Abstract
In this work, we obtain algebraic conditions which assure the Fredholm solvability of an abstract differential equation of elliptic type. In this respect, our work can be considered as an extension of Yakubov’s results to the case of boundary conditions containing a linear operator. Although essential technical, this extension is not straight forward as we show it below. The obtained abstract result is applied to a non regular boundary value problem for a second order partial differential equation of an elliptic type in a cylindrical domain. It is interesting to note that the problems considered in cylindrical domains are not coercive.
Highlights
Consider in the space Lp(0, 1; H) where H is a Hilbert space, the boundary value problem for the second order abstract differential equation
We apply the abstract results obtained to a non regular boundary value problem for a concrete partial differential equation in a cylinder
We study an homogeneous abstract differential equation, we prove the isomorphism and the non coercive estimates for the solution, an estimates which is not explicit with respect to the spectral parameter
Summary
Let H1, H be Hilbert spaces such that H1 ⊂ H with continuous injection. Let E0, E1 be two Banach spaces, continuously embedded in the Banach space E, the pair {E0, E1} is said an interpolation couple. E0 + E1 = {u : u ∈ E, ∃uj ∈ Ej , j = 0, 1, with u = u0 + u1} , and the functional u. The interpolation space for the couple {E0, E1} is defined, by the K-method, as follows (E0, E1)θ,p = u : u ∈ E0 + E1, u θ,p =. H(A) is the domain of A provided with the hilbertian norm u.
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