Huygens' Operators and Hadamard's Conjecture

点击翻译按钮获取中文摘要

ABSTRACT A lacuna of a linear hyperbolic differential operator is adomain inside the propagation cone where its fundamental solution vanishes identically. The study of lacunas for generalhyperbolic equations was initiated by I.G.Petrovsky (1945).Extending and clarifying his results, Atiyah, Bott and Garding(1970-73) developed a profound and complete theory for hyperbolicoperators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. The investigation of this classical problem with a view ofgeneralizing the Petrovsky-Atiyah-Bott-Garding theory is one of the primary purposes of my project. In particular, theefforts will be aimed at resolving the old question ofJ.Hadamard on explicit determination of second order analyticwave operators satisfying Huygens' principle on Minkowskispaces. Some new conjectures and problems (mostly from Analysis and Algebraic Geometry) arising in connection with the theoryof lacunas will be also explored. The study of propagation of waves in continuous media is one ofthe fundamental problems in Mathematical Physics with many important applications in natural sciences and engineering. Of special interest (both from practical and theoretical point of view) is the question of when the waves may propagate without diffusion to allow the possibility of transmitting `clean-cut' (sharp) signals. Well-studied in homogeneous spaces this question remains largely open in general. My project aims to develop new mathematical tools and techniques to investigate this difficult problem in the case ofinhomogeneous and anisotropic media. The results sought are of fundamentalinterest and significance in mathematical theory of wave propagation and may have applications in related physical disciplines including thetheory of electromagnetic and acoustic waves, space communicationtechnologies, magnetohydrodynamics, crystal optics, etc.