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Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems

Authors: Michael Beals and Michael Reed
Journal: Trans. Amer. Math. Soc. 285 (1984), 159-184
MSC: Primary 35S05; Secondary 35J60, 35L70
MathSciNet review: 748836
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Abstract: The authors develop a calculus of pseudodifferential operators with nonsmooth coefficients in order to study the regularity of solutions to linear equations $ P\,(x,D)\,u = f$. The regularity theorems are similar to those of Bony, but the calculus and the methods of proof are quite different. We apply the linear results to study the regularity properties of solutions to quasilinear partial differential equations.

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  • [1] M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients, Comm. Pure Appl. Math. 35 (1982), 169-184. MR 644021 (83h:35119)
  • [2] M. Beals, C. Fefferman and R. Grossman, Strictly pseudoconvex domains in $ {C^n}$, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 125-322. MR 684898 (85a:32025)
  • [3] J.-M. Bony, Calcul symbolique et propagation des singularitées pour les équations nonlinéaires, Exposé XXII, Séminaire Goulaouic-Schwartz, Ecole Polytechnique, Paris, 1980. MR 600707 (83h:35076)
  • [4] -, Calcul symbolique et propagation des singularitées pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246. MR 631751 (84h:35177)
  • [5] Shu Xing Chen, Pseudo-differential operators with finitely smooth symbol and their application to quasi-linear equations, Nonlinear Anal. 6 (1982), 1193-1206. MR 683840 (84a:35302)
  • [6] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977. MR 0473443 (57:13109)
  • [7] L. Hörmander, Linear differential operators, Actes Congr. Internat. Math., vol. 1 (Nice, 1970), Gauthier-Villars, Paris, 1971, pp. 121-133. MR 0513000 (58:23766)
  • [8] -, On the extistence and regularity of solutions of linear pseudo-differential equations, Enseign. Math. (2) 17 (1971), 99-163. MR 0331124 (48:9458)
  • [9] L. Nirenberg, Lectures on linear partial differential equations, CBMS Regional Conf. Ser. in Math., no. 17, Amer. Math. Soc., Providence, R.I., 1973. MR 0450755 (56:9048)
  • [10] J. Rauch, Singularities of solutions to semilinear wave equations, J. Math. Pures Appl. 58 (1979), 299-308. MR 544255 (83c:35078)
  • [11] J. Rauch and M. Reed, Jump discontinuities of semilinear strictly hyperbolic systems in two variables: creation and propagation, Comm. Math. Phys. 81 (1981), 203-227. MR 632757 (82m:35104)
  • [12] -, Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J. 49 (1982), 397-475. MR 659948 (83m:35098)
  • [13] -, Striated solutions of semilinear, two speed wave equations, preprint, 1984.
  • [14] M. Taylor, Pseudo-differential operators, Princeton Univ. Press, Princeton, N.J., 1981.

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