Partial Hölder continuity for Q-valued energy minimizing maps

We consider multivalued maps between Ω ? ?^N open (N ≥ 2) and a smooth, compact Riemannian manifold 𝒩 locally minimizing the Dirichlet energy. An interior partial Hölder regularity results in the spirit of R. Schoen and K. Uhlenbeck is presented. Consequently a minimizer is Hölder continuous outside a set of Hausdorff dimension at most N ? 3. Almgren's original theory includes a global interior Hölder continuity result if the minimizers are valued into some ?^m. It cannot hold in general if the target is changed into a Riemannian manifold, since it already fails for “classical” single valued harmonic maps.     

The Wegner estimate and the integrated density of states for some random operators

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local H?lder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on theL ^p-theory of the spectral shift function (SSF), forp ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace idealI _p, for 0p ≤ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local H?lder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.     

Local Hölder Continuity Property of the Densities of Solutions of SDEs with Singular Coefficients

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and Hölder continuous drift has a Hölder continuous density function. This result complements recent results of Fournier–Printems (Bernoulli 16(2):343–360, 2010), where the density is shown to exist if both coefficients are Hölder continuous, and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.     

Time regularity of the densities for the Navier–Stokes equations with noise

We prove that the density of the law of any finite-dimensional projection of solutions of the Navier–Stokes equations with noise in dimension three is Hölder continuous in time with values in the natural space L ¹. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected, at least with the known regularity, diffusive scaling.     

Stability for the Multi-Dimensional Borg-Levinson Theorem with Partial Spectral Data

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues λ _k and the normal derivatives ?φ _k /?ν of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to O(k ^?α) with α ? 1, then we still have Hölder stability.     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

Hölder Flow and Differentiability for SDEs with Nonregular Drift

We prove the existence of a stochastic flow of Hölder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties. We also show that the solution map x → X ^x is differentiable in a weak sense.     

Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity q > 1 and with natural growth

In this paper we deal with the Hölder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity q > 1 and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global Hölder continuous in the case of the dimension n = q without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets $\Omega_0 \subset \OmegaIn this paper we deal with the H?lder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second-order discontinuous elliptic systems with nonlinearity q > 1 and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global H?lder continuous in the case of the dimension n = q without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets , are always empty for n = q. Moreover we show that also for 1 < q < 2, but q close enough to 2, the solutions are global H?lder continuous for n = 2.      

Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems

We investigate Hölder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Hölder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Hölder continuous, while they have the two sided lower Hölder continuity property for less regular constraints. Finally, we provide sufficient conditions for Hölder type regularity of optimal controls.     

On Boundary Regularity of the Navier–Stokes Equations

Abstract

We study boundary regularity of weak solutions of the Navier–Stokes equations in the half-space in dimension n ≥ 3. We prove that a weak solution u which is locally in the class L ^{p, q} with 2/p + n/q = 1, q > n near boundary is Hölder continuous up to the boundary. Our main tool is a pointwise estimate for the fundamental solution of the Stokes system, which is of independent interest.     

Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>: the case of compact Kähler manifolds

We prove that on compact Kähler manifolds solutions to the complex Monge–Ampère equation, with the right-hand side in L ^p , p > 1, are Hölder continuous.     

The Hardy inequality and the heat equation with magnetic field in any dimension

In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schrödinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrödinger operator on the (d ? 1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrödinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.     

Continuity of integrated density of states — independent randomness

In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the ν dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly α-Hölder continuous, 0 < α ≤ 1, then the density of states is also uniformly α-Hölder continuous.     

Normal solvability of linear elliptic problemsRésolubilité normale des problèmes elliptiques linéaires

The paper is devoted to general linear elliptic problems in Hölder spaces. We consider unbounded domains and define limiting problems at infinity. We give a necessary and sufficient condition of normal solvability through uniqueness of solutions of limiting problems. We study a structure of spaces dual to Hölder spaces and specify the subspace of functionals, which provide the condition of normal solvability. This allows us to prove that for Fredholm operators all limiting operators are invertible. To cite this article: V. Volpert, A. Volpert, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 457–462.     

Cahn-Hilliard stochastic equation: Strict positivity of the density

This paper deals with the Cahn-Hilliard stochastic equation driven by a space-time white noise with a non-linear diffusion coefficient. Using new lower estimate of the kernel, we prove the "local" existence of the density without non-degeneracy condition in a case of Hölder continuous trajectories, and we show that the density of any vector is lower bounded by a strictly positive continuous function under a non-degeneracy condition.     

Partial Hölder regularity for elliptic systems with non-standard growth

We study partial Hölder regularity for elliptic systems with non-standard growth. We consider general systems with Orlicz growth and discontinuous coefficient factor, for which we prove that their weak solutions are partially Hölder continuous for any Hölder exponents. In addition, we also obtain a similar result for systems with double phase growth.     

Embedding Theorems for Hölder Classes Defined on p-Adic Linear Spaces

We prove analogues of P. L. Ul’yanov and V. A. Andrienko results concerning embeddings of L^q Hölder spaces into Lebesgue spaces L^r or L^r Hölder spaces in the case 1 ≤ q < r ≤ 2 for functions defined on p-adic linear spaces. The conditions presented in these theorems are sharp. Also we give necessary and sufficient conditions for such embeddings in the case 1 ≤ q < r < ∞ that generalize recent results of S. S. Platonov.

     

On the Closure of Smooth Compactly Supported Functions in Weighted Hölder Spaces

Mathematical Notes - The closure of the set of smooth compactly supported functions in a weighted Hölder space on ?n, n ≥ 1, with a weight controlling the behavior at the point at...     

Commutators and singular integral operators in Clifford analysis

In this paper we develop a method for setting the compactness of the commutator relative to the singular integral operator acting on Hölder continuous functions over Ahlfors David regular surfaces in R ⁿ⁺¹ . This method is based on the essential use of the monogenic decomposition of Hölder continuous functions. We also set forth explicit representations of the adjoints of the singular Cauchy type integral operators, relative to a total subset of real functionals.     

Hölder Index at a Given Point for Density States of Super-<Emphasis Type="Italic">α</Emphasis>-Stable Motion of Index 1+<Emphasis Type="Italic">β</Emphasis>

A Hölder regularity index at given points for density states of (α,1,β)-superprocesses with α>1+β is determined. It is shown that this index is strictly greater than the optimal index of local Hölder continuity for those density states.