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On well-posedness of the nonlocal boundary value problems for elliptic equations. (English) Zbl 1055.35018
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 - 1 2 u(x)-a(x 2 ) 2 2 u(x)+δu(x)=f(x) in the square [0,1]×[0,1] subject to periodic boundary conditions.
35B30Dependence of solutions of PDE on initial and boundary data, parameters
34G10Linear ODE in abstract spaces
35J25Second order elliptic equations, boundary value problems