180 research outputs found
Semilinear geometric optics with boundary amplification
We study weakly stable semilinear hyperbolic boundary value problems with
highly oscillatory data. Here weak stability means that exponentially growing
modes are absent, but the so-called uniform Lopatinskii condition fails at some
boundary frequency in the hyperbolic region. As a consequence of this
degeneracy there is an amplification phenomenon: outgoing waves of amplitude
O(\eps^2) and wavelength \eps give rise to reflected waves of amplitude
O(\eps), so the overall solution has amplitude O(\eps). Moreover, the
reflecting waves emanate from a radiating wave that propagates in the boundary
along a characteristic of the Lopatinskii determinant. An approximate solution
that displays the qualitative behavior just described is constructed by solving
suitable profile equations that exhibit a loss of derivatives, so we solve the
profile equations by a Nash-Moser iteration. The exact solution is constructed
by solving an associated singular problem involving singular derivatives of the
form \partial_{x'}+\beta\frac{\partial_{\theta_0}}{\eps}, being the
tangential variables with respect to the boundary. Tame estimates for the
linearization of that problem are proved using a first-order calculus of
singular pseudodifferential operators constructed in the companion article
\cite{CGW2}. These estimates exhibit a loss of one singular derivative and
force us to construct the exact solution by a separate Nash-Moser iteration.
The same estimates are used in the error analysis, which shows that the exact
and approximate solutions are close in on a fixed time interval
independent of the (small) wavelength \eps. The approach using singular
systems allows us to avoid constructing high order expansions and making small
divisor assumptions
Propagation des singularités pour une classe d'opérateurs à caractéristiques multiples et résolubilité locale
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2011 ElsevierFor functions from the Sobolev space H^s(\Omega), 1/2 < s < 3/2 , definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the generalized co-normal derivative inherent non-uniqueness are given. It is shown, that the canonical co-normal derivatives, although de¯ned on a more narrow function class than the generalized ones, are continuous extensions of the classical co-norma derivatives. Some new results about trace operator estimates and Sobolev spaces haracterizations, are also presented
Convoluted -cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines
Convoluted -cosine functions and semigroups in a Banach space setting
extending the classes of fractionally integrated -cosine functions and
semigroups are systematically analyzed. Structural properties of such operator
families are obtained. Relations between convoluted -cosine functions and
analytic convoluted -semigroups, introduced and investigated in this paper
are given through the convoluted version of the abstract Weierstrass formula
which is also proved in the paper. Ultradistribution and hyperfunction sines
are connected with analytic convoluted semigroups and ultradistribution
semigroups. Several examples of operators generating convoluted cosine
functions, (analytic) convoluted semigroups as well as hyperfunction and
ultradistribution sines illustrate the abstract approach of the authors. As an
application, it is proved that the polyharmonic operator
acting on with appropriate boundary
conditions, generates an exponentially bounded -convoluted cosine
function, and consequently, an exponentially bounded analytic
-convoluted semigroup of angle for suitable
exponentially bounded kernels and $K_{n+1}.
Generalized Bergman kernels on symplectic manifolds
We study the near diagonal asymptotic expansion of the generalized Bergman
kernel of the renormalized Bochner-Laplacian on high tensor powers of a
positive line bundle over a compact symplectic manifold. We show how to compute
the coefficients of the expansion by recurrence and give a closed formula for
the first two of them. As consequence, we calculate the density of states
function of the Bochner-Laplacian and establish a symplectic version of the
convergence of the induced Fubini-Study metric. We also discuss generalizations
of the asymptotic expansion for non-compact or singular manifolds as well as
their applications. Our approach is inspired by the analytic localization
techniques of Bismut-Lebeau.Comment: 48 pages. Add two references on the Hermitian scalar curvatur
Pseudodifferential multi-product representation of the solution operator of a parabolic equation
By using a time slicing procedure, we represent the solution operator of a
second-order parabolic pseudodifferential equation on as an infinite
product of zero-order pseudodifferential operators. A similar representation
formula is proven for parabolic differential equations on a compact Riemannian
manifold. Each operator in the multi-product is given by a simple explicit
Ansatz. The proof is based on an effective use of the Weyl calculus and the
Fefferman-Phong inequality.Comment: Comm. Partial Differential Equations to appear (2009) 28 page
Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients
This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK
- …