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

Summary: Consider on $\left(0,1\right)$ the boundary value problem

$\begin{array}{cc}& Lu=-a\left(x\right){u}^{\left[2\right]}\left(x\right)+A\left(x\right)u\left(x\right)+{A}_{1}\left(x\right){u}^{\left[1\right]}\left(x\right)+{A}_{2}\left(x\right)u\left(x\right)=f,\hfill \\ & {L}_{1}u=\sum _{k=0}^{{m}_{1}}{\alpha }_{k}{u}^{\left[k\right]}\left(0\right)=0,\phantom{\rule{1.em}{0ex}}{L}_{2}u=\sum _{k=0}^{{m}_{2}}{\beta }_{k}{u}^{\left[k\right]}\left(1\right)=0\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(*\right)$

in ${L}_{p}\left(0,1;E\right)$, where ${u}^{\left[i\right]}={\left[{x}^{{\gamma }_{1}}{\left(1-x\right)}^{{\gamma }_{2}}\frac{d}{dx}\right]}^{i}u\left(x\right)$, $0\le {\gamma }_{i}<1$, ${m}_{k}\in \left\{0,1\right\}$; ${\alpha }_{k}$ and ${\beta }_{k}$ are complex numbers, $A$ and ${A}_{i}\left(x\right)$ 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 ODE in abstract spaces 35J25 Second order elliptic equations, boundary value problems 35J70 Degenerate elliptic equations 35K65 Parabolic equations of degenerate type 34B15 Nonlinear boundary value problems for ODE
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