Cn−1-estimate of the solution of the Dirichlet problem with L2-boundary data

1991 ◽  
pp. 142-167
Author(s):  
Jan Chabrowski
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data

On the Solution of the Dirichlet Problem with Rational Holomorphic Boundary Data

2006 ◽  
Vol 5 (2) ◽  
pp. 445-457 ◽  
Author(s):  
Peter Ebenfelt ◽  
Michael Viscardi
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data

On Dirichlet problem for Beltrami equations in finitely connected domains

2021 ◽  
Vol 34 ◽  
pp. 85-99
Author(s):  
Ihor Petkov ◽  
Vladimir Ryazanov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Boundary Behavior ◽  
Beltrami Equation ◽  
Elliptic Systems ◽  
Beltrami Equations ◽  
Prime Ends ◽  
Wide Circle ◽  
Continuous Boundary ◽  
Homeomorphic Solution

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.

scholarly journals Dirichlet problem for Poisson equations in Jordan domains

2018 ◽  
Vol 32 ◽  
pp. 30-41
Author(s):  
Vladimir Gutlyanskii ◽  
Vladimir Ryazanov ◽  
Eduard Yakubov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Continuous Functions ◽  
Logarithmic Capacity ◽  
Hölder Continuous ◽  
Arbitrary Boundary ◽  
Poisson Equations ◽  
Local Properties ◽  
Continuous Boundary ◽  
Angular Limits

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

The Dirichlet problem for the mimimal hypersurface equation with Lipschitz continuous boundary data in a Riemannian manifold

2020 ◽  
Vol 307 (1) ◽  
pp. 1-12
Author(s):  
Arì Aiolfi ◽  
Giovanni da Silva Nunes ◽  
Lisandra Sauer ◽  
Rodrigo Soares
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Boundary Data ◽  
Lipschitz Continuous ◽  
Continuous Boundary

scholarly journals Dirichlet problem with LP-boundary data

1987 ◽  
Vol 36 (1) ◽  
pp. 173-175
Author(s):  
Thomas Hoffmann-Walbeck
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data

Note on the Dirichlet problem with L2-boundary data

1982 ◽  
Vol 40 (1) ◽  
pp. 91-108 ◽  
Author(s):  
Jan Chabrowski
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data
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