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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.

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
Full Text: DOI
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