Mikhail Kamenskii, Paul Raynaud de Fitte, Ngai-Ching Wong, Margarita Zvereva, A model of deformations of a discontinuous Stieltjes string with a nonlinear boundary condition
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DOI: 10.23952/jnva.5.2021.5.08
Volume 5, Issue 5, 1 October 2021, Pages 737-759
Abstract. Variational methods are used to study a model of the deformation of a discontinuous Stieltjes string (a chain of strings held together by springs) located along the segment . The model is described by the integro-differential equation
with derivatives with respect to the measure $\mu$ generated by a given strictly increasing function
on the segment
, where the function
determines the deformation of the string,
characterizes the elasticity of the string, the functions
and
describe the elastic response of the external environment and the external load, respectively. The integral
is understood in the generalized sense according to Stieltjes. We are looking for solutions
in the class of
-absolutely continuous functions on
, whose derivatives have bounded variation on
. We assume that one of the boundary conditions is nonlinear and has the form
where
denotes the outward normal cone at the point
to the segment
. This condition arises due to the presence of the limiter
on the motion of the elastically fixed right end of the string (by a spring with elasticity
) so that
. In this paper, necessary and sufficient conditions for the minimization of the energy functional of the Stieltjes string system are established, the critical loads at which the contact of the end of the string with the boundary points of the limiter occurs are determined, and the dependence of the solution on the length of the limiter is studied.
How to Cite this Article:
Mikhail Kamenskii, Paul Raynaud de Fitte, Ngai-Ching Wong, Margarita Zvereva, A model of deformations of a discontinuous Stieltjes string with a nonlinear boundary condition, J. Nonlinear Var. Anal. 5 (2021), 737-759.