Degenerate elliptic problems

(a) As we saw in Chapter I any first-order elliptic system of two real equations may be put into complex form 3.1https://www.w3.org/1998/Math/MathML">awz¯+cwz+bw¯z+dw¯z¯+a0w+b0w¯=fhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn3_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with ellipticity condition: 3.2https://www.w3.org/1998/Math/MathML">4∣ac¯−b¯d2−|a|2+|c|2−|b|2−|d|22>0.https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn3_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> We consider (3.1) in the whole complex plane C without contour Γ = Γ₀ + Γj H 1-Γ_m, which divides this plane into bounded multiple connected domain G + and its exterior G¯. If the coefficients in G⁺ and G¯ satisfy the condition https://www.w3.org/1998/Math/MathML">||a2−|b|2|>|ac¯−b¯d|+|ad¯−b¯c∣,https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn0335.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> then we may eliminate ŵ_z and write (3.1) as 3.3https://www.w3.org/1998/Math/MathML">wz¯−q1wz−q2wz¯+a1w+b1w¯=f1https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn3_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with https://www.w3.org/1998/Math/MathML">q1+q2<1https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn0336.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> on C⁺ u C ‘, In this case, if q₁, q₂ are measurable bounded functions and a₁, a₂ are L_p, ₂ (ℂ)-functions with p > 2, then equation (3.3) has a unique solution continuous on the whole plane for any f ∈ L_{p, 2}(ℂ), p > 2.