Related papers: On sharp constants in one-dimensional embedding th…
It is not uncommon in analysis that existence of extremal objects is obtained via an iterative procedure: we start from a given admissible object, then modify it, then modify again etc... If being extremal means maximimizing a real valued…
We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.
Our main purpose in this paper is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi-Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger-Moser…
In a previous paper, the author proved the existence of extremal function for the Moser-Trudinger inequality on a compact manifold. In the this paper, we will give a new proof of one of the key proposition.
We extend the classical Heisenberg uncertainty principle to a fractional $L^p$ setting by investigating a novel class of uncertainty inequalities derived from the fractional Schr\"odinger equation. In this work, we establish the existence…
The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction…
We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the…
We construct an example of a real-valued continuous non-constant function $f$ defined on a connected complete metric space $X$ such that every point of $X$ is a point of local minimum or local maximum for $f$. The space $X$ is connected but…
A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…
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…
A well known conjecture states that constant functions are extremizers of the $L^2 \to L^6$ Tomas-Stein extension inequality for the circle. We prove that functions supported in a $\sqrt{6}/80$-neighbourhood of a pair of antipodal points on…
Motivated by recent work on ordinal embedding (Kleindessner and von Luxburg, 2014), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons.…
The problem is considered as to whether a monotone function defined on a subset P of a Euclidean space can be strictly monotonically extended to the whole space. It is proved that this is the case if and only if the function is {\em…
This paper focus on the symmetry and symmetry breaking about the second order Hydrogen Uncertainty Principle. \emph{Firstly}, by choosing a suitable test function, we give a negative answer to the conjecture presented by Cazacu, Flynn and…
Let $V$ be a symmetric convex body in $\R^m$. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in $V$ and discuss certain properties of the extremal functions. Markov-type inequalities…
We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $\text{Re}…
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic…
We perform conformal perturbation theory by marginal operators to first order. A suitable renormalization method is needed that makes the conformal invariance of the deformed correlation functions manifest. Combining the embedding space…
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is…
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of…