Differential equations in Hilbert spaces and applications to boundary value problems in nonsmooth domains

AbstractWe give existence and uniqueness results in weighted Sobolev spaces for a solution of a first order differential equation with operator coefficients of type ∂u∂t − Au(t) = ƒ(t), on the line R. This equation is an abstract version of elliptic boundary value problems in infinite cylinders (see [V. A. Kondratiev, Trans. Moscow Math. Soc.16 (1967), 227–313; V. G. Maz'ya and B. A. Plamenevskii, Trans. Moscow Math. Soc.1 (1980), 49–97; V. G. Maz'ya and B. A. Plamenevskii, Amer. Math. Soc. Transl. Ser. 2123 (1984), 1–56; V. G. Maz'ya and B. A. Plamenevskii. J. Soviet Math.9 (1978), 750–764; V. G. Maz'ya and B. A. Plamenevskii, Amer. Math. Soc. Transl. Ser. 2123 (1984), 57–88]); therefore, our results translate the classical ones to this setting. The difference between two solutions belonging to different weighted Sobolev spaces is a finite linear combination of singular functions depending on the eigenvalues of A and on the corresponding eigenspaces. We compute the coefficients of these singular functions, because we can give explicitly the Laurent series of the resolvent of A near an eigenvalue of the operator A. We also give an abstract polynomial resolution, which corresponds to the resolution of an elliptic boundary value problem in an infinite cone with polynomial right-hand sides. In practice, it allows one to pass from weighted Sobolev spaces to classical Sobolev spaces. Finally, we consider some applications of this abstract theory