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Degenerate differential operators with parameters. (English) Zbl 1181.35112

The author studies nonlocal boundary value problems for regular degenerate differential-operator equations with parameters. A form of a nonlocal boundary value problem is given by \[ L_0(\lambda,t)u=-tu^{|2|}(x)+(A+\lambda)u(x)=0, \text{ together with} \]
\[ L_1u= \alpha_0t^{\theta_1}u^{\lfloor m_1\rfloor}(0)= f_1\quad L_2u=\beta_0 t^{\theta_2}u^{\lfloor m_1\rfloor}(1)=f_2, \] where \(m_k\in [0,1]\); \(\alpha_k,\beta_k,\delta_{kj}\) are complex numbers, \(A\) is, generally speaking, an unbounded operator in \(E\). One of the big theorems proven is as follows: Let \(A\) be a positive operator in a Banach space \(E\) for \(\vartheta\in (0,\pi]\), \(0\leq\nu<1-1/p\), \(p\in(1,\infty)\), \(0<t<t_0< \infty\), \(\alpha_0\neq 0\), \(\beta_0\neq 0\). The problem above for \(f_k\in E_k\), \(|\arg\lambda|\leq\pi-\vartheta\), for sufficiently large \(|\lambda|\) and \(t\), has a unique solution \(u\) belonging to the space, \(W^{[2]}_{p, \gamma}(0,1;E(A),E)\), and has a very coercive uniform estimate given in the paper. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valued \(L_p\) spaces are given for these problems. In applications, the nonlocal boundary value problems for degenerated elliptic partial differential equations and for systems of elliptic equations with parameters are studied on cylindrical domains.

MSC:

35J70 Degenerate elliptic equations
35J57 Boundary value problems for second-order elliptic systems
47F05 General theory of partial differential operators
34G10 Linear differential equations in abstract spaces
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
46E40 Spaces of vector- and operator-valued functions
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References:

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