Rectifiable Jordan Curve Research Articles

The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

Assume you are given a finite configuration Gamma of disjoint rectifiable Jordan curves in {mathbb {R}}^n. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus of at most p which span Gamma . While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations Gamma , where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in {mathbb {R}}^n but in general proper metric spaces. In particular, the existence of an area minimizer is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus p bounding a given configuration Gamma need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.

Inverse potential problems in divergence form for measures in the plane

We study inverse potential problems with source term the divergence of some unknown (ℝ3-valued) measure supported in a plane;e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetizationμby penalizing the measure-theoretic total variation norm ∥μ∥TV, and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane thatTV-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTV-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree-like. We note that such magnetizations can be recoveredvia TV-regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.

On Hilbert and Riemann problems for generalized analytic functions and applications

The research of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. The present paper is devoted to various theorems on the existence of nonclassical solutions of the Hilbert and Riemann boundary value problems with arbitrary measurable data for generalized analytic functions by Vekua and the corresponding applications to the Neumann and Poincare problems for generalized harmonic functions. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. First of all, here it is proved theorems on the existence of solutions to the Hilbert boundary value problem with arbitrary measurable data for generalized analytic functions in arbitrary Jordan domains with rectifiable boundaries in terms of the natural parameter and angular (nontangential) limits, moreover, in arbitrary Jordan domains in terms of harmonic measure and principal asymptotic values. Moreover, it is established the existence theorems on solutions for the appropriate boundary value problems of Hilbert and Riemann with arbitrary measurable data along the so-called Bagemihl–Seidel systems of Jordan arcs terminating at the boundary in arbitrary domains whose boundaries consist of finite collections of rectifiable Jordan curves. On this basis, it is established the corresponding existence theorems for the Poincare boundary value problem on the directional derivatives and, in particular, for the Neumann problem with arbitrary measurable data to the Poisson equations.

An Equivalence Theorem on Minimum Sheltering Speed for Non-convex Habitats

The paper is devoted to a new family of variational problems for differential inclusions, motivated by the protection of human and wildlife habitats when an invasive environmental disaster occurs. Indeed, the mathematical model consists of a differential inclusion describing the expansion of invaded region over time, and an artificial barrier that cannot be penetrated by the invasive agent is erected to shield the habitat, which serves as the control strategy and is characterized as a one-dimensional rectifiable set. We consider an isotropic case that the disaster spreads with uniform speed in all directions, and develop an equivalence result on the minimum construction speed required for implementing a sheltering strategy determined by a rectifiable Jordan curve. By measuring the invaded portion of barriers over time, it is shown that each connected non-convex habitat admits an equivalent habitat that contains the given habitat and their minimum speeds are equal. In particular, the boundary of an equivalent habitat can be partitioned into two arcs: an arc is locally convex, and the other is part of the boundary of illuminated area. This leads to a corollary on the existence of admissible sheltering strategies for non-convex habitats.

Cauchy Integral Operators Involving Higher Order Lipschitz Classes in the Poly-Analytic Function Theory

In this note, we consider poly-analytic Cauchy integral operators introduced in a rather natural way on the higher order Lipschitz classes. We prove some boundary properties of the function represented by such an integral operator in the general context of rectifiable Jordan curves.

A note on Faber operator

For a rectifiable Jordan curve Γ with complementary domains D and D*, Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces B p (1 < p < ∞) of analytic functions in the unit disk and in the inner domain D, respectively. We point out that the conjecture is not true in the special case p = 2, and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.

Immersed Solutions of Plateau’s Problem for Piecewise Smooth Boundary Curves with Small Total Curvature

We provide a new proof of the classical result that any closed rectifiable Jordan curve $${\Gamma \subset \mathbb{R}^3}$$ being piecewise of class C 2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Γ is smaller than 6π. In contrast to the methods due to Osserman (Ann Math 91(2):550–569, 1970), Gulliver (Ann Math 97(2):275–305, 1973) and Alt (Math Z 127:333–362, 1972, Math Ann 201:33–35, 1973), our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau’s problem for polygonal boundary curves, provided by the first author’s accomplishment (The Plateau problem, Fuchsian equations and the Riemann–Hilbert problem. Memoires de la Soc. Math. Fr. (to appear) arXiv: 1003.0978) of Garnier’s ideas in (Annales scientifiques de l’E.N.S. 45:53–144, 1928).

Riemann boundary-value problem on an open rectifiable Jordan curve. II

Riemann boundary-value problem on an open rectifiable Jordan curve. II

Riemann boundary-value problem on an open rectifiable jordan curve. I

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.

Estimates of gradient and of Jacobian of harmonic mappings defined in the unit disk

Let $F:\mathbb T\to \gamma$ be a bounded measurable function of the unit circle $\mathbb T$ onto a rectifiable Jordan curve $\gamma$ with the length $|\gamma |$, and let $w=P[F]$ be its harmonic extension to the unit disk $\mathbb U$. By using the arc length parametrization of $\gamma$ we obtain the following results: (i) If $F$ is a quasi-homeomorphism and $1\le p<2$, the $L^p$-norm of the Hilbert-Schmidt norm of the gradient of $w$ is bounded as follows: $\|D(w)\|_{p}\le \frac {|\gamma |}{4\sqrt 2}(\frac {16}{\pi (2-p)})^{1/p}$. (ii) If $F$ is $p$-Lipschitz continuous and $\gamma$ is Dini smooth, then the Jacobian of $w$ is bounded in $\mathbb U$ by a constant $C(p,\gamma )$. The first result is an extension of a recent result of Verchota and Iwaniec, and Martin and Sbordone, while the second result is an extension of a classical result of Martio where $\gamma =\mathbb T$.

MULTIPLIER THEOREMS IN WEIGHTED SMIRNOV SPACES

The analogues of Marcinkiewicz multiplier theorem and Littlewood-Paley theorem are proved for p-Faber series in weighted Smirnov spaces defined on bounded and unbounded components of a rectifiable Jordan curve.

The exterior Plateau problem in higher codimension

We prove existence theorems for two-dimensional, noncompact, complete minimal surfaces in ℝn of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within a bounded distance from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity flat ends as well as of surfaces with polynomial-type nonflat ends.

On the asymptotics of orthogonal polynomials on the curve with a denumerable mass points

We investigate the asymptotic behavior of orthogonal polynomials with respect to a measure of the type \(\sigma =\alpha +\gamma \), where \(\alpha \) is a measure concentrated on a rectifiable Jordan curve and \(\gamma \) is an infinite discrete measure.

Extremal polynomials with varying measures

We investigate the strong asymptotics for Lp-extremal polynomials with respect to varying measures on a rectifiable Jordan curve perturbed by a finite Blaschke sequence of point masses outside the curve Mathematics Subject Classification: 42C05, 30E10

Unstable extremal surfaces of the “Shiffman functional” spanning rectifiable boundary curves

In this paper we derive a sufficient condition for the existence of extremal surfaces of a parametric functional \(\mathcal{J}\) with a dominant area term, which do not furnish global minima of \(\mathcal{J}\) within the class \(\mathcal{C}^*(\Gamma )\) of H1,2-surfaces spanning an arbitrary closed rectifiable Jordan curve \(\Gamma\subset \mathbb{R}^3\) that merely has to satisfy a chord-arc condition. The proof is based on the “mountain pass result” of (Jakob in Calc Var 21:401–427, 2004) which yields an unstable \(\mathcal{J}\)-extremal surface bounded by an arbitrary simple closed polygon and Heinz’ ”approximation method” in (Arch Rat Mech Anal 38:257–267, 1970). Hence, we give a precise proof of a partial result of the mountain pass theorem claimed by Shiffman in (Ann Math 45:543–576, 1944) who only outlined a very sketchy and partially incorrect proof.

Absolute uniqueness of minimal surfaces bounded by contours with a one-to-one projection onto a plane

In this paper the uniqueness of the always existing Douglas-Rado solution to Plateau’s Problem of finding a minimal surface of the type of the disc to a given rectifiable Jordan Curve Γ in ℝ3 is considered in combination with the question: When is the solution surface also a graph? So only contours with a one-to-one projection onto a plane are candidates. The uniqueness is generalized to all genera and orientations and the non orientable case and then called “absolute”. It is demonstrated, that if it we have a continuous family of catenoids as support surfaces or barriers at the non convex projected part of the C 2 contour, assured by the non negativity of a function of the radii of the necks of the catenoids and the contour, the Douglas-Rado solution is absolute unique and a graph. Especially there are no further restrictions to the projection of the contour than C 2-smoothness. As easy first consequences, the earlier result of Sauvigny, using Scherk’s surface as support surfaces for certain contours with a non convex projection, is generalized to absolute uniqueness and the problem initiating, famous uniqueness and existence result in case of a convex projection of Rado, Kneser and Meeks is also carried over to absolute uniqueness. Finally, by a generic example using our mentioned main fruit of investigation, the non necessity of any curvature bound, any bound on function norms, any height bound, and Williams local tangential Lipschitz condition for the whole contour for the existence of a solution for the minimal surface equation or uniqueness of parametric minimal surfaces is shown. This covers more than all known sufficient conditions, i.e. the well known 4π curvature bound of Nitsche, Sauvigny’s uniform concavity-, Lau’s smallness in the C 0,1-norm condition and Meeks restriction of the deviation from the plane in the C 2-norm. The article is strongly geometrical in spirit so enabling the use of geometrical imagination and intuition in the realms of results and techniques from many different, including very advanced and abstract, mathematical fields. As the proofs are very complete, transparent and proceed in little steps, covering also new elementary results, the article is predestined for educational employment. Especially for its use of the maximum principle in a very sophistcated, long run, merged with topology, way and the new setting of surfaces with a Riemannian structure and usual derivations for handling the non-orientable case.

Piecewise-continuous Riemann boundary-value problem on a rectifiable curve

We extend classes of closed rectifiable Jordan curves and given functions in the theory of the piecewise-continuous Riemann boundary-value problem and the characteristic singular integral equation with Cauchy kernel related to this problem.

Approximation Properties of Julia Polynomials

Let G be a finite simply connected domain in the complex plane ℂ, bounded by a rectifiable Jordan curve L, and let w = ϕ0 (z) be the Riemann conformal mapping of G onto D(0, r0) := {w : |w| < r0}, normalized by the conditions ϕ0 (z0) = 0, ϕ'0 (z0) = 1.

Rational approximation in Orlicz spaces on Carleson curves

AbstractWe define some subclasses of Orlicz spaces of functions and establish herea direct theorem of the approximation theory by rational functions. 1 Introduction and main results Let Γ be a rectifiable Jordan curve in the complex plane C and let G:= IntΓ,G − := ExtΓ. Without loss of generality we suppose that 0 ∈ G.Further let T :={w∈ C : |w| = 1}, U:= IntT,U − := ExtT.We denote by ϕand ϕ 1 the conformalmappings of G − and Gonto U − normalized by the conditionsϕ(∞) = ∞, lim z→∞ ϕ(z)z>0andϕ 1 (0) = ∞, lim z→0 zϕ 1 (z) >0respectively and let ψand ψ 1 be the inverse mappings of ϕand ϕ 1 .Let also L p (Γ) and E p (G) (1 ≤ p<∞) be the Lebesgue space of measurablecomplex valued functions on Γ and the Smirnov class of analytic functions in Grespectively. Since Γ is rectifiable, we have ϕ 0 ∈ E 1 (G − ), ϕ 0 1 ∈ E 1 (G) and ψ 0 , Received by the editors June 2003.Communicated by R. Delanghe.1991 Mathematics Subject Classification : Primary 30E10, 41A10, 41A20. Secondary 41A25,46E30.Key words and phrases : Carleson curve, Cauchy singular operator, Faber polynomials, Orliczspace, Smirnov-Orlicz class.Bull. Belg. Math. Soc. 12(2005), 223–234

On Asymptotics of Lp Extremal Polynomials on a Complex Curve Plus an Infinite Number of Points

We study, for all p>0 and under certain conditions the asymptotic behavior of Lp extremal polynomials with respect to the measure α=β+Y. Where, β denotes a positive Szego measure on the closed rectifiable Jordan curve E in the complex plane and γ has an infinity of masses in the region exterior to the curve E.