scholarly journals The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Dirichlet Problem ◽  
Classical Solution ◽  
Laplace Equation ◽  
Boundary Data ◽  
Exterior Domains ◽  
Compatibility Conditions ◽  
Closed Curves ◽  
Asymptotic Formulae ◽  
Well Posed ◽  
Solution Gradient

We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. In the formulation of the problem, we do not require compatibility conditions for Dirichlet's boundary data at the tips of the cracks. However, if boundary data satisfies the compatibility conditions at the tips of the cracks, then this is a particular case of our problem. The cases of both interior and exterior domains are considered. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a solution is obtained. It is shown that weak solution of the problem does not typically exist, though the classical solution exists. The asymptotic formulae for singularities of a solution gradient at the tips of the cracks are presented.

scholarly journals The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Helmholtz Equation ◽  
Exterior Domain ◽  
Index Zero ◽  
Compatibility Conditions ◽  
Unique Classical Solution ◽  
Asymptotic Formulae ◽  
Solution Gradient

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

scholarly journals On the high order convergence of the difference solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 893-901 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Ahlam Abdussalam
Keyword(s):  
Approximate Solution ◽  
Dirichlet Problem ◽  
Laplace Equation ◽  
Grid Point ◽  
Order Convergence ◽  
Compatibility Conditions ◽  
Second Derivatives ◽  
Boundary Functions ◽  
High Order Convergence ◽  
The Difference

The boundary functions ?j of the Dirichlet problem, on the faces ?j, j = 1,2,..., 6 of the parallelepiped R are supposed to have seventh derivatives satisfying the H?lder condition and on the edges their second, fourth and sixth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh-u of the approximate solution uh at each grid point (x1,x2,x3), a pointwise estmation O(?h6) is obtained, where ?= ?(x1,x2,x3) is the distance from the current grid point to the boundary of R; h is the grid step. The solution of difference problems constructed for the approximate values of the first and pure second derivatives converge with orders O(h6 ?ln h?) and O(h5+?), 0 < ? < 1, respectivly.

scholarly journals The Dirichlet problem for the two-dimensional Laplace equation in a multiply connected domain with cuts

2000 ◽  
Vol 43 (2) ◽  
pp. 325-341 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Dirichlet Problem ◽  
Classical Solution ◽  
Potential Theory ◽  
Laplace Equation ◽  
Connected Domain ◽  
Fredholm Equation ◽  
Two Dimensional ◽  
Multiply Connected Domain ◽  
Plane Region ◽  
Multiply Connected

AbstractThe Dirichlet problem for the Laplace equation in a connected-plane region with cuts is studied. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable.

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.

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