The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n???1)/n?p?≤?2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation ??Δu?+?Vu?=?0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H ^p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.     

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving <Emphasis Type="Italic">k</Emphasis>-Modulus of Smoothness

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H ^σ X(IR ⁿ ) with order of smoothness σ?∈?(0, n), modelled upon rearrangement invariant Banach function spaces X(IR ⁿ ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR ⁿ ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W ^k?+?1 L ^n/k(logL) ^α (IR ⁿ ) and W ^k L ^n/k(logL) ^α (IR ⁿ ) into generalized Hölder spaces.     

Hölder exponents for the pluricomplex green function of Julia type sets

Assume that the pluricomplex Green function V _E of some compact set E ? ? ^N is Hölder continuous and define the Hölder exponent of the set E to be the supremum over all such exponents, with which V _E is Hölder continuous. We give some lower bounds of the Hölder exponents for the filled-in and composite Julia sets of polynomial mappings.     

The local Hölder exponent for the entropy of real unimodal maps

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ (equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Hölder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Hölder exponent equals exactly, up to a factor log 2, the value of the function at that point. This confirms a conjecture of Isola and Politi (1990), and extends a similar result for the dimension of invariant subsets of the circle.     

Hermitian decomposition of continuous functions on a fractal surface

In this paper the Théodoresco transform is used to show that, under additional assumptions, each Hölder continuous function f defined on the boundary Γ of a fractal domain Ω ? ?²ⁿ can be expressed as f = Ψ⁺ ? Ψ^?, where Ψ^± are Hölder continuous functions on Γ and Hermitian monogenically extendable to Ω and to ?²ⁿ ? (Ω ∪ Γ) respectively.     

Regularity of harmonic maps from polyhedra to CAT(1) spaces

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the \((n-2)\)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points \(x \in X^{(k)}\) with \(k \le n-2\), we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of \(x \in X\).     

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.     

Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible.     

Hölder continuity of solutions of an elliptic equation uniformly degenerating on part of the domain

We consider a second-order divergence elliptic equation in a domain D divided by a hyperplane in two parts. The equation is uniformly elliptic in one of these parts and is uniformly degenerate with respect to a small parameter ? in the other. We show that each solution is Hölder continuous in D with Hölder exponent independent of ?.     

Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent

In this paper, we prove the Hölder continuity of solutions of parabolic equations containing the p(x, t)-Laplacian. The degree p must satisfy the so-called logarithmic condition.     

Some Fourier series with gaps

We examine diverse local and global aspects of the family of Fourier series ∑n ^?α e(n ^k x). In particular, combining number theoretical and harmonic analytic arguments, we study differentiability, Hölder continuity, spectrum of singularities and fractal dimension of the graph.     

Necessary and Sufficient Conditions for the Solvability of the Neumann and the Regularity Problems in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of Some Schrödinger Equations on Lipschitz Domains

Let n ≥? 3and Ω be a bounded Lipschitz domain in \(\mathbb {R}^{n}\). Assume that the non-negative potential V belongs to the reverse Hölder class \(RH_{n}(\mathbb {R}^{n})\) and p ∈ (2, ∞). In this article, two necessary and sufficient conditions for the unique solvability of the Neumann and the Regularity problems of the Schrödinger equation ? Δu + V u =? 0 in Ω with boundary data in L ^p , in terms of a weak reverse Hölder inequality with exponent p and the unique solvability of the Neumann and the Regularity problems with boundary data in some weighted L ² space, are established. As applications, for any p ∈ (1, ∞), the unique solvability of the Regularity problem for the Schrödinger equation ? Δu + V u =?0 in the bounded (semi-)convex domain Ω with boundary data in L ^p is obtained.     

Local Hölder regularity for set-indexed processes

In this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff–Merzbach. The first key result is a Kolmogorov-like Hölder-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as X_U ? X_V, hence we considered different notions of Hölder-continuity. Then, the localization of these properties leads to various definitions of Hölder exponents, which we compare to one another.     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.     

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

It is shown that, for any compact set K ? ? ⁿ (n ? 2) of positive Lebesgue measure and any bounded domain G ? K, there exists a function in the Hölder class C^1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.     

Regularity results for an optimal design problem with quasiconvex bulk energies

Regularity results for equilibrium configurations of variational problems involving both bulk and surface energies are established. The bulk energy densities are uniformly strictly quasiconvex functions with quadratic growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u, E), partial Hölder continuity of the gradient of the deformation u is proved, and partial regularity of the boundary of the minimal set E is obtained.     

Boundary value problems for a higher-order parabolic equation with growing coefficients

We establish the unique solvability of boundary value problems in Hölder function classes for a linear parabolic equation of order 2m in noncylindrical domains of the class C ^{2m ? 1,α} , possibly unbounded (with respect to x as well as t), with nonsmooth (with respect to t) lateral boundary under the condition that the lower-order coefficients and the right-hand side of the equation can grow in a certain way when approaching the parabolic boundary of the domain and the leading coefficients may fail to satisfy the Dini condition near this boundary.     

On the optimal robust solution of IVPs with noisy information

We investigate the optimal solution of systems of initial-value problems with smooth right-hand side functions f from a Hölder class \(F^{r,\varrho }_{\text {reg}}\), where r ≥ 0 is the number of continuous derivatives of f, and ? ∈ (0, 1] is the Hölder exponent of rth partial derivatives. We consider algorithms that use n evaluations of f, the ith evaluation being corrupted by a noise δ_i of deterministic or random nature. For δ ≥ 0, in the deterministic case the noise δ_i is a bounded vector, ∥δ_i∥≤δ. In the random case, it is a vector-valued random variable bounded in average, (E(∥δ_i∥^q))^1/q ≤ δ, q ∈ [1, + ∞). We point out an algorithm whose L_p error (p ∈ [0, + ∞]) is O(n ^{? (r + ?)} + δ), independently of the noise distribution. We observe that the level n ^{? (r + ?)} + δ cannot be improved in a class of information evaluations and algorithms. For ε > 0, and a certain model of δ-dependent cost, we establish optimal values of n(ε) and δ(ε) that should be used in order to get the error at most ε with minimal cost.     

Hölder seminorm preserving linear bijections and isometries

Let (X, d) be a compact metric and 0 < α < 1. The space Lip ^α (X) of Hölder functions of order α is the Banach space of all functions ? from X into \(\mathbb{K}\) such that ∥?∥ = max{∥?∥_∞, L(?)} < ∞, where

$L(f) = sup\{ \left| {f(x) - f(y)} \right|/d^\alpha (x,y):x,y \in X, x \ne y\} $

is the Hölder seminorm of ?. The closed subspace of functions ? such that

$\mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0$

is denoted by lip ^α (X). We determine the form of all bijective linear maps from lip ^α (X) onto lip ^α (Y) that preserve the Hölder seminorm.

     

Regularity of solutions of the model Venttsel’ problem for quasilinear parabolic systems with nonsmooth in time principal matrices

The Venttsel’ problem in the model statement for quasilinear parabolic systems of equations with nondiagonal principal matrices is considered. It is only assumed that the principal matrices and the boundary condition are bounded with respect to the time variable. The partial smoothness of the weak solutions (Hölder continuity on a set of full measure up to the surface on which the Venttsel’ condition is defined) is proved. The proof uses the A(t)-caloric approximation method, which was also used in [1] to investigate the regularity of the solution to the corresponding linear problem.