Related papers: A critical point theorem on a closed ball and some…
A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
The classical Gauss--Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We…
An initial-value problem for an ordinary differential equation of the first order, is considered. It is supposed that the right-hand side of the equation is a continuous function defined on a set consisting of an open set and a part of its…
The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator…
A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As…
On the two dimensional sphere, we consider axisymmetric critical points of an isoperimetric problem perturbed by a long-range interaction term. When the parameter controlling the nonlocal term is sufficiently large, we prove the existence…
This paper is a continuation of \cite{Lu1}. In Part I, applying the new splitting theorems developed therein we generalize previous some results on computations of critical groups and some critical point theorems to weaker versions. In Part…
We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the…
We prove that if a continuous piecewise-smooth map on $\mathbb{R}^n$ is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important…
The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.
The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is…
In critical loop models, there exist diagonal fields with arbitrary conformal dimensions, whose $3$-point functions coincide with those of Liouville theory at $c\leq 1$. We study their $N$-point functions, which depend on the $2^{N-1}$…
A fundamental open question asking whether all real-valued strongly quasiconvex functions defined on $\mathbb R^n$ are necessarily continuous, akin to their convex counterparts, is answered in detail in this paper. Among other things, we…
We consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected.…
In the present article we describe a class of algebraic curves on which rational functions of two arguments may reach all their possible limiting values. We also solve a similar question for functions that can be represented as a uniform…
This note aims at obtaining a variational characterization of complex structures by means of a calculus of variations for real vector bundle valued differential forms, and outlines a perspective to study existence questions via functionals…
We investigate the boundary behavior of holomorphic functions with respect to a family of curves in a domain of finite type. This work is a generalization of \u{C}irka's classical result on the unit ball and it supplements the result by…
We obtain an upper bound for the number of critical points of the systole function on $\mathcal{M}_g$. Besides, we obtain an upper bound for the number of those critical points whose systole is smaller than a constant.
The existence of radial solutions of a nonlinear Dirichlet problem in a ball is translated to the language of Mechanics, i.e. to requirements on the time of motion of a particle in an external potential and under the action of a viscosity…