Green‘s Function and Poisson Integral in a Circle Disk for Strongly Elliptic Systems with Constant Coefficients

The paper deals with Dirichlet problem for a homogeneous strongly elliptic second-order system with constant coefficients, in other words, for a partial differential equation of the following kind L_τ,σf = 0 where f is a complex-valued function, and L_τ,σ= (∂∂τ∂²)I+σ(τ∂∂+∂²)C. Here δδ are Cauchy --- Riemann operators; I is an identity operator; C:z--- z̅ is a complex conjugation operator; τ,σ are such parameters, that τσ ∈ (--1,1). For such systems, integral formulas of the Poisson type, Green's function and solutions of Dirichlet problem in a circle and an ellipse of a special form are obtained. The L_τ,σ operator is a perturbation of Laplace operator Δ, and the Dirichlet problem solution for the equation L_τ,σf = 0 is obtained as a sum of a series in powers of the parameter σ. Functions that are coefficients of the corresponding series can be found by solving the "recurrent" sequence of Dirichlet problems for the ordinary Laplace and Poisson equations.

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