Why the definition of harmonic function needs twice “ continuously ” differentiable?












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Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$

However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?










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  • $begingroup$
    See en.wikipedia.org/wiki/Harmonic_function
    $endgroup$
    – Kavi Rama Murthy
    Feb 2 at 23:13
















4












$begingroup$


Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$

However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?










share|cite|improve this question









$endgroup$












  • $begingroup$
    See en.wikipedia.org/wiki/Harmonic_function
    $endgroup$
    – Kavi Rama Murthy
    Feb 2 at 23:13














4












4








4


1



$begingroup$


Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$

However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?










share|cite|improve this question









$endgroup$




Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$

However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?







real-analysis harmonic-analysis






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share|cite|improve this question











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asked Feb 2 at 23:10









J.GuoJ.Guo

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  • $begingroup$
    See en.wikipedia.org/wiki/Harmonic_function
    $endgroup$
    – Kavi Rama Murthy
    Feb 2 at 23:13


















  • $begingroup$
    See en.wikipedia.org/wiki/Harmonic_function
    $endgroup$
    – Kavi Rama Murthy
    Feb 2 at 23:13
















$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13




$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13










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