Department of Mathematics
University of Bologna
Piazza di Porta San Donato 5, I-40126, Bologna, Italy
email: giovanni.dore@unibo.it
Giovanni Dore
Abstract:
We study the well-posedness in the space of continuous functions of
the Dirichlet boundary value problem for a homogeneous linear
second-order differential equation u''+Au = 0, where A is a linear closed
densely defined operator in a Banach space.
We give necessary conditions for the well-posedness, in terms of the
resolvent operator of A. In particular we obtain an estimate on the norm of
the resolvent at the points k^2, where k is a positive integer, and we show that
this estimate is the best possible one, but it is not sufficient for the
well-posedness of the problem.
Moreover we characterize the bounded operators for which the problem is well-posed.
Submitted August 19, 2018. Published October 29, 2020.
Math Subject Classifications: 34G10.
Key Words: Boundary value problem; differential equations in Banach spaces.
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Giovanni Dore Department of Mathematics University of Bologna Piazza di Porta San Donato 5, I-40126, Bologna, Italy email: giovanni.dore@unibo.it |
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