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Microlocal Analysis

  • Chapter
Partial Differential Equations IV

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 33))

Abstract

The microlocal analysis is the local analysis in cotangent bundle space. The remarkable progress made in the theory of linear partial differential equations over the past two decades is essentially due to the extensive applicaton of the microlocalization idea. The Hamiltonian systems, canonical transformations, Lagrange manifolds and other concepts, used in theoretical mechanics for examining processes in the phase space, have in recent years become the central objects of the theory of differential equations. For example, the evolution of singularities of solutions of a differential equation is described most naturally in terms of Lagrange manifolds and Hamiltonian systems, the solvability conditions are formulated in terms of the behaviour of integral curves of the Hamiltonian system whose Hamiltonian function serves as the characteristic form, the class of pseudodifferential equations arises in a natural way from that of differential equations under the action of canonical transformations, the class of subelliptic operators is defined by means of the Poisson brackets, etc. The difficulty faced in the microlocal analysis is connected with the principle of uncertainty which does not permit us to localize a function in any neighbourhood of a point of the cotangent space.

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Egorov, Y.V. (1993). Microlocal Analysis. In: Egorov, Y.V., Shubin, M.A. (eds) Partial Differential Equations IV. Encyclopaedia of Mathematical Sciences, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09207-1_1

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