Phase Space Analysis of Partial Differential Equations

This collection of original articles and surveys treats linear and nonlinear aspects of the theory of partial differential equations. Phase space analysis methods, also known as microlocal analysis, have yielded striking results over the past years and have become one of the main tools of investigation. Equally important is their role in many applications to physics, for example, in quantum and spectral theory.

Key topics:

* The Cauchy problem for linear and nonlinear hyperbolic equations

* Scattering theory

* Inverse problems

* Hyperbolic systems

* Gevrey regularity of solutions of PDEs

* Analytic hypoellipticity

and unique features:

* Original articles are self-contained with full proofs

* Survey articles give a quick and direct introduction to selected topics evolving at a fast pace

Graduate students at various levels as well as researchers in PDEs and related fields will find this an excellent resource.

Trace theorem on the Heisenberg group on homogeneous hypersurfaces.- Strong unique continuation and finite jet determination for Cauchy-Riemann mappings.- On the Cauchy problem for some hyperbolic operator with double characteristics.- On the differentiability class of the admissible square roots of regular nonnegative functions.- The Benjamin—Ono equation in energy space.- Instabilities in Zakharov equations for laser propagation in a plasma.- Symplectic strata and analytic hypoellipticity.- On the backward uniqueness property for a class of parabolic operators.- Inverse problems for hyperbolic equations.- On the optimality of some observability inequalities for plate systems with potentials.- Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach.- Non-effectively hyperbolic operators and bicharacteristics.- On the Fefferman-Phong inequality for systems of PDEs.- Local energy decay and Strichartz estimates for the wave equation with time-periodic perturbations.- An elementary proof of Fedi?’s theorem and extensions.- Outgoing parametrices and global Strichartz estimates for Schrödinger equations with variable coefficients.- On the analyticity of solutions of sums of squares of vector fields.