## 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

### Keywords:

averaging regularity

### Citations:

Zbl 0652.47031; Zbl 0658.35083
Full Text:

### References:

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