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The Method of Stationary Phase and Applications

In: Methods for Partial Differential Equations

Author

Listed:
  • Marcelo R. Ebert

    (University of São Paulo, Department of Computing and Mathematics)

  • Michael Reissig

    (TU Bergakademie Freiberg, Institute of Applied Analysis)

Abstract

In Sections 12.1.2 and 12.1.3 we derived L p − L q decay estimates on and away from the conjugate line for solutions to the Cauchy problem for the heat equation. The basic tools of the approach are tools from the theory of Fourier multipliers, Young’s inequality and embedding theorems. This approach can not be applied to the free wave equation. The goal to derive L p − L q decay estimates for solutions to the Cauchy problem for the wave equation requires a deeper understanding of oscillating integrals with localized amplitudes in different parts of the extended phase space. In particular, L ∞ − L ∞ estimates of such integrals are of interest. One basic tool to get such estimates is the method of stationary phase. We will apply this method to prove L p − L q decay estimates for solutions to the Cauchy problems for the free wave equation, for the Schrödinger equation and for the plate equation. The key lemmas are Littman-type lemmas in the form of Theorems 16.3.1 and 16.8.1. All these tools and interpolation arguments together yield L p − L q estimates on the conjugate line.

Suggested Citation

  • Marcelo R. Ebert & Michael Reissig, 2018. "The Method of Stationary Phase and Applications," Springer Books, in: Methods for Partial Differential Equations, chapter 0, pages 241-269, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-66456-9_16
    DOI: 10.1007/978-3-319-66456-9_16
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