Solutions of the dirichlet problem for elliptic systems in a disk

1992 ◽  
Vol 44 (10) ◽  
pp. 1197-1203 ◽  
Author(s):  
V. P. Burskii
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Systems

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 The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO

2019 ◽  
Vol 12 (3) ◽  
pp. 605-720 ◽  
Author(s):  
José María Martell ◽  
Dorina Mitrea ◽  
Irina Mitrea ◽  
Marius Mitrea
Keyword(s):  
Dirichlet Problem ◽  
Half Space ◽  
Elliptic Systems

scholarly journals Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

2021 ◽  
Author(s):  
Gioconda Moscariello ◽  
Giulio Pascale
Keyword(s):  
Dirichlet Problem ◽  
Unit Ball ◽  
Elliptic System ◽  
Positive Constant ◽  
Elliptic Systems ◽  
Second Order ◽  
Identity Matrix ◽  
Bmo Coefficients

AbstractWe consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ d i v Λ exp ( - | x | ) - log | x | I D u = d i v F + g in B . Here B denotes the unit ball of $$\mathbb {R}^n$$ R n , for $$n > 2$$ n > 2 , centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$ W 1 , 2 ( B , R n × n ) , g is a vector in $$L^2(B, \mathbb {R}^n)$$ L 2 ( B , R n ) and $$\Lambda $$ Λ is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$ u ∈ W 0 1 , 2 ( B , R n ) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$ Λ is not large enough.

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