70,048 research outputs found
Symmetry approach in boundary value problems
The problem of construction of the boundary conditions for nonlinear
equations is considered compatible with their higher symmetries. Boundary
conditions for the sine-Gordon, Jiber-Shabat and KdV equations are discussed.
New examples are found for the Jiber-Shabat equation.Comment: 7 pages, LaTe
Geodesic boundary value problems with symmetry
This paper shows how left and right actions of Lie groups on a manifold may
be used to complement one another in a variational reformulation of optimal
control problems equivalently as geodesic boundary value problems with
symmetry. We prove an equivalence theorem to this effect and illustrate it with
several examples. In finite-dimensions, we discuss geodesic flows on the Lie
groups SO(3) and SE(3) under the left and right actions of their respective Lie
algebras. In an infinite-dimensional example, we discuss optimal
large-deformation matching of one closed curve to another embedded in the same
plane. In the curve-matching example, the manifold \Emb(S^1, \mathbb{R}^2)
comprises the space of closed curves embedded in the plane
. The diffeomorphic left action \Diff(\mathbb{R}^2) deforms the
curve by a smooth invertible time-dependent transformation of the coordinate
system in which it is embedded, while leaving the parameterisation of the curve
invariant. The diffeomorphic right action \Diff(S^1) corresponds to a smooth
invertible reparameterisation of the domain coordinates of the curve. As
we show, this right action unlocks an important degree of freedom for
geodesically matching the curve shapes using an equivalent fixed boundary value
problem, without being constrained to match corresponding points along the
template and target curves at the endpoint in time.Comment: First version -- comments welcome
Nonharmonic analysis of boundary value problems
In this paper we develop the global symbolic calculus of pseudo-differential
operators generated by a boundary value problem for a given (not necessarily
self-adjoint or elliptic) differential operator. For this, we also establish
elements of a non-self-adjoint distribution theory and the corresponding
biorthogonal Fourier analysis. We give applications of the developed analysis
to obtain a-priori estimates for solutions of operators that are elliptic
within the constructed calculus.Comment: 54 pages, updated version, to appear in IMR
Pseudospectra of Semiclassical Boundary Value Problems
We consider operators where is a constant vector field, in
a bounded domain and show spectral instability when the domain is expanded by
scaling. More generally, we consider semiclassical elliptic boundary value
problems which exhibit spectral instability for small values of the
semiclassical parameter h, which should be thought of as the reciprocal of the
Peclet constant. This instability is due to the presence of the boundary: just
as in the case of , some of our operators are normal when
considered on R^d. We characterize the semiclassical pseudospectrum of such
problems as well as the areas of concentration of quasimodes. As an
application, we prove a result about exit times for diffusion processes in
bounded domains. We also demonstrate instability for a class of spectrally
stable nonlinear evolution problems that are associated to these elliptic
operators.Comment: 43 pages, 6 figure
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