On Hölder continuity of approximate solutions to parametric equilibrium problems

We establish verifiable sufficient conditions for Hölder continuity of approximate solutions to parametric equilibrium problems, when solutions may be not unique. Many examples are provided to illustrate the need of considering approximate solutions instead of exact solutions and the essentialness of the imposed assumptions. As applications, we derive this Hölder continuity for constrained minimization, variational inequalities and fixed point problems.     

Convergence of Rothe's Method in Hölder Spaces

The convergence of Rothe's method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe's approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.     

Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems

In this paper, we establish the Hölder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set-valued. Our main assumptions are weaker than those in the literature, and the results extend and improve the recent ones. Furthermore, as an application of Hölder continuity, we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters.     

Remarks on Hölder continuity for solutions of the p-Laplace evolution equations

We study the interior Hölder regularity problem for the gradient of solutions of the p-Laplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa.     

On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Hölder coefficients in a Hölder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Hölder coefficients in a Hölder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

Livsic theorems for hyperbolic flows

We consider Hölder cocycle equations with values in certain Lie groups over a hyperbolic flow. We extend Livsic's results that measurable solutions to such equations must, in fact, be Hölder continuous.

     

Global Minimization Algorithms for Holder Functions

This paper deals with the one-dimensional global optimization problem where the objective function satisfies a Hölder condition over a closed interval. A direct extension of the popular Piyavskii method proposed for Lipschitz functions to Hölder optimization requires an a priori estimate of the Hölder constant and solution to an equation of degree N at each iteration. In this paper a new scheme is introduced. Three algorithms are proposed for solving one-dimensional Hölder global optimization problems. All of them work without solving equations of degree N. The case (very often arising in applications) when a Hölder constant is not given a priori is considered. It is shown that local information about the objective function used inside the global procedure can accelerate the search signicantly. Numerical experiments show quite promising performance of the new algorithms.     

Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs

We here establish an a priori Hölder estimate of Krylov and Safonov type for the viscosity solutions of a degenerate quasilinear elliptic PDE of non-divergence form. The diffusion matrix may degenerate when the norm of the gradient of the solution is small: the exhibited Hölder exponent and Hölder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient. In particular, the given estimate is independent of the regularity of the coefficients. As in the original paper by Krylov and Safonov, the proof relies on a probabilistic interpretation of the equation.     

A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.     

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift

We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Hölder continuous drift term. We prove the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Itô-Tanaka type. The proof requires non-standard elliptic estimates in Hölder spaces. As an application of the stochastic flow, we obtain a Bismut-Elworthy-Li type formula for the first derivatives of the associated diffusion semigroup.     

A Subspace of Hölder Space Consisting Only of Nonsmoothest Functions

In the first part of this paper, we give a complete answer to an old question of the geometric theory of Banach spaces; namely, we construct an infinite-dimensional closed subspace of Hölder space such that each function not identically zero is not smoother at each point than the nonsmoothest function in Hölder space. In the second part, using constructions from the first part, we show that the set of functions from Hölder space which are smoother on a set of positive measure than the nonsmoothest function is a set of first category in this space.     

Evaluating Improvements for Spacings of Order Statistics

We evaluate sharp upper bounds for the consecutive spacings of order statistics from an i.i.d. sample, measured in scale units generated by various central absolute moments of the parent distribution. The bounds are based on the projection method combined with the Hölder inequalities. We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically and compare them with other existing bounds.     

The equivalence of generalized Hölder and Cesáro matrices

We show that the generalized Hölder and Cesáro matrices of order α > −1 are equivalent. We also show that the corresponding is true for doubly infinite generalized Hölder and Cesáro matrices.     

Controlling rough paths

We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Hölder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons’ theory of rough paths in Hölder topology.     

Reverse Hölder inequalities for singular parabolic equations near the boundary

We show that weak solutions to a singular parabolic partial differential equation globally belong to a higher Sobolev space than assumed a priori. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. The results extend to singular parabolic systems as well. Motivation for studying reverse Hölder inequalities comes partly from applications to regularity theory.     

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.     

Regularity Properties of Some Stochastic Volterra Integrals with Singular Kernel

We prove the Hölder continuity of some stochastic Volterra integrals, with singular kernels, under integrability assumptions on the integrand. Some applications to processes arising in the analysis of the fractional Brownian motion are given. The main tool is the embedding of some Besov spaces into some sets of Hölder continuous functions.     

The 1-d stochastic wave equation driven by a fractional Brownian sheet

In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one- dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the fractional Brownian sheet.