A Picard-Lindel\"of theorem for smooth PDE

Paolo Giordano; Lorenzo Luperi Baglini

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arXiv · 2211.01118

A Picard-Lindel\"of theorem for smooth PDE

Paolo Giordano·Lorenzo Luperi Baglini

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Abstract

We prove that Picard-Lindel\"of iterations for an arbitrary smooth normal Cauchy problem for PDE converge if we assume a suitable Weissinger-like sufficient condition. This condition includes both a large class of non-analytic PDE or initial conditions, and more classical real analytic functions. The proof is based on a Banach fixed point theorem for contractions with loss of derivatives. From the latter, we also prove an inverse function theorem for locally Lipschitz maps with loss of derivatives in arbitrary graded Fr\'echet spaces.

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Paolo Giordano, Lorenzo Luperi Baglini. 2022-11-02. A Picard-Lindel\"of theorem for smooth PDE. https://arxiv.org/abs/2211.01118

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