Dirichlet problem for second-order abstract differential equations
Texas State University, Department of Mathematics
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 k2, 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.
Boundary value problem, Differential equations in Banach spaces
Dore, G. (2020). Dirichlet problem for second-order abstract differential equations. <i>Electronic Journal of Differential Equations, 2020</i>(107), pp. 1-16.
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