Related papers: On the minimum of $\sigma$-Brjuno functions
We consider the mixed local/nonlocal semilinear equation \begin{equation*} -\epsilon^{2}\Delta u +\epsilon^{2s}(-\Delta)^s u +u=u^p\qquad \text{in } \Omega \end{equation*} with zero Dirichlet datum, where $\epsilon>0$ is a small parameter,…
Motivated by establishing Neumann Talenti type comparison results, we concern the minimization of the following shape functional under volume constraint: \begin{align*} T(\Omega):=\inf\left\{\frac12 \int_{\Omega} |\nabla u|^2\,dx…
Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak…
We discuss certain recent mathematical advances, mainly due to Perelman, in the theory of Ricci flows and their relevance for renormalization group (RG) flows. We consider nonlinear sigma models with closed target manifolds supporting a…
A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold…
We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an…
In this article we derive, using the Lagrange inversion theorem and applying twice the Fa\`a di Bruno formula, an expression of the minimum of the Gamma function $\Gamma$ as an expansion in powers of the Euler-Mascheroni constant $\gamma$.…
We systematically study several versions of the disjunction and the existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm{B}$, $\Delta(\mathrm{B})$, and $\Sigma(\mathrm{B})$ of formulas of modal…
We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions…
We consider the simplest optimal control problem with one nonregular mixed inequality constraint, i.e. when its gradient in the control can vanish on the zero surface. Using the Dubovitskii--Milyutin theorem on the approximate separation of…
Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given…
We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{X}, d, \mu)$ satisfying a doubling condition and…
Local minimizers of integral functionals of the calculus of variations are analyzed under growth conditions dictated by different lower and upper bounds for the integrand. Growths of non-necessarily power type are allowed. The local…
For $\sigma\in (0,+\infty)$, denote by $B_\sigma^{\infty}$ the Bernstein space (of type $\sigma$) of all entire functions of exponential type $\leq \sigma$ bounded on real axis $\R$. Let $I_d\subset \R$ be a segment of length $d>0$. We…
The aim of this paper is to present an original approach that takes advantage from the geometric features of strictly convex functions to tackle the problem of finding the minimum from another perspective. The general idea is that near the…
A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed "locally perturbating set") are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the…
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at…
Global minimization is a fundamental challenge in optimization, especially in machine learning, where finding the global minimum of a function directly impacts model performance and convergence. This article introduces a novel optimization…
We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex…
In this paper it is shown that any regular critical point of the Mumford-Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the L^1-topology.