Related papers: Quantitative stratification of $F$-subharmonic fun…

We introduce an approach to study homogenisation of a large class of singular SPDEs of the form $$ \partial_t u_\varepsilon - \nabla\cdot {A}(x/\varepsilon,t/\varepsilon^2) \nabla u_\varepsilon = F(x/\varepsilon , t/\varepsilon^2,…

In this paper, we study the stationary solutions of semilinear elliptic equation with singular nonlinearity $$ \Delta u=u^{-p}+f,\,\,u\geq 0\text{ in }\Omega\subset\mathbb{R}^n, $$ where $ n\geq 2 $, $ p>1 $, $ \Omega $ is a bounded domain,…

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex…

Let $X$ be a compact K\"ahler manifold and $\theta$ a smooth closed $(1,1)$-real form representing a big cohomology class $\alpha \in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that…

We study the stratification of the singular locus of four dimensional $\mathcal{N}=2$ Coulomb branches. We present a set of self-consistency conditions on this stratification which can be used to extend the classification of scale-invariant…

Let $E$ be an arbitrary closed set on the unit circle $\partial \mathbb{D}$, u be a harmonic function on the unit disk $\mathbb{D}$ satisfying $|u(z)|\lesssim (1-|z|)^\gamma \rho^{-q}(z)$ where $\rho(z)= \mathop{\rm dist}(z, E)$, $\gamma$,…

Suppose that $p \in (1,\infty]$, $\nu \in [1/2,\infty)$, $\mathcal{S}_\nu = \left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |\phi| < \frac{\pi}{2\nu}\right\}$, where $\phi$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and…

We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-\Delta)^\gamma u=fu^p\quad\text{in }\mathbb R^n\setminus\Sigma $$ which are singular at $\Sigma$. Here $\gamma…

Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms…

In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\colon\R\to {\cal{S}}^{m-1}$ such that $|u_k|_{\dot H^{1/2}(\R,{\cal{S}}^{m-1})}\le C.$ More precisely we show that there exist a weak…

First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly…

In this paper we study a class of functions that appear naturally in some equidistribution problems and that we call $F$-harmonic. These are functions of the universal cover of a closed and negatively curved which possess an integral…

In this survey, we examine the properties of the top dimensional singular set $\text{sing}_{*}(u)$, including several theorems, geometric properties, and relation to homogeneous degree zero minimizers.

Given a finite subset $\Sigma\subset\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma;q)=\big\{\sum_{n=0}^\infty…

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form \[ \mathcal{F}[u] := \int_{\Omega} f \bigg( \frac{1}{2} \bigl( \nabla u(x) + \nabla u(x)^T \bigr) \bigg)\,\mathrm{d} x, \qquad u : \Omega…

In this paper, we consider the following variational problem: \begin{eqnarray*} \inf_{u\in…

This note is based on a series of lectures delivered in Kyoto University. This note surveys the homogeneous Besov space $\dot{B}^s_{pq}$ on ${\mathbb R}^n$ with $1 \le p,q \le \infty$ and $s \in {\mathbb R}$ in a rather self-contained…

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a stratified framework. In many applications, there is no a priori information on what points should…

The following ``Key Lemma'' plays an important role in Parusinski's work on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer n, there is a finite set of homogeneous symmetric polynomials…

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation $$ -\Delta u-u=Q|u|^{p-1}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim_{|x|\to0}u(x)=+\infty, $$ where $N\geq 2$, $p>1$ and the…