Lipschitz Curves Research Articles

Smoothness of unordered curves in two-dimensional strongly competitive systems

It is known that in two-dimensional systems of ODEs of the form $\dotx^i=x^if^i(x)$ with ${\partial f^i}/{\partial x^j} < 0$ (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unorder

Semi-Discrete Galerkin Approximations for the Single-Layer Equation on Lipschitz Curves

Semi-Discrete Galerkin Approximations for the Single-Layer Equation on Lipschitz Curves

Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves

The paper presents a theory of Fourier transforms of bounded holomorphic functions defined in sectors. The theory is then used to study singular integral operators on star-shaped Lipschitz curves, which extends the result of Coifman-McIntosh-Meyer on the

A holomorphic extension result*

In [7] we obtained, as a consequence of the Fourier transform theory developed in the paper, a sufficient and necessary condition on a sequence (b n)ψl ∞ for the function to be holomorphicaily extensible to a heart-shaped region containing the set , and dominated by C/|1-z| when z is near 1 in the region. This note generalizes this result to the cases when . It also includes corresponding results for series of negative powers and for Laurent series as well. The theory has applications to singular and fractional integrals on closed Lipschitz curves which are closely related io boundary value problems in Lipschitz domains.

Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves

Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves

A Uniqueness Result for a Generalized Radon Transform

There exist five families of Lipschitz curves on the unit square such that any continuous function is uniquely defined by the values of its integral (properly defined) along these curves. We present this uniqueness result as a consequence of the Kolmogorov superposition theorem.

Perturbation of the domain and regularity of the solutions of the bipolar fluid flow equations in polygonal domains

In this paper, we consider the isothermal, incompressible, viscous, bipolar fluid flow equation, which is a mathematical model of viscous fluid motion with non-linear and higher order viscosity. We investigate the question of stability of the solutions to the bipolar equations with respect to perturbations of the boundary of the domain, and show that in general the solutions are not stable with respect to perturbations of the boundary by Lipschitz curves. We also study the regularity of the solution to the bipolar equations in a polygonal domain Ω. We show that near a corner of the polygonal domain, if F ϵ L 2(Ω) , then any weak solution w ϵ H 2(Ω) ∩ H 0 1(Ω) may be written as the sum w = w reg + w sing , where w reg ϵ H loc 4 is the regular part, and w sing is the singular part which is not in H loc 4 and whose precise behavior depends on the interior angle of the corner. We also provide an explicit characterization of the local singularities in terms of the interior angle of the corner. The local singularities are calculated using the symbolic manipulation software package (Maple), and sharp a-priori estimates are then used to show that there are no other singularities.

On the resolvents of dyadic paraproducts

We consider the boundedness of certain singular integral operators that arose in the study of Sobolev spaces on Lipschitz curves, [P1]. The standard theory available (David and Journe's T1 Theorem, for instance; see [D]) does not apply to this case becuase the operators are not necessarily Calderon-Zygmund operators, [Ch]. One of these operators gives an explicit formula for the resolvent at ? = 1 of the dyadic paraproduct, [Ch].

The Besov and Triebel-Lizorkin spaces with high order on the Lipschitz curves

The Besov spacesB p α,μ (Γ) and Triebel-Lizorkin spacesF p α,μ (Γ) with high order α∈R on a Lipschitz curve Γ are defined. when 1≤p≤∞, 1≤q≤∞. To compare to the classical case a difference characterization of such spaces in the case |α|<1 is given also.

Fourier Multipliers on Lipschitz Curves

We develop the theory of Fourier multipliers acting on ${L_p}(\gamma )$ where $\gamma$ is a Lipschitz curve of the form $\gamma = \{ x + ig(x)\}$ with $\left \| g\right \| _\infty < \infty$ and $\left \| g\prime \right \| _\infty < \infty$ . The aim is to better understand convolution singular integrals $B$ defined naturally on such curves by \[ Bu(z) = {\text {p.v.}}\int _\gamma {\varphi (z - \zeta )u(\zeta )d\zeta } \] for almost all $z \in \gamma$ .

Fourier multipliers on Lipschitz curves

We develop the theory of Fourier multipliers acting on L p ( γ ) {L_p}(\gamma ) where γ \gamma is a Lipschitz curve of the form γ = { x + i g ( x ) } \gamma = \{ x + ig(x)\} with ‖ g ‖ ∞ &gt; ∞ \left \| g\right \| _\infty &gt; \infty and ‖ g ′ ‖ ∞ &gt; ∞ \left \| g\prime \right \| _\infty &gt; \infty . The aim is to better understand convolution singular integrals B B defined naturally on such curves by \[ B u ( z ) = p.v. ∫ γ φ ( z − ζ ) u ( ζ ) d ζ Bu(z) = {\text {p.v.}}\int _\gamma {\varphi (z - \zeta )u(\zeta )d\zeta } \] for almost all z ∈ γ z \in \gamma .

Two Elementary Proofs of the L 2 Boundedness of Cauchy Integrals on Lipschitz Curves

Two Elementary Proofs of the L 2 Boundedness of Cauchy Integrals on Lipschitz Curves

Two elementary proofs of the 𝐿² boundedness of Cauchy integrals on Lipschitz curves

Two elementary proofs of the 𝐿² boundedness of Cauchy integrals on Lipschitz curves

Estimates of analytic functions in Jordan domain

The classical estimates for analytic functions in a disc are carried over to functions which are analytic in Jordan domains whose boundaries are Lipschitz and Radon curves. Singular integrals, maximal estimates, and Lusin's inequality are considered. Analytic functions, the moduli of whose boundary values satisfy the conditions of B. Muckenhoupt are studied in detail. Properties of conformal maps and their level lines connected with the Muckenhoupt conditions on the moduli of the boundary derivatives are considered. This consideration lets one give “real” proofs of various distortion theorems for conformal maps of Lipschitz domains.

Some new function spaces and their applications to Harmonic analysis

In this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications. These spaces are the “tent spaces.” They provide the natural setting for the study of such things as maximal functions (the relevant space here is T ∞ p ), and also square functions (where the space T 2 p is relevant). As such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis. Thus they are closely related to L p and Hardy spaces, important parts of whose theory become corollaries of the description of tent spaces. Also, as (“Proc. Conf. Harmonic Analysis, Cortona,” Lect. Notes in Math. Vol. 992, Springer-Verlag, Berlin/New York,1983), already indicated where these spaces first appeared explicitly, the tent spaces can be used to simplify some of the results related to the Cauchy integral on Lipschitz curves, and multilinear analysis. In retrospect one can recognize that various ideas important for tent spaces had been used, if only implicitly, for quite some time. Here one should mention Carleson's inequality, its simplifications and extensions, the theory of Hardy spaces, and atomic decompositions.

Applications of the Cauchy integral on Lipschitz curves

Applications of the Cauchy integral on Lipschitz curves

Cauchy integrals on Lipschitz curves and related operators

In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the L(p)-boundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [(1976) "Commutateurs d'intégrales singulières:" Analyse harmonique d'Orsay n(o) 211, Université Paris XI] on the continuity of the so-called commutator operators.

The porous medium equation in one dimension

We consider a second order nonlinear degenerate parabolic partial differential equation known as the porous medium equation, restricting our attention to the case of one space variable and to the Cauchy problem where the initial data are nonnegative and have compact support consisting of a bounded interval. Solutions are known to have compact support for each fixed time. In this paper we study the lateral boundary, called the interface, of the support P [ u ] P[u] of the solution in R 1 × ( 0 , T ) {R^1} \times (0,T) . It is shown that the interface consists of two monotone Lipschitz curves which satisfy a specified differential equation. We then prove results concerning the behavior of the interface curves as t approaches zero and as t approaches infinity, and prove that the interface curves are strictly monotone except possibly near t = 0 t = 0 . We conclude by proving some facts about the behavior of the solution in P [ u ] P[u] .