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Zbl 1229.34093
Shakhmurov, Veli B.
Regular degenerate separable differential operators and applications. (English)
[J] Potential Anal. 35, No. 3, 201-222 (2011). ISSN 0926-2601; ISSN 1572-929X/e

Summary: Consider on $(0,1)$ the boundary value problem $$\aligned & Lu=-a(x)u^{[2]}(x)+A(x)u(x)+A_1(x)u^{[1]}(x)+A_2(x)u(x)=f,\\ & L_1u=\sum^{m_1}_{k=0}\alpha_ku^{[k]}(0)=0,\quad L_2u=\sum^{m_2}_{k=0}\beta_ku^{[k]}(1)=0\endaligned\tag*$$ in $L_p(0,1;E)$, where $u^{[i]}=\left[x^{\gamma_1}(1-x)^{\gamma_2}\frac{d}{dx}\right]^iu(x)$, $0\le\gamma_i<1$, $m_k\in\{0,1\}$; $\alpha_k$ 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 2000:: *34G10 Linear ODE in abstract spaces
35J25 Second order elliptic equations, boundary value problems
35J70 Elliptic equations of degenerate type
35K65 Parabolic equations of degenerate type
34B15 Nonlinear boundary value problems of ODE

Keywords: degenerate differential-operator equations; semigroups of operators; Banach-valued function spaces; separability; Fredholmness; operator-valued Fourier multipliers; interpolation of Banach spaces

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Zentralblatt MATH Berlin [Germany]