10,472 research outputs found
Towards a Theory of Molecular Forces between Deformed Media
A macroscopic theory for the molecular or Casimir interaction of dielectric
materials with arbitrarily shaped surfaces is developed. The interaction is
generated by the quantum and thermal fluctuations of the electromagnetic field
which depend on the dielectric function of the materials. Using a path integral
approach for the electromagnetic gauge field, we derive an effective Gaussian
action which can be used to compute the force between the objects. No
assumptions about the independence of the shape and material dependent
contributions to the interaction are made. In the limiting case of flat
surfaces our approach yields a simple and compact derivation of the Lifshitz
theory for molecular forces. For ideal metals with arbitrarily deformed
surfaces the effective action can be calculated explicitly. For the general
case of deformed dielectric materials the applicability of perturbation theory
and numerical techniques to the evaluation of the force from the effective
action is discussed.Comment: 15 pages, 1 figur
Deformations of constant mean curvature 1/2 surfaces in H2xR with vertical ends at infinity
We study constant mean curvature 1/2 surfaces in H2xR that admit a
compactification of the mean curvature operator. We show that a particular
family of complete entire graphs over H2 admits a structure of infinite
dimensional manifold with local control on the behaviors at infinity. These
graphs also appear to have a half-space property and we deduce a uniqueness
result at infinity. Deforming non degenerate constant mean curvature 1/2
annuli, we provide a large class of (non rotational) examples and construct
(possibly embedded) annuli without axis, i.e. with two vertical, asymptotically
rotational, non aligned ends.Comment: 35 pages. Addition of a half-space theore
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file
Some monotonicity results for minimizers in the calculus of variations
We obtain monotonicity properties for minima and stable solutions of general
energy functionals of the type under the
assumption that a certain integral grows at most quadratically at infinity. As
a consequence we obtain several rigidity results of global solutions in low
dimensions
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