abstract: An existence theorem for solutions of non-linear transmission problems is proved using ... more abstract: An existence theorem for solutions of non-linear transmission problems is proved using certain special estimates of inverses of Toeplitz operators. A framework for construction of analytic discs attached to singular targets is described, and a mechanism of creating such discs is presented.
... From the viewpoint ofcontrol theory these theorems can be interpreted as bang-bang principles... more ... From the viewpoint ofcontrol theory these theorems can be interpreted as bang-bang principles. ... where (w,],,, denotes the family of solutions to W(~)E Mr with wind, w = 0. These results also can be viewed as a bang-bang principle, namely, if w(zO)~?R(zO). ...
ABSTRACT The solvability of a wide class of nonlinear boundary value problems of Riemann–Hilbert ... more ABSTRACT The solvability of a wide class of nonlinear boundary value problems of Riemann–Hilbert type of generalized analytic functions (in the sense of Vekua) is shown. Applications to nonlinear boundary value problems of Poincaré type for elliptic equations of second order are discussed.
The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. ... more The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem (due to Beurling) contains a number of gaps which seem inherent in Beurling’s geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurling’s geometric method with a number of new analytic tools, notably H p -space techniques and methods from the theory of Riemann-Hilbert-Poincaré problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.
We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions... more We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles can be prescribed, so the solutions include locally univalent as well as branched packings. The proof of existence rests on an application of Brouwer's fixed point theorem and a global parameterization of the differentiable manifold of circle packings. We also present some first results on the uniqueness of solutions. In the last two sections we propose an algorithm for the numerical solution of the problem, which is based on an embedded Newton method, and report on some test calculations.
We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions... more We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles can be prescribed, so the solutions include locally univalent as well as branched packings. The proof of existence rests on an application of Brouwer's fixed point theorem and a global parameterization of the differentiable manifold of circle packings. We also present some first results on the uniqueness of solutions. In the last two sections we propose an algorithm for the numerical solution of the problem, which is based on an embedded Newton method, and report on some test calculations.
Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we p... more Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we prove new existence and uniqueness results for solutions of nonlinear Riemann-Hilbert problems with noncompact restriction curves.
In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problem... more In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problems with non-smooth target manifold. His main result is a characterization of solutions to Riemann-Hilbert problems as extremal functions in certain function classes. However, a complete analogy to corresponding results for problems with smooth target manifold holds only for a subclass of the toplogical target manifolds introduced by Belch, which are called normal. The conjecture that this subclass coincides with the whole class of topological target manifolds was left unproved. In the present paper we give a (counter-)example of a topological target manifold for which the solution set of the Riemann-Hilbert problem is in some sense bigger than in the smooth case. The problem to characterize normal topological target manifolds in geometric terms arises now as a challenging question of ongoing research.
abstract: An existence theorem for solutions of non-linear transmission problems is proved using ... more abstract: An existence theorem for solutions of non-linear transmission problems is proved using certain special estimates of inverses of Toeplitz operators. A framework for construction of analytic discs attached to singular targets is described, and a mechanism of creating such discs is presented.
... From the viewpoint ofcontrol theory these theorems can be interpreted as bang-bang principles... more ... From the viewpoint ofcontrol theory these theorems can be interpreted as bang-bang principles. ... where (w,],,, denotes the family of solutions to W(~)E Mr with wind, w = 0. These results also can be viewed as a bang-bang principle, namely, if w(zO)~?R(zO). ...
ABSTRACT The solvability of a wide class of nonlinear boundary value problems of Riemann–Hilbert ... more ABSTRACT The solvability of a wide class of nonlinear boundary value problems of Riemann–Hilbert type of generalized analytic functions (in the sense of Vekua) is shown. Applications to nonlinear boundary value problems of Poincaré type for elliptic equations of second order are discussed.
The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. ... more The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem (due to Beurling) contains a number of gaps which seem inherent in Beurling’s geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurling’s geometric method with a number of new analytic tools, notably H p -space techniques and methods from the theory of Riemann-Hilbert-Poincaré problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.
We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions... more We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles can be prescribed, so the solutions include locally univalent as well as branched packings. The proof of existence rests on an application of Brouwer's fixed point theorem and a global parameterization of the differentiable manifold of circle packings. We also present some first results on the uniqueness of solutions. In the last two sections we propose an algorithm for the numerical solution of the problem, which is based on an embedded Newton method, and report on some test calculations.
We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions... more We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles can be prescribed, so the solutions include locally univalent as well as branched packings. The proof of existence rests on an application of Brouwer's fixed point theorem and a global parameterization of the differentiable manifold of circle packings. We also present some first results on the uniqueness of solutions. In the last two sections we propose an algorithm for the numerical solution of the problem, which is based on an embedded Newton method, and report on some test calculations.
Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we p... more Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we prove new existence and uniqueness results for solutions of nonlinear Riemann-Hilbert problems with noncompact restriction curves.
In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problem... more In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problems with non-smooth target manifold. His main result is a characterization of solutions to Riemann-Hilbert problems as extremal functions in certain function classes. However, a complete analogy to corresponding results for problems with smooth target manifold holds only for a subclass of the toplogical target manifolds introduced by Belch, which are called normal. The conjecture that this subclass coincides with the whole class of topological target manifolds was left unproved. In the present paper we give a (counter-)example of a topological target manifold for which the solution set of the Riemann-Hilbert problem is in some sense bigger than in the smooth case. The problem to characterize normal topological target manifolds in geometric terms arises now as a challenging question of ongoing research.
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Papers by Elias Wegert