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 $\beta$ 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}, $x'$ 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 $L^\infty$ 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

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 CC-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

Convoluted $C$-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated $C$-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted $C$-cosine functions and analytic convoluted $C$-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 $(-\Delta)^{2^{n}},$ $n\in {\mathbb N},$ acting on $L^{2}[0,\pi]$ with appropriate boundary conditions, generates an exponentially bounded $K_{n}$-convoluted cosine function, and consequently, an exponentially bounded analytic $K_{n+1}$-convoluted semigroup of angle $\frac{\pi}{2},$ for suitable exponentially bounded kernels $K_{n}$ 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 $\R^n$ 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