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Averaging regularity results for PDEs under transversality assumptions. (English) Zbl 0832.35020

Summary: Let \(u= u(x, y)\) and \(f= f(x, y)\) be two functions related by a PDE \(P(x, y, D_x, D_y) u= f\); the regularity of the \(y\)-average \(\int u(x,\cdot) dy\) as a function of \(x\) is investigated knowing that of \(u\) and \(f\). Our method consists in reducing \(P\) to a microlocal normal form under a natural transversality assumption. The 2-microlocal regularity of \(u\) is also determined knowing that of \(f\). These results are then applied to a homogenization problem. This article generalizes the results of [the second author, P.-L. Lions, B. Perthame and R. Sentis, J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)] and [the second author, C. R. Acad. Sci., Paris, Ser. I 305, 801-804 (1987; Zbl 0658.35083)] on velocity averaging and homogenization for kinetic equations.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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