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Regular degenerate separable differential operators and applications. (English) Zbl 1229.34093
Summary: Consider on $$(0,1)$$ the boundary value problem
\begin{aligned} & Lu=-a(x)u^{[2]}(x)+A(x)u(x)+A_1(x)u^{[1]}(x)+A_2(x)u(x)=f,\\ & L_1u=\sum^{m_1}_{k=0}\alpha_ku^{[k]}(0)=0,\quad L_2u=\sum^{m_2}_{k=0}\beta_ku^{[k]}(1)=0\end{aligned}\tag{*}
in $$L_p(0,1;E)$$, where $$u^{[i]}=\left[x^{\gamma_1}(1-x)^{\gamma_2}\frac{d}{dx}\right]^iu(x)$$, $$0\leq\gamma_i<1$$, $$m_k\in\{0,1\}$$; $$\alpha_k$$ and $$\beta_k$$ 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:
 34G10 Linear differential equations in abstract spaces 35J25 Boundary value problems for second-order elliptic equations 35J70 Degenerate elliptic equations 35K65 Degenerate parabolic equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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