high-dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such…
We study the parabolic obstacle problem $$\lap u-u_t=f\chi_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1$$ and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular…
We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share…
We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $\gamma(x)$ and how it changes. Under a mild continuity assumption on…
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator…
We show that given $\alpha \in (0, 1)$ there is a constant $c=c(\alpha) > 0$ such that any planar $(\alpha, 2\alpha)$-Furstenberg set has Hausdorff dimension at least $2\alpha + c$. This improves several previous bounds, in particular…
Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular…
A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain…
For a certain class of genuinely nonlinear two-by-two planar hyperbolic systems we show that any classical solution on a smoothly bounded domain has nontangential boundary limits except on a set whose Hausdorff dimension is bounded by some…
In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable…
We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_\varphi:(x,t)\mapsto(x+\alpha,t+\varphi(x))$ where $Tx=x+\alpha$ is an irrational rotation on the circle $\T$ and…
We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and H\"{o}lder continuous in time. For the limiting free boundary problem, we analyse the…
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 consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for…
The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing…
The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal…