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Zbl 1055.35018
Ashyralyev, Allaberen
On well-posedness of the nonlocal boundary value problems for elliptic equations. (English)
[J] Numer. Funct. Anal. Optimization 24, No. 1-2, 1-15 (2003). ISSN 0163-0563; ISSN 1532-2467/e

The author considers the problem $-u''+Au = f$ in $[0,1]$ subject to the periodicity condition $u(0)=u(1)$ , $u'(0)=u'(1)$. Here $u$ is Banach space valued and $A$ is a positive operator. Working within a space of weighted Hölder norms the author proves coercive solvability. The abstract result is applied to elliptic problems like $-\partial_1^2u(x) - a(x_2) \partial_2^2 u(x) + \delta u(x) = f(x)$ in the square $[0,1] \times [0,1]$ subject to periodic boundary conditions.
[Norbert Weck (Essen)]

MSC 2000:: *35B30 Dependence of solutions of PDE on initial and boundary data
34G10 Linear ODE in abstract spaces
35J25 Second order elliptic equations, boundary value problems

Keywords: ordinary differential equations in Banach spaces; periodic boundary conditions; elliptic equations

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Zentralblatt MATH Berlin [Germany]