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Exponential majorization applied to a non-linear cauchy (goursat) problem for functions of gevrey nature

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Summary

Let f(y, x, z) be a continuous function defined in some neighbourhood of the origin in R3. The function f is supposed to be of Gevrey class d≥1 in the x- and z-variables. If γ<β, and γ+αd≤β then it is proved that

$$D_y^\beta = f(y,x,D_y^\gamma D_x^\alpha u),D_y^i u(0,x) = 0, 0 \leqslant j< \beta,$$

has a unique continuous solution of Gevrey class d in the x-variable.

References

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    A. Friedman,A new proof and generalizations of the Cauchy-Kowalevski theorem. Trans. Am. Math. Soc. 98 (1961) pp. 1–20.

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    M. Gevrey,Sur la nature analytiqe des solutions des équations aux dérivées partielles Ann. Ec. Norm. Sup. (3) 35 (1918) pp. 127–190.

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    N. A. Lednev,A new method for solving partial differential equations Mat. Sbornik 22 (64) (1948), pp. 205–264, (Russian).

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    J. Leray-Y. Ohya,Equations et systèmes non-lineaires, hyperboliques non-stricts Math. Ann. 170 (3) (1967), pp. 167–205.

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    J. Persson,New proofs and generalizations of two theorems by Lednev for Goursat problems To appear in Math. Ann.

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    -- --,Exponential majorization and global Goursat problems To appear in Math. Ann.

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    G. Talenti,Sul problema di Cauchy per le equazioni a derivate parziali Ann. Mat. Pura ed Appl. LXVII (1965) pp. 365–394.

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Persson, J. Exponential majorization applied to a non-linear cauchy (goursat) problem for functions of gevrey nature. Annali di Matematica 78, 259–267 (1968) doi:10.1007/BF02415117

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Keywords

  • Continuous Function
  • Continuous Solution
  • Gevrey Class
  • Unique Continuous Solution