On a homeomorphic solution to a multidimensional analog of the Beltrami equation

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.

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Existence of solutions to a multidimensional analog of the Beltrami equation

Siberian Mathematical Journal ◽

10.1007/bf01054219 ◽

1989 ◽

Vol 30 (1) ◽

pp. 79-87

Author(s):

I. V. Zhuravlev

Keyword(s):

Existence Of Solutions ◽

Beltrami Equation ◽

Multidimensional Analog

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On the existence of solutions of the Beltrami equations with conditions on inverse dilatations

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2021-18-2-7 ◽

2021 ◽

Vol 18 (2) ◽

pp. 243-254

Author(s):

Evgeny Sevost’yanov

Keyword(s):

Sobolev Class ◽

Existence Of Solutions ◽

Continuous Solution ◽

Beltrami Equation ◽

Beltrami Equations ◽

Homeomorphic Solution

We have found one of possible conditions under which the degenerate Beltrami equation has a continuous solution of the Sobolev class. This solution is H\"{o}lder continuous in the ''weak'' (logarithmic) sense with the exponent power $\alpha=1/2.$ Moreover, it belongs to the class $W^{1, 2}_{\rm loc}.$ Under certain additional requirements, it can also be chosen as a homeomorphic solution. We give an appropriate example of the equation that satisfies all the conditions of the main result of the article, but does not have a homeomorphic Sobolev solution.

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Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

Georgian Mathematical Journal ◽

10.1515/gmj-2020-2074 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Roland Duduchava

Keyword(s):

Boundary Value Problems ◽

Mixed Boundary ◽

Boundary Value ◽

Beltrami Equation ◽

Bessel Potential ◽

Convolution Operators ◽

Classical Setting ◽

Type Boundary ◽

Mellin Convolution ◽

Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{1}{p}, 1<p<\infty. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain \Omega_{\alpha}\subset\mathbb{R}^{2} of magnitude 𝛼. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J.27 (2020), 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \mathbb{R}^{+} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys.60 (2013), 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl.443 (2016), 2, 707–731].

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On the Hilbert boundary-value problem for Beltrami equations with singularities

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-4-2 ◽

2020 ◽

Vol 17 (4) ◽

pp. 484-508

Author(s):

Vladimir Gutlyanskii ◽

Vladimir Ryazanov ◽

Eduard Yakubov ◽

Artyem Yefimushkin

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Beltrami Equation ◽

General Nature ◽

Countable Collection ◽

Smooth Curves ◽

Beltrami Equations ◽

Jordan Curves ◽

Equations Of Mathematical Physics ◽

Hilbert Boundary Value Problem

We investigate the Hilbert boundary-value problem for Beltrami equations $\overline\partial f=\mu\partial f$ with singularities in generalized quasidisks $D$ whose Jordan boundary $\partial D$ consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves. Generally speaking, generalized quasidisks do not satisfy the standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring-Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations. In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain $D$ of a general nature. As usual, a point in $\overline D$ is called a singularity of the Beltrami equation, if the dilatation quotient $K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its neighborhoods. Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincar\'{e} boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

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Virtual Element Method for the Laplace-Beltrami equation on surfaces

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2017040 ◽

2018 ◽

Vol 52 (3) ◽

pp. 965-993 ◽

Cited By ~ 8

Author(s):

Massimo Frittelli ◽

Ivonne Sgura

Keyword(s):

Numerical Experiments ◽

Beltrami Equation ◽

Convergence Result ◽

Point Of View ◽

Polygonal Approximation ◽

Virtual Element Method ◽

Element Space ◽

First Order ◽

Virtual Element ◽

Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

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