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In this chapter we present the classical stability proof of John Theorem, which was partly introduced regarding uniqueness. The theorem provides a logarithmic stability estimate for the Cauchy problem with a non-characteristic initial surface in linear PDEs with analytic coefficients. While the continuous dependence here is modest due to the strong regularity assumptions, John showed that for certain equations, such as the wave equation in two or more spatial dimensions, the logarithmic stability cannot be improved.

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The John Stability Theorem for the Cauchy Problem for PDEs with Analytic Coefficients

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  • 发表日期：

    2025年

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  • Sergio Vessella

摘要

In this chapter we present the classical stability proof of John Theorem, which was partly introduced regarding uniqueness. The theorem provides a logarithmic stability estimate for the Cauchy problem with a non-characteristic initial surface in linear PDEs with analytic coefficients. While the continuous dependence here is modest due to the strong regularity assumptions, John showed that for certain equations, such as the wave equation in two or more spatial dimensions, the logarithmic stability cannot be improved.