On the Cauchy problem for first-order elliptic systems

Abdunabi Abdukarimov Tashkent State Architectural Construction University

In this note we present a formula for recovering the solution of systems of firstorder differential equations of elliptic type of a special class in a domain on the basis of their values on the boundary.

Let A_{l×n}(x) denote the class of matrices D(x^{T})(x^{T} = (x_{1}, ..., x_{m})^{T} the transposed column vector of x ) with constant coefficients in C^{1} for which

D^{∗}(x^{T})D(x′) = ϵ((| x |^{2} +λ)U^{0},

where D^{∗}(x^{T}) is the Hermitian conjugate of the matrix D^{∗}(x^{T}) and ϵ(x) is a diagonal matrix , U^{0} is the unit vector.

We denote by G_{ρ} a domain in R^{m} whose boundary consists of part of the - surface of the cone

y_{1}^{2} + ... + y_{m−1}^{2} = τy_{m}^{2}, (τ = tan π/2ρ)

and a smooth surface S lying inside the cone.

We consider the system of differential equations ( 1 )

(1)  D(\frac{∂}{∂x})u(x) = 0,

where the characteristic matrix

D(x^{T}) ∈ A_{l×n}(x).

In this note we consider the Cauchy problem in the formulation of M.M. Lavrent’ev [ 1 ] . Suppose u(x) ∈ C^{1}(G_{ρ}) ∩ C(G_{ρ}) and u(x) satisfies (1) , and let u_{δ}(x) be continuous approximation to u(x) on S , i . e . ,

max | u(x) − u_{δ}(x) |< δ, 0 < δ < 1.

We want recover u ( x ) in G_{ρ}.

Theorem . Suppose u ( x ) is a solution of ( 1 ) and

| u(y) |= M, y ∈ T = ∂G_{ρ} \ S,

where M is a given positive number . Set

u_{σ}(x) = ∫_{S} N_{σ}(x, y)u(y)ds_{y}, x ∈ G_{ρ}.

Then

| u(x) − u_{σ}(x) |≤ MC(x)C(σ)exp(−σγ^{ρ}), x ∈ G_{ρ},

where

C(x) = C_{ρ} ∫_{S} \frac{ds}{r^{m−1}} ,

C(σ) = \{ \begin{matrix}σ^{m} for m = 2k + 1, k ≥ 1, \\ σ^{m−1} for m = 2k, k ≥ 2. \end{matrix}

Corollari . The limit equality

lim_{σ→∞} u(x) = u(x), x ∈ G_{ρ},

is satisfied uniformly in each compact set in G_{ρ}.

Bibliography

1 . M . M . Lavrent’ ev , On some ill - posed problems of mathematical physics , Izdat . Sibirsk . Otdel Acad . Nauk . SSSR , Novosibirsk , 1962; English transl ., Springer - Verlag , Berlin , 1967 .