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The wave front set of the solution of a simple initial-boundary value problem with glancing rays

  • F. G. Friedlander (a1)

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1. The singularities of solutions of linear partial differential equations are most appropriately described in terms of wave front sets of distributions. (For the definition and properties of wave front sets, see (4) and (5).) In particular, let Ω ⊂ RN be an open set and let P be a linear partial differential operator defined on Ω, with smooth coefficients, real principal symbol p(x, ξ), and simple characteristics. Then if u′ (Ω) and Pu = 0, the wave front set of u, WF(u), is a set in the cotangent bundle T*Ω (minus the zero section) that is contained in p−1(0) and is invariant under the flow generated in p−1(0) by the Hamiltonian vector field (gradξp, − gradxp). In other words, WF(u) is made up of bicharacteristic strips. This is a special case of an important theorem due to Hörmander (5), and Duistermaat and Hörmander (1), which generalizes classical results on the relation between discontinuities of solutions of a partial differential equation and the characteristics of the equation.

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(1)Duistermaat, J. J. and Hörmander, L.Fourier Integral Operators II. Acta Math. 128 (1972), 183259.
(2)Erdélyi, A.Asymptotic expansions (Dover; New York, 1956).
(3)Friedlander, F. G.Sound pulses (Cambridge University Press, 1958).
(4)Hörmander, L.Fourier Integral Operators I. Acta Math. 127 (1971), 79183.
(5)Hörmander, L.On the existence and regularity of solutions of linear pseudo-differential equations. L'Enseignement Math. 17 (1971), 99163.
(6)Nirenberg, L.Lectures on linear partial differential equations. Regional Conference Series 17, A.M.S., 1973.
(7)Zauderer, E.Domains of dependence for mixed problems for wave equations. Arch. Rat. Mech. and Anal. 15 (1964), 6978.

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