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Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations

Zhuangchu LuoHua ChenChanggui Zhang — 2012

Annales de l’institut Fourier

In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at ( t , x ) = ( 0 , 0 ) C 2 . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the k -summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

On nonisometric isospectral connected fractal domains.

Brian D. SleemanChen Hua — 2000

Revista Matemática Iberoamericana

A fundamental question raised by M. Kac in 1966 is: Must two isospectral planar domains necessarily be isometric? Following a long history of investigation C. Gordon, D. L. Webb and S. Wolpert in 1992 finally proved that the answer is no. By using the idea of transposition maps one can construct a wide class of planar domains with piecewise continuous boundaries which are isospectral but non-isometric. In this note we study the Kac question in relation to domains with fractal boundaries and by following...

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