Holder Continuous Coefficients Research Articles

Approximate solution of a dominant singular integral equation with conjugation

UDC 517.5In the present paper, the method of successive approximations and Faber polynomials are used to derive an approximate solution of a dominant singular integral equation with Holder continuous coefficients and conjugation on the Lyapunov curve. Moreover, conditions of convergence in the and spaces are presented.

Weak Error for the Euler Scheme Approximation of Degenerate Diffusions with Nonsmooth Coefficients

We study the weak error associated with the Euler scheme of Kolmogorov like degenerate diffusion processes with nonsmooth bounded coefficients. More precisely, we consider the case of Holder continuous homogeneous coefficients.

XÂY DỰNG LƯỢC ĐỒ XẤP XỈ ỔN ĐỊNH CHO PHƯƠNG TRÌNH VI PHÂN NGẪU NHIÊN KHÔNG ÔTÔNÔM VỚI HỆ SỐ KHUẾCH TÁN LIÊN TỤC HOLDER

XÂY DỰNG LƯỢC ĐỒ XẤP XỈ ỔN ĐỊNH CHO PHƯƠNG TRÌNH VI PHÂN NGẪU NHIÊN KHÔNG ÔTÔNÔM VỚI HỆ SỐ KHUẾCH TÁN LIÊN TỤC HOLDER

Continuity of the optimal stopping boundary for two-dimensional diffusions

We first show that a smooth fit between the value function and the gain function at the optimal stopping boundary for a two-dimensional diffusion process implies the absence of boundary’s discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of Holder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in the elliptic case. The method of proof relies upon regularity results for the second-order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.

Failure of $$L^2$$ L 2 boundedness of gradients of single layer potentials for measures with zero low density

Consider a totally irregular measure \(\mu \) in \({{\mathbb {R}}}^{n+1}\), that is, the upper density \(\limsup _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) is positive \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\), and the lower density \(\liminf _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) vanishes \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\). We show that if \(T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu (y)\) is an operator whose kernel \(K(\cdot ,\cdot )\) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Holder continuous coefficients, then \(T_\mu \) is not bounded in \(L^2(\mu )\). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.

Strictly hyperbolic equations with coefficients low-regular in time and smooth in space

We consider the Cauchy problem for strictly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is \(L^2\) well-posed in the case of Lipschitz continuous coefficients in time, \(H^s\) well-posed in the case of Log-Lipschitz continuous coefficients in time (with an, in general, finite loss of derivatives) and Gevrey well-posed in the case of Holder continuous coefficients in time (with an, in general, infinite loss of derivatives). Here, we use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. The well-known results for Lipschitz, Log-Lipschitz and Holder coefficients are recovered.

Strong rate of tamed Euler–Maruyama approximation for stochastic differential equations with Hölder continuous diffusion coefficient

We study the strong rate of convergence of the tamed Euler–Maruyama approximation for one-dimensional stochastic differential equations with superlinearly growing drift and Holder continuous diffusion coefficients.

The Leray-Schauder Degree as Topological Method Solution of Nonlinear Elliptic Equations

In the present paper using precise results on the solutions of linear elliptic differential operators with Holder continuous coefficient as well as a variant of the Lery - Schauder method and the gal of this paper to find an adequate degree theory for the infinite dimensional setting and to extend the theory of homotopy classes of maps form <img width="17" height="16" src="http://article.sciencepublishinggroup.com/journal/141/1411129/image001.png" /> to <img width="17" height="16" src="http://article.sciencepublishinggroup.com/journal/141/1411129/image001.png" /> to homotopy classes of maps on infinite dimensional spaces.

Pathwise uniqueness for an SPDE with Hölder continuous coefficient driven by $\alpha $-stable noise

In this paper we study the pathwise uniqueness of nonnegative solution to the following stochastic partial differential equation with Holder continuous noise coefficient: \[ \frac{\partial X_t(x)} {\partial t}=\frac{1} {2} \Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L} _t(x),\quad t>0, ~x\in \mathbb{R} , \] where for $1 0$.

Optimal partial regularity for nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups

This paper is concerned with the interior regularity for nonlinear sub-elliptic systems with Dini continuous coefficients under superquadratic controllable growth conditions in Carnot groups. We adapt the technique of the $\mathcal{A}$ -harmonic approximation to the case of sub-elliptic systems in divergence form, and we show a partial regularity result for weak solutions. In particular, our result is optimal in the sense that in the case of Holder continuous coefficients we obtain directly the optimal Holder exponent for the horizontal gradient of weak solutions on its regular set.

Unbiased simulation of stochastic differential equations using parametrix expansions

In this article, we consider an unbiased simulation method for multidimensional diffusions based on the parametrix method for solving partial differential equations with Holder continuous coefficients. This Monte Carlo method which is based on an Euler scheme with random time steps, can be considered as an infinite dimensional extension of the Multilevel Monte Carlo method for solutions of stochastic differential equations with Holder continuous coefficients. In particular, we study the properties of the variance of the proposed method. In most cases, the method has infinite variance and therefore we propose an importance sampling method to resolve this issue.

On the boundary behavior of positive solutions of elliptic differential equations

Let $u$ be a positive harmonic function in the unit ball $B_1 \subset \mathbb{R}^n$ and let $\mu$ be the boundary measure of $u$. Consider a point $x\in \partial B_1$ and let $n(x)$ denote the unit normal vector at $x$. Let $\alpha$ be a number in $(-1,n-1]$ and $A \in [0,+\infty) $. We prove that $u(x+n(x)t)t^{\alpha} \to A$ as $t \to +0$ if and only if $\frac{\mu({B_r(x)})}{r^{n-1}} r^{\alpha} \to C_\alpha A$ as $r\to+0$, where ${C_\alpha= \frac{\pi^{n/2}}{\Gamma(\frac{n-\alpha+1}{2})\Gamma(\frac{\alpha+1}{2})}}$. For $\alpha=0$ it follows from the theorems by Rudin and Loomis which claim that a positive harmonic function has a limit along the normal iff the boundary measure has the derivative at the corresponding point of the boundary. For $\alpha=n-1$ it concerns about the point mass of $\mu$ at $x$ and it follows from the Beurling minimal principle. For the general case of $\alpha \in (-1,n-1)$ we prove it with the help of the Wiener Tauberian theorem in a similar way to Rudin's approach. Unfortunately this approach works for a ball or a half-space only but not for a general kind of domain. In dimension $2$ one can use conformal mappings and generalise the statement above to sufficiently smooth domains, in dimension $n\geq 3$ we showed that this generalisation is possible for $\alpha\in [0,n-1]$ due to harmonic measure estimates. The last method leads to an extension of the theorems by Loomis, Ramey and Ullrich on non-tangential limits of harmonic functions to positive solutions of elliptic differential equations with Holder continuous coefficients.

Singular Integral Operators on an Open Arc in Spaces with Weight

The aim of this work is to study a singular integral operator \({\mathbf{A}=aI+bS_\Gamma}\) with the Cauchy operator SΓ (SIO) and Holder continuous coefficients a, b in the space \({\mathbb{H}^0_\mu(\Gamma,\rho)}\) of Holder continuous functions with an power “Khvedelidze” weight. The underlying curve is an open arc. It is well known, that such operator is Fredholm if and only if, along with the ellipticity condition \({a^2(t)-b^2(t)\not=0,\, t\in\Gamma}\), the “Gohberg–Krupnik arc condition” is fulfilled (see Duduchava, in Dokladi Akademii Nauk SSSR 191:16–19, 1970). Based on the Poincare–Beltrami formula for a composition of singular integral operators and the celebrated Muskhelishvili formula describing singularities of Cauchy integral, the formula for a composition of weighted singular integral operators is proved. Using the obtained composition formula and the localization, the Fredholm criterion of the SIO is derived in a natural way, by looking for the regularizer of the operator \({\mathbf{A}}\) and equating to 0 non-compact operators. The approach is space-independent and this is demonstrated on similar results obtained for SIOs with continuous coefficients in the Lebesgue spaces with a “Khvedelidze” weight \({\mathbb{L}_p(\Gamma,\rho)}\), investigated earlier by Gohberg and Krupnik (Studia Mathematica 31:347–362, 1968; One Dimensional Singular Integral Operators II, Operator Theory, Advances and Applications, vol. 54, chapter IX, 1979) with a different approach.

Schauder theory for Dirichlet elliptic operators in divergence form

Let A be a strongly elliptic operator of order 2m in divergence form with Holder continuous coefficients of exponent $${\sigma \in (0,1)}$$ defined in a uniformly C 1+σ domain Ω of $${\mathbb{R}^n}$$ . Regarding A as an operator from the Holder space of order m + σ associated with the Dirichlet data to the Holder space of order −m + σ, we show that the inverse (A − λ)−1 exists for λ in a suitable angular region of the complex plane and estimate its operator norms. As an application, we give a regularity theorem for elliptic equations.

Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities

In this work the authors deal with linear second order partial differential operators of the following type $H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)$ where $X_{1},X_{2},\ldots,X_{q}$ is a system of real Hormander's vector fields in some bounded domain $\Omega\subseteq\mathbb{R}^{n}$, $A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix such that $\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\text{}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})$ for a suitable constant $\lambda>0$ a for some real numbers $T_{1} < T_{2}$. Table of Contents: Introduction. Part I: Operators with constant coefficients: Overview of Part I; Global extension of Hormander's vector fields and geometric properties of the CC-distance; Global extension of the operator $H_{A}$ and existence of a fundamental solution; Uniform Gevray estimates and upper bounds of fundamental solutions for large $d\left(x,y\right)$; Fractional integrals and uniform $L^{2}$ bounds of fundamental solutions for large $d\left(x,y\right)$; Uniform global upper bounds for fundamental solutions; Uniform lower bounds for fundamental solutions; Uniform upper bounds for the derivatives of the fundamental solutions; Uniform upper bounds on the difference of the fundamental solutions of two operators. Part II: Fundamental solution for operators with Holder continuous coefficients: Assumptions, main results and overview of Part II; Fundamental solution for $H$: the Levi method; The Cauchy problem; Lower bounds for fundamental solutions; Regularity results. Part III: Harnack inequality for operators with Holder continuous coefficients: Overview of Part III; Green function for operators with smooth coefficients on regular domains; Harnack inequality for operators with smooth coefficients; Harnack inequality in the non-smooth case; Epilogue; References. (MEMO/204/961)

Non-Linear Rational Riemann-Hilbert-Problems with Circular Target Curves

This article may be considered as a continuation of [3], where we studied non-linear Riemann-Hilbert problems with circular target curves ¦w − c¦ = r and Holder continuous coefficients c and r. Here we assume that c and r2 are rational functions and emphasize algorithmic and numerical aspects. It is shown that all solutions of the problem are rational and can be obtained by solving an interpolation problem of (generalized) Nevanlinna-Pick type. This problem is in turn reduced to a linear system, which leads to efficient (and stable) numerical methods. Special emphasis is on the Laurent case, which is of importance in applications. We propose an a- posteriori estimate which allows one to verify the accuracy of the approximate solution and report on some test calculations.

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Holder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Holder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.

Addendum to “Concentration and Lack of Observability of Waves in Highly Heterogeneous Media”

In [1] we introduced a class of 1−d wave equations with rapidly oscillating Holder continuous coefficients for which the classical boundary observability property fails. We also established that these examples could be used to contradict Strichartz-type inequalities for the wave equation with low regularity coefficients. The object of this addendum is to further analyze this issue. As we will see, the argument in [1] only provides sharp counter-examples to the Strichartz estimates when the coefficient ρ belongs to L ∞. We carefully analyze these counter-examples for Holder continuous coefficients. We also give a new application of our construction which shows that some eigenfunction estimates for elliptic operators due to Sogge can fail when coefficients are not smooth enough.

ON HILBERT BOUNDARY VALUE PROBLEMS WITH RADICAL

The Hilbert boundary value problem Re {λ(t)Ψ+(t)p}=c(t),t∈Lof normal type with Holder continuous coefficients is discussed, where L is the unit circle |t| = 1, p ≥ 2 is any definite integer, Ψ+(t) is the boundary value of the unknown function Ψ(z) holomorphic in |z| < 1 with single-valued continuous Ψ+(t)p on L.

Reconsideration on homogeneous quadratic Riemann boundary value problem

The homogeneous quadratic riemann boundary value problem (1) with Holder continuous coefficients for the normal case was considered by the author in 1997. But the solutions obtained there are incomplete. Here its general method of solution is obtained.