Related papers: Korenblum-Type Extremal Problems in Bergman Spaces

It is shown that, for any given $p\ge5$, $A>0$ and $B>0$, the exact upper bound on $\mathsf{E}|\sum X_i|^p$ over all independent zero-mean random variables (r.v.'s) $X_1,\ldots,X_n$ such that $\sum\mathsf{E}X_i^2=B$ and…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $m$ a positive integer, we remark that there is a constant $C$ so that, for all $r\in\lbrack1,\frac {2m}{m+1}],$ the supremum of the ratio between the $\ell_{r}$ norm of the coefficients of…

In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$. If $s$ is a positive integer satisfying $2s<n$, we let $P_g^s$ be the GJMS operator of order $2s$ in $M$. We investigate in this paper the extremal values taken by fixed…

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces…

For open radial sets $\Omega\subset \mathbb{R}^N$, $N\geq 2$ we consider the nonlinear problem \[ (P)\quad Iu=f(|x|,u) \quad\text{in $\Omega$,}\quad u\equiv 0\quad \text{on $\mathbb{R}^N\setminus \Omega$ and }\lim_{|x|\to\infty} u(x)=0, \]…

We prove a conjecture of B. Gr\"unbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on…

We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $\lambda_1$ of the Dirichlet…

he equation $-\Delta u = \lambda e^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $\lambda = 2(N-2)$. If $N\ge 4$ we show that…

A simple method is shown to provide optimal variational bounds on $f$-divergences with possible constraints on relative information extremums. Known results are refined or proved to be optimal as particular cases.

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with…

In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the…

An extremal point of a positive threshold Boolean function $f$ is either a maximal zero or a minimal one. It is known that if $f$ depends on all its variables, then the set of its extremal points completely specifies $f$ within the universe…

We prove that in many cases the existence of an extremal metric for some Laplace eigenvalue in a conformal class allows to find extremal metrics in conformal classes close by. As a consequence and as part of the arguments we obtain…

We extend Berge's Maximum Theorem to allow for incomplete preferences. We first provide a simple version of the Maximum Theorem for convex feasible sets and a fixed preference. Then, we show that if, in addition to the traditional…

Aleksandrov-Bakelman-Pucci maximum principles are studied for a class of fully nonlinear integro-differential equations of order $\sigma\in [2-\varepsilon_0,2)$, where $\varepsilon_0$ is a small constant depending only on given parameters.…

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v &=& \gamma G(u,v)…

We consider some second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at…

The Landau-Kolmogorov problem consists of finding the upper bound $M_k$ for the norm of intermediate derivative $|f^{(k)}|$, when the bounds $|f| \le M_0$ and $|f^{(n)}| \le M_n$, for the norms of the function and of its higher derivative,…