By Schaaf W.L.

ISBN-10: 0873531205

ISBN-13: 9780873531207

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N + 1. 4(m + 2) Acknowledgement The authors wish to thank both Cathleen Morawetz for numerous stimulating discussions and the Courant Institute for its hospitality during the period in which this work was begun. 42 References [1] S. Agmon, L. Nirenberg, and M. H. Protter, A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type, Comm. Pure Appl. Math. 6 (1953), 455-470. [2] J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator, Duke Math.

In fact, one can show that most higher order derivatives vanish (such as ηxi xk , ξxi k y with i = k and ξxi j xk with i = j = k = i in the case N ≥ 3) and that ηx1 x1 = . . ηxN xN , which is then used to show that ξ i , η are quadratic polynomials in x with y dependent coefficients. 14). In short, the argument is similar to that used in the classical computation for the (constant coefficient) wave operator and the principal difficulty here is to allow for the y dependence everywhere. 15) where α0 is a new constant.

On the eigenfunctions of the equation ∆u + λf (u) = 0. Soviet Math. Dokl. 6 (1965), 1408-1411. [28] Protter, M. H. Uniqueness theorems for the Tricomi problem, J. Rat. Mech. Anal. 2 (1953), 107-114. [29] J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. 138 (1993), 503-518. [30] J. Shatah and M. Struwe, “Geometric Wave Equations”, Courant Lecture Notes in Mathematics 2, American Mathematical Society, Providence, RI, 1998. [31] W. Strauss, “Nonlinear Wave Equations”, CBMS Regional Conference Series in Mathematics 73, American Mathematical Society, Providence, RI, 1989.

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