相关论文: Self-improving estimates of growth of subharmonic …

We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $\underset{x\in\Omega}{\mbox{ess-sup}}\hspace{0.03cm} f\left(\frac{x}{\varepsilon}, Du\right)$, when $\Omega$ is a bounded…

We study the complexity of computing majority as a composition of local functions: \[ \text{Maj}_n = h(g_1,\ldots,g_m), \] where each $g_j :\{0,1\}^{n} \to \{0,1\}$ is an arbitrary function that queries only $k \ll n$ variables and $h :…

We study the size of the set of points where the $\alpha$-divided difference of a function in the H\"older class $\Lambda_\alpha$ is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which…

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to…

Let $\Omega\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partial\Omega$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc…

We obtain estimates in the corona theorem for the algebra of analytic functions in the unit disc whose nth derivative is bounded, and its subalgebras defined by the boundary continuity of the nth derivative. The corona theorem for such…

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the…

In this paper we study $2$nd order $L^\infty$ variational problems, through seeking to minimise a supremal functional involving the Hessian of admissible functions as well as lower-order terms. Specifically, given a bounded domain…

We study the approximation properties of a harmonic function $u \in H\sp{1-k}(\Omega)$, $k > 0$, on relatively compact sub-domain $A$ of $\Omega$, using the Generalized Finite Element Method. For smooth, bounded domains $\Omega$, we obtain…

Let $\Omega\subset \mathbb{R}^n $ be any open set and $u$ be a weak supersolution of $\mathcal{L}u=c(x)g(|u|)\frac{u}{|u|}$ where \[\mathcal{L}u(x)=\text{p.v.} \int_{\mathbb{R}^n} g\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)…

In this paper it is shown that if $\Omega \subset \mathbb{R}^N$ is an open, bounded Lipschitz set, and if $f: \Omega \times \mathbb{R}^{d \times N \times N} \rightarrow [0, \infty)$ is a continuous function with $f(x, \cdot)$ of linear…

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space $Hol (\D,\Omega)$ of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case…

We show higher integrability of minimisers of functionals \[ I(u) = \int_{\Omega} f(x,u(x)) ~\mathrm{d}x \] subject to a differential constraint $\mathscr{A} u=0$ under natural $p$-growth and $p$-coercivity conditions for $f$ and regularity…

We consider the class of convex minimization problems, composed of a self-concordant function, such as the $\log\det$ metric, a convex data fidelity term $h(\cdot)$ and, a regularizing -- possibly non-smooth -- function $g(\cdot)$. This…

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise…

We develop and use some key concepts of potential theory, such as balayage and duality between measures and their potentials, to study the distribution of masses of subharmonic functions while restrictions to their growth near the boundary…

In min-min optimization or max-min optimization, one has to compute the gradient of a function defined as a minimum. In most cases, the minimum has no closed-form, and an approximation is obtained via an iterative algorithm. There are two…

Given a Lipschitz function $f:\{1,...,d\}^\mathbb{N} \to \mathbb{R}$, for each $\beta>0$ we denote by $\mu_\beta$ the equilibrium measure of $\beta f$ and by $h_\beta$ the main eigenfunction of the Ruelle Operator $L_{\beta f}$. Assuming…

We prove a sharp decay of capacity of sublevel sets of a $(\omega,m)$-subharmonic functions on a $n$-dimensional compact Hermitian manifold $(X,\omega)$ which generalizes the case $m=n$ as well as the case $1\leq m\leq n$ on a compact…

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give…