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On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

In a cholesteric liquid crystal the director field n ( x , y , z ) tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz...

On a class of functional boundary value problems for the equation x'' = f(t,x,x',x'',λ)

Svatoslav Staněk (1994)

Annales Polonici Mathematici

The Leray-Schauder degree theory is used to obtain sufficient conditions for the existence and uniqueness of solutions for the boundary value problem x'' = f(t,x,x',x'',λ), α(x) = 0, β(x̅) = 0, γ(x̿)=0, depending on the parameter λ. Here α, β, γ are linear bounded functionals defined on the Banach space of C⁰-functions on [0,1] and x̅(t) = x(0) - x(t), x̿(t)=x(1)-x(t).

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