Viscosity Solutions to Second Order Parabolic PDEs on Riemannian Manifolds

In this work we consider viscosity solutions to second order parabolic PDEs u _t +F(t,x,u,du,d ² u)=0 defined on complete Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold M has nonnegative sectional curvature, we get the finest results. If one additionally requires F to depend on d ² u in a uniformly continuous manner, the assumptions on curvature can be thrown away.     

On Jackson's Inequality for Generalized Moduli of Continuity

Mathematical Notes -     

Optimal Estimates with Moduli of Continuity

In this paper we obtain estimates with optimal constants for the pointwise approximation of functions by linear positive functional, with the aid of a new second order modulus with a parameter, as well as with the aid of the first order modulus of the derivatives of functions.     

Inequalities for Moduli of Continuity in Abstract Banach Spaces

Suppose that X is a Banach space, K denotes the set of real numbers R or the set of nonnegative real numbers R _{+}, is a family of linear operators from X into X such that T _0=I is the identity operator in X, for all , and there exists M such that for all . The expression is called the rth order modulus of continuity of an element x with step h in the space X with respect to the family A(K). The properties of are studied. Bibliography: 3 titles.     

On Some Properties of Multiple Moduli of Continuity

For functions of several variables a property is established similar to the well-known result of S. B. Stechkin about the metric property of almost increasing functions. In the case of one variable the proof is easier than the known one.     

On Moduli of Continuity for Local Times of Fractional Stable Processes

We establish estimates for moduli of continuity of the local times of a class of self-similar stable processes. Our arguments yield also sharp lower bounds for the Hausdorff measure of the level sets of these processes. Finally, we prove the continuity in law of the local times with respect to the self-similarity index for a wide class of self-similar stable processes.     

Analyticity of Higher-Order Moduli of Continuity of Real-Analytic Functions

The Perel'man's result according to which the first modulus of continuity of any real-analytic function f is a function analytic in a certain neighborhood of the origin is generalized to the case of arbitrary moduli of continuity of higher order.     

Existence of Solutions for Vector Optimization on Hadamard Manifolds

In this paper, a relationship between a vector variational inequality and a vector optimization problem is given on a Hadamard manifold. An existence of a weak minimum for a constrained vector optimization problem is established by an analogous to KKM lemma on a Hadamard manifold.     

Equiconvergence Rate in Terms of General Moduli of Continuity for Differential Operators

In this paper the ordinary nonselfadjoint differential operator L, defined by a differential expression with nonzero nonsmooth coefficient of the (n? l)-th derivative and two-point Birkhoff-regular boundary conditions, is considered. The problem of uniform equicon-vergence of the expansion of a given function in series in terms of the eigenfunctions and associated functions of L and in terms of the trigonometric system is investigated. Estimates of the equiconvergence rate in terms of moduli of continuity of the expanded function and the coefficient of the (n ? l)-th derivative are obtained.     

Stable Solutions of Elliptic Equations on Riemannian Manifolds

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows us to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.     

On Topological Index of Solutions for Variational Inequalities on Riemannian Manifolds

In this paper, based on the fixed point index theory for a class of -multivalued maps on absolute neighbourhood retracts, we introduce the notion of index of solvability for a variational inequality on a Riemannian manifold involving a multivalued vector field. We describe the main properties of this topological characteristic and use it to justify the existence of a solution for a variational inequality problem. As application, the problem of optimization of a non-smooth functional on a Hadamard manifold is considered.     

Symmetry Results for Positive Solutions of Some Elliptic Equations on Manifolds

We adapt the method of moving planes in order to obtainsymmetry results for positive solutions of some elliptic equations onmanifolds, in the case where our problem satisfies certain symmetryhypothesis. We also obtain monotonicity results using the slidingmethod.     

大偏差与lp-值Wiener过程在H(o)lder范数下的泛函连续模

本文在Holder范数生成的强拓扑下,建立了l~2-值Wiener过程的大偏差公式,从而得到了l~2-值与l~p-值Wiener过程在Holder范数下的泛函连续模.     

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.     

One Inequality for the Moduli of Continuity of Fractional Order Generated by Semigroups of Operators

Ukrainian Mathematical Journal - A new inequality for the moduli of continuity of fractional order generated by semigroups of operators is obtained. This inequality yields a generalization to the...     

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

For Schr?dinger operators (including those with magnetic fields) with singular scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann–Schwinger equation. Submitted: September 20, 2005. Revised: July 20, 2006. Accepted: October 31, 2006.     

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,