Natural Domain Of Definition Research Articles

A Note on the Jacobian Problem of Coifman, Lions, Meyer and Semmes

Coifman, Lions, Meyer and Semmes asked in 1993 whether the Jacobian operator and other compensated compactness quantities map their natural domain of definition onto the real-variable Hardy space H1(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^1({\\mathbb {R}}^n)$$\\end{document}. We present an axiomatic, Banach space geometric approach to the problem in the case of quadratic operators. We also make progress on the main open case, the Jacobian equation in the plane.

Donaldson Functional in Teichmüller Theory

Abstract In this paper, we define a Donaldson type functional whose Euler–Lagrange equations are a system of differential equations, which corresponds to Hitchin’s self-duality equations for a suitable choice of Higgs bundle on closed Riemann surfaces. The main challenge of this functional is its lack of regularity and lack of compactness when defined in its natural domain of definition. Though a standard variational approach cannot directly be applied, we provide the appropriate analytical tools that make Donaldson functional treatable by a variational viewpoint. We prove that this functional admits a unique critical point corresponding to its global minimum. As an immediate consequence, we find that this system of self-duality equations admits a unique solution. Among the applications in geometry of this fact, we obtain a parametrization of closed constant mean curvature immersions in hyperbolic manifolds (possibly incomplete), and their moduli spaces.

Poisson equation beyond its natural domain of definition

This note is concerned with the Poisson equation in the unit disk of the complex plane. Our setting is in the Sobolev space \mathcal W^{1,p} (\mathbb D) with exponent 1 < p < \infty . Such a setting with p \neq 2 is referred to as beyond the natural domain of definition. The novelty lies in the use of a singular integral called Beurling Transform.

Tongues in degree 4 Blaschke products

The goal of this paper is to investigate the family of Blasche products , which is a rational family of perturbations of the doubling map. We focus on the tongue-like sets which appear in its parameter plane. We first study their basic topological properties and afterwards we investigate how bifurcations take place in a neighborhood of their tips. Finally we see how the fixed tongue extends beyond its natural domain of definition.

A pre-order on operators with positive real part and its invariance under linear fractional transformations

A pre-order and equivalence relation on the class of Hilbert space operators with positive real part are introduced, in correspondence with similar relations for contraction operators defined by Yu.L. Shmul'yan in [6]. It is shown that the pre-order, and hence the equivalence relation, is preserved by certain linear fractional transformations. As an application, the operator relations are extended to the class C(U) of Carathéodory functions on the unit disc D of C whose values are operators on a finite dimensional Hilbert space U. With respect to these relations on C(U) it turns out that the associated linear fractional transformations of C(U) preserve the equivalence relation on their natural domain of definition, but not necessarily the pre-order, paralleling similar results for Schur class functions in [3].

On continuity properties of monotone operators beyond the natural domain of definition

We study boundary value problems associated to a nonlinear elliptic system of partial differential equations. The leading second order elliptic operator provides L2-coerciveness and has at most linear growth with respect to the gradient. Incorporating properties of Lipschitz approximations of Sobolev functions we are able to show that the problems in consideration are well-posed, in the sense of Hadamard, in W1,p for all \({p \in (p_0,2]}\) for certain \({p_0 \in [3/2,2)}\) . For simplicity we restrict ourselves to the Dirichlet problem.

Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(ℝ+), we introduce the modified strong dyadic integral J α and the fractional derivative D (α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(ℝ+). We find a countable set of eigenfunctions of the operators D (α) and J α, α > 0. We also prove the relations D (α)(J α(f)) = f and J α(D (α)(f)) = f under the condition that $$\smallint _{\mathbb{R}_ + } f(x)dx = 0$$ . We show the unboundedness of the linear operator $$J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$$ , where L J α is its natural domain of definition. A similar assertion is proved for the operator $$D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$$ . Moreover, for a function f ∈ L(ℝ+) and a given point x ∈ ℝ+, we introduce the modified dyadic derivative d (α)(f)(x) and the modified dyadic integral j α(f)(x). We prove the relations d (α)(J α(f))(x) = f(x) and j α(D (α)(f)) = f(x) at each dyadic Lebesgue point of the function f.

The p-Harmonic transform beyond its natural domain of definition

The p-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in R n . They originate from the study of the p-harmonic type equation di v |⊇u| p-2 ⊇u = divf, where f: Ω → Ρ n is a given vector field in L q (Ω, R n ) and u is an unknown function of Sobolev class W 1,p 0 (Ω, R n ), p + q = pq. The p-harmonic transform H p : L q (Ω,R n ) → L p (Ω,R n ) assigns to f the gradient of the solution: H p f = ⊇u ∈ L p (Ω, R n ). More general PDE's and the corresponding nonlinear operators are also considered. We investigate the extension and continuity properties of the p-harmonic transform beyond its natural domain of definition. In particular, we identify the exponents A > 1 for which the operator H p : L λq (Ω, R n ) → L λP (Ω, R n ) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution ⊇u ∈ L λp (Ω, R n ) fails when A exceeds certain critical value. In case p = n = dim Ω, there is no uniqueness in W 1,λn (R n ) for any A > 1.