## Coerciveness inequality for nonlocal boundary value problems for second order abstract elliptic differential equations.(English)Zbl 1126.35308

The paper deals mainly with boundary value problems for second order abstract differential equations of type $- u'' (x) +A u(x) + A(x) u(x) = f, \quad x \in (0,1),$
$\delta u(0) = f_1, \quad u'(1) + B u(0) = f_2,$ in Hilbert spaces, where $$A$$, $$A(x)$$, $$B$$, are linear operators and $$\delta$$ is a complex number: a priori estimates, solvability, completeness of generalised eigenfunctions. The results are applied to second order elliptic partial differential equations in cylindrical domains.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 34G10 Linear differential equations in abstract spaces 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 47F05 General theory of partial differential operators 47N20 Applications of operator theory to differential and integral equations
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