Related papers: A zero-one law for improvements to Dirichlet's the…

We consider positive solutions to $\displaystyle -\Delta_p u=\frac{1}{u^\gamma}+f(u)$ under zero Dirichlet condition in the half space. Exploiting a prio-ri estimates and the moving plane technique, we prove that any solution is monotone…

We construct and study the one-parameter semigroup of $\sigma$-finite measures ${\cal L}^{\theta}$, $\theta>0$, on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a…

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the…

We investigate matrices satisfying the Kreiss condition $$\|(zI-T)^{-1}\|\le\cfrac{K}{|z|-1}, \hspace{0.7 cm} |z|>1, $$ with $K$ lying arbitrarily close to $1.$ We provide lower bounds for the power growth of such matrices, which complement…

Can the behavior of a random matrix be improved by modifying a small fraction of its entries? Consider a random matrix $A$ with i.i.d. entries. We show that the operator norm of $A$ can be reduced to the optimal order $O(\sqrt{n})$ by…

In this paper, we establish an improved version of a saddle point theorem ([4]) removing a weak lower semicontinuity assumption at all. We then revisit some of the applications of that theorem in the light of such an improvement. For…

For $p>1$, we introduce the cutoff Sobolev inequality on general metric measure spaces, and prove that there exists a metric measure space endowed with a $p$-energy that satisfies the chain condition, the volume regular condition with…

This paper shows that a wide class of effective learning rules -- those that improve a scalar performance measure over a given time window -- can be rewritten as natural gradient descent with respect to a suitably defined loss function and…

Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,\Lambda_t)_{t\geq0}$. Suppose that for any Schwartz function $\varphi$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}}…

We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to…

We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of…

Let $\varphi$, $\psi\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^{4}(\mathbb{D})$ satisfies the following: (1) the…

The so-called {\it zero number diminishing property} (or {\it zero number argument}) is a powerful tool in qualitative studies of one dimensional parabolic equations, which says that, under the zero- or non-zero-Dirichlet boundary…

Let $x\ge 2$. The $\psi$-form of the prime number theorem is $\psi(x) =\sum\sb{n \le x}\Lambda(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$. Tur\'an proved in 1950…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half plane $\mathbb{R}^2_+$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $\omega\in…

This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain…

Consider a real matrix $\Theta$ consisting of rows $(\theta_{i,1},\ldots,\theta_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}\theta_{i,1}+\cdots+x_{n}\theta_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small…

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-\Delta_p u = \lambda m(x)|u|^{p-2}u + \eta a(x)|u|^{q-2}u + f(x)$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where $q<p$. Under…

For plurisubharmonic functions $\varphi$ and $\psi$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $\varphi$ at the origin vanishes, we show that the mass of the origin with…

We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive…