707 research outputs found
Gap solitons in almost periodic one-dimensional structures
We consider almost periodic stationary nonlinear Schr\"odinger equations in
dimension . Under certain assumptions we prove the existence of nontrivial
finite energy solutions in the strongly indefinite case. The proof is based on
a carefull analysis of the energy functional restricted to the so-called
generalized Nehari manifold, and the existence and fine properties of special
Palais-Smale sequences. As an application, we show that certain one dimensional
almost periodic photonic crystals possess gap solitons for all prohibited
frequencies
Variational methods in relativistic quantum mechanics
This review is devoted to the study of stationary solutions of linear and
nonlinear equations from relativistic quantum mechanics, involving the Dirac
operator. The solutions are found as critical points of an energy functional.
Contrary to the Laplacian appearing in the equations of nonrelativistic quantum
mechanics, the Dirac operator has a negative continuous spectrum which is not
bounded from below. This has two main consequences. First, the energy
functional is strongly indefinite. Second, the Euler-Lagrange equations are
linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral
gap (between the negative and positive continuous spectra). Moreover, since we
work in the space domain R^3, the Palais-Smale condition is not satisfied. For
these reasons, the problems discussed in this review pose a challenge in the
Calculus of Variations. The existence proofs involve sophisticated tools from
nonlinear analysis and have required new variational methods which are now
applied to other problems
A first order system least squares method for the Helmholtz equation
We present a first order system least squares (FOSLS) method for the
Helmholtz equation at high wave number k, which always deduces Hermitian
positive definite algebraic system. By utilizing a non-trivial solution
decomposition to the dual FOSLS problem which is quite different from that of
standard finite element method, we give error analysis to the hp-version of the
FOSLS method where the dependence on the mesh size h, the approximation order
p, and the wave number k is given explicitly. In particular, under some
assumption of the boundary of the domain, the L2 norm error estimate of the
scalar solution from the FOSLS method is shown to be quasi optimal under the
condition that kh/p is sufficiently small and the polynomial degree p is at
least O(\log k). Numerical experiments are given to verify the theoretical
results
Partial expansion of a Lipschitz domain and some applications
We show that a Lipschitz domain can be expanded solely near a part of its
boundary, assuming that the part is enclosed by a piecewise C1 curve. The
expanded domain as well as the extended part are both Lipschitz. We apply this
result to prove a regular decomposition of standard vector Sobolev spaces with
vanishing traces only on part of the boundary. Another application in the
construction of low-regularity projectors into finite element spaces with
partial boundary conditions is also indicated
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