Lder Spaces Research Articles

Optimal control in the boundary value problem for elliptic equations with degeneration

The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special H\"{o}lder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will bethe solution to the given problem.
 The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.

The Approximation of Weighted Hölder Functions by Fourier-Jacobi Polynomials to the Singular Sturm-Liouville Operator

In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.

The DB Boundary Condition in Electromagnetic Scattering Revisited

For the solution of the scattering problem for time-harmonic electromagnetic waves with boundary conditions for the normal components of both the electric and the magnetic field, an integral equation method proposed by Gülzow in 1988 is reconsidered. It is made more concise, made symmetric with respect to the electric and magnetic field, and also extended from the classical Hölder spaces to a contemporary Sobolev space setting. For this regime, new reciprocity principles for scattering of plane waves and dipole fields are established and a related far field operator is discussed. Finally, the corresponding inverse scattering problem to recover the shape of the scatterer from far field data is discussed with the main emphasis on uniqueness results.

Smoothness of Generalized Solutions of the Neumann Problem for a Strongly Elliptic Differential-Difference Equation on the Boundary of Adjacent Subdomains

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

Low Mach number limit of multidimensional steady flows on the airfoil problem

In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter $\varepsilon$, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is on the H\"{o}lder space and is unique. Moreover, the convergence rate is of order $\varepsilon^2$. It is noticeable that, due to the feature of the airfoil problem, the extra force dominates the asymptotic decay rate of the compressible flow to the infinity. And the effect of extra force vanishes in the limiting process from compressible flows to the incompressible ones, as the Mach number goes to zero.

Hölderian weak invariance principle under the Maxwell and Woodroofe condition

We investigate the weak invariance principle in H{\"o}lder spaces under some reinforcement of the Maxwell and Woodroofe condition.

Numerical Analysis of Nonlocal Fracture Models in Hölder Space

In this work, we calculate the convergence rate of the finite difference approximation for a class of nonlocal fracture models. We consider two point force interactions characterized by a double well potential. We show the existence of a evolving displacement field in H\"{o}lder space with H\"{o}lder exponent $\gamma \in (0,1]$. The rate of convergence of the finite difference approximation depends on the factor $C_s h^\gamma/\epsilon^2$ where $\epsilon$ gives the length scale of nonlocal interaction, $h$ is the discretization length and $C_s$ is the maximum of H\"older norm of the solution and its second derivatives during the evolution. It is shown that the rate of convergence holds for both the forward Euler scheme as well as general single step implicit schemes. A stability result is established for the semi-discrete approximation. The H\"older continuous evolutions are seen to converge to a brittle fracture evolution in the limit of vanishing nonlocality.

Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces

We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $C^{1,\alpha }$ to $L^\infty _tC^{1,\alpha }_x$ for any $0<\alpha <1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $c^{1,\alpha }$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.

Global well-posedness of the MHD equations in a homogeneous magnetic field

In this paper, we study the MHD equations with small viscosity and resistivity coefficients, which may be different. This is a typical setting in high temperature plasmas. It was proved that the MHD equations are globally well-posed if the initial velocity is close to 0 and the initial magnetic field is close to a homogeneous magnetic field in the weighted H\"{o}lder space, where the closeness is independent of the dissipation coefficients.

A remark on a theorem of Berkes

Let x be a standard Wiener process on \([0,\ 1]\), starting at the origin and with all its sample functions lying in the H\(\ddot{\text {o}}\)lder space \(H_{\alpha }^0\) of exponent \(\alpha ,\ 0< \alpha < \frac{1}{2}\). Define for \(n = 1, 2, \ldots \) processes \(x_n\) and \(z_n :\ x_n(t) = \frac{x(nt)}{\sqrt{n}}\) and \(z_n(t) = \frac{x(nt)}{\sqrt{n}f(n)},\ t \in [0,\ 1]\) where \(f(n) \rightarrow \infty \) and \(\frac{f(n)}{\sqrt{\log \log n}} \rightarrow 0\). In this paper, we show that the limit set of each of the random sequences \((x_n)\) and (\(z_n\)) of elements in \(H_{\alpha }^0\) in its norm topology is all of the space \(H_{\alpha }^0\).

Value Function Regularity in Option Pricing Problems Under a Pure Jump Model

In this paper, we consider an option pricing problem in a pure jump model where the process X(t) models the logarithm of the stock price. By the Schauder fixed point theorem, we show the existence and uniqueness of the solutions in H\(\ddot{o}\)lder spaces for the European and American option pricing problems respectively. Due to the estimates of fractional heat kernel, we give the regularity of the value functions \(u_{E}(t,x)\) and \(u_{A}(t,x)\) of the European option and the American option respectively.

On the Cauchy problem for backward stochastic partial differential equations in Hölder spaces

This paper is concerned with solution in H\"{o}lder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic spatial functionals which take values in Banach spaces of random (vector) processes. We define suitable functional H\"{o}lder spaces for them and give some inequalities among these H\"{o}lder norms. The existence, uniqueness as well as the regularity of solutions are proved for BSPDEs, which contain new assertions even on deterministic PDEs.

Statistical estimation of jump rates for a piecewise deterministic Markov processes with deterministic increasing motion and jump mechanism

We consider the class of Piecewise Deterministic Markov Processes (PDMP), whose state space is $\R_{+}^{*}$, that possess an increasing deterministic motion and that shrink deterministically when they jump. Well known examples for this class of processes are Transmission Control Protocol (TCP) window size process and the processes modeling the size of a marked {\it Escherichia coli} cell. Having observed the PDMP until its $n$th jump, we construct a nonparametric estimator of the jump rate $\lambda$. Our main result is that for $D$ a compact subset of $\R_{+}^{*}$, if $\lambda$ is in the H{\{o}}lder space ${\mathcal H}^s({\mathcal D})$, the squared-loss error of the estimator is asymptotically close to the rate of $n^{-s/(2s+1)}$. Simulations illustrate the behavior of our estimator.

On Invertibility of Functional Operators with Shift in Weighted H&amp;#246;lder Spaces

In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems with renewable resources.

Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted H&amp;#246;lder Spaces

In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted Hölder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted Hölder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

A priori estimates for nonvariational operators modeled on Hörmander's vector fields with drift

For a nonvariational operator structured on H o rmander's vector fields with drift, where the matrix of coffiecients is real, symmetric and uniformly positive, we prove local a priori estimates on the second order derivatives with respect to the vector fields, in H o lder spaces if the coecients are Holder continuous, in Lp spaces if the coefficients are bounded, measurable and locally VMO.

Stability of complex hyperbolic space under curvature-normalized Ricci flow

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little H\"{o}lder spaces on a complete noncompact manifold and establish their interpolation properties.

HÖLDER ESTIMATES FOR THE CAUCHY-RIEMANN EQUATION ON PARAMETERS

Let <TEX>$\{\Omega_{\tau}\}_{\tau{\in}I}$</TEX> be a family of strictly convex domains in <TEX>$\mathbb{C}^n$</TEX>. We obtain explicit estimates for the solution of the <TEX>$\bar{\partial}$</TEX>-equation on <TEX>$\Omega{\times}I$</TEX> in H<TEX>$\ddot{o}$</TEX>lder space. We also obtain explicit point-wise derivative estimates for the <TEX>$\bar{\partial}$</TEX>-equation both in space and parameter variables.

Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an $L$-shaped domain

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional $L$-shaped domain $\Omega$ subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space $C^{2/3}(\bar \Omega )$ and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle $3\pi /2$. We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.

Stable solvability of Pad� difference schemes for parabolic equations in H�lder spaces

We study Pade difference scheme for the approximate solution of the Cauchy problem for parabolic equations generated by the Pade fractions Rj,l, of exponential approximation. We establish an estimate of the coerciveness of the difference schemes for j=l−2,l −1, or even j=l in a smaller space than C0α(E).