Geometrical optics and related topics

This volume contains 14 research papers which are expanded versions of conferences given at a meeting held in Cortona, Italy in the fall of '96. The topics include blowup questions for quasilinear equations in 2-d, time decay of waves in L p , uniqueness results for systems of conservation laws in 1-d, concentration effects for critical nonlinear wave equations, diffraction of nonlinear waves, propagation of singularities in scattering theory, and caustics for semilinear oscillations. Other topics linked to microlocal analysis are Sobolev spaces in Weyl-Hormander calculus, local solvability for pseudodifferential equations, and hypoellipticity for highly degenerate operators. Included as well are a result on uniqueness for the Cauchy problem under partial analyticity assumptions and an article on the regularity of solutions for the characteristic initial boundary value problem. For each of the above topics, there are sharp new results as well as a description of the state-of-the-art by a renowned expert in the field. Various methods related to nonlinear geometrical optics are a transversal theme of several articles. Pseudodifferential techniques are used to tackle classical PDE problems such as Cauchy uniqueness. Most of the papers contain detailed proofs which are accessible to graduate students and active researchers in connected areas. Series: Progress in Nonlinear Differential Equations and Their Applications, Volume 32 Contents: Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions: an outline of the proof - S. Alinhac Concentration effects in critical nonlinear wave equation and scattering theory - H. Bahouri and P Gerard Lower semicontinuity of weighted path length in BV - P. Baiti and A Bressan Time decay of L p norms for solutions of the wave equation on exterior domains - M. Beals Sobolev embeddings in Weyl-Hormander calculus - J.-Y. Chemin and C.-J. Xu About the Cauchy problem for a system of conservation laws - C. Cheverry Global existence of the solutions and formation of singularities for a class of hyperbolic systems - D. Del Santo, V. Georgiev and E. Mitidieri A class of solvable operators- N. Dencker On the uniqueness of the Cauchy problem under partial analyticity assumptions - L. Hormander Nonlinear wave diffraction - J.K. Hunter Caustics for dissipative semilinear oscillations - J.-L. Joly, G. Metivier, and J. Rauch Geometric optics and the bottom of the spectrum - R.B. Melrose Hypoellipticity for a class of infinitely degenerate elliptic operators - Y. Morimoto and T. Morioka Regularity of solutions to characteristic boundary value problem for symmetric systems - T. Nishitani and M. Takayama