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Regular degenerate separable differential operators and applications. (English) Zbl 1229.34093

Summary: Consider on (0,1) the boundary value problem

Lu=-a(x)u [2] (x)+A(x)u(x)+A 1 (x)u [1] (x)+A 2 (x)u(x)=f,L 1 u= k=0 m 1 α k u [k] (0)=0,L 2 u= k=0 m 2 β k u [k] (1)=0(*)

in L p (0,1;E), where u [i] =x γ 1 (1-x) γ 2 d dx i u(x), 0γ i <1, m k {0,1}; α k and β k are complex numbers, A and A i (x) are linear operators in a Banach space E.

Several conditions for separability, Fredholmness and resolvent estimates in L p -spaces are given. As applications, the degenerate Cauchy problem for parabolic equations, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on a cylindrical domain are studied.

MSC:
34G10Linear ODE in abstract spaces
35J25Second order elliptic equations, boundary value problems
35J70Degenerate elliptic equations
35K65Parabolic equations of degenerate type
34B15Nonlinear boundary value problems for ODE
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