Geometrical Optics and Related Topics (Progress in Nonlinear Differential Equations and Their Applications (32))

This book contains fourteen research papers which are expanded versions of conferences given at a meeting held in September 1996 in Cortona, Italy. The topics include blowup questions for quasilinear equations in two dimensions, time decay of waves in LP, uniqueness results for systems of conservation laws in one dimension, concentra­ tion effects for critical nonlinear wave equations, diffraction of nonlin­ ear waves, propagation of singularities in scattering theory, caustics for semi-linear oscillations. Other topics linked to microlocal analysis are Sobolev embedding theorems in Weyl-Hormander calculus, local solv­ ability for pseudodifferential equations, hypoellipticity for highly degen­ erate operators. The book also contains a result on uniqueness for the Cauchy problem under partial analyticity assumptions and an article on the regularity of solutions for characteristic initial-boundary value problems. On each topic listed above, one will find new results as well as a description of the state of the art. Various methods related to nonlinear geometrical optics are a transversal theme of several articles. Pseu­ dodifferential techniques are used to tackle classical PDE problems like Cauchy uniqueness. We are pleased to thank the speakers for their contributions to the meeting: Serge Alinhac, Mike Beals, Alberto Bressan, Jean-Yves Chemin, Christophe Cheverry, Daniele Del Santo, Nils Dencker, Patrick Gerard, Lars Hormander, John Hunter, Richard Melrose, Guy Metivier, Yoshinori Morimoto, and Tatsuo Nishitani. The meeting was made possible in part by the financial support of a European commission pro­ gram, "Human capital and mobility CHRX-CT94-044" Front Matter....Pages i-vii Blowup of small data solutions for a class of quasilinear wave equations in two dimensions: an outline of the proof....Pages 1-15 Concentration effects in critical nonlinear wave equation and scattering theory....Pages 17-30 Lower semicontinuity of weighted path length in BV....Pages 31-58 Time decay of L p norms for solutions of the wave equation on exterior domains....Pages 59-77 Sobolev embeddings in Weyl-Hörmander calculus....Pages 79-93 About the Cauchy problem for a system of conservation laws....Pages 95-116 Global existence of the solutions and formation of singularities for a class of hyperbolic systems....Pages 117-140 A class of solvable operators....Pages 141-177 On the uniqueness of the Cauchy problem under partial analyticity assumptions....Pages 179-219 Nonlinear wave diffraction....Pages 221-243 Caustics for dissipative semilinear oscillations....Pages 245-266 Geometric optics and the bottom of the spectrum....Pages 267-293 Hypoellipticity for a class of infinitely degenerate elliptic operators....Pages 295-317 Regularity of solutions to characteristic boundary value problem for symmetric systems....Pages 319-361 Back Matter....Pages 363-366