Electron. J. Differential Equations, Vol. 2020 (2020), No. 107, pp. 116.
Dirichlet problem for secondorder abstract differential equations
Giovanni Dore
Abstract:
We study the wellposedness in the space of continuous functions of
the Dirichlet boundary value problem for a homogeneous linear
secondorder differential equation u''+Au = 0, where A is a linear closed
densely defined operator in a Banach space.
We give necessary conditions for the wellposedness, 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
wellposedness of the problem.
Moreover we characterize the bounded operators for which the problem is wellposed.
Submitted August 19, 2018. Published October 29, 2020.
Math Subject Classifications: 34G10.
Key Words: Boundary value problem; differential equations in Banach spaces.
DOI: 10.58997/ejde.2020.107
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Giovanni Dore
Department of Mathematics
University of Bologna
Piazza di Porta San Donato 5, I40126, Bologna, Italy
email: giovanni.dore@unibo.it

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