Ngô Quốc Anh

April 16, 2020

Restriction of gradient, Laplacian, etc on level sets and applications

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:10

This topic is devoted to proofs of several interesting identities involving derivatives on level sets. First, we start with the case of gradient. We shall prove

The first identity

\displaystyle \partial_\nu f = \pm |\nabla f|

on the level set

\displaystyle \big\{ x \in \mathbf R^n : f(x) =0 \big\}.

The above identity shows that while the right hand side involves the value of f in a neighborhood, however, the left hand side indicates that only the normal direction is affected. Heuristically, any change of f along the level set does not contribute to any derivative of f, namely, on the boundary of the level set, the norm of \nabla f is actually the normal derivative \partial_\nu f. Therefore, the only direction taking into derivatives of f is in the normal direction and this should be true for higher-order derivatives of f.

Next we prove the following

The second identity

\displaystyle \partial_\nu \big(x \cdot \nabla f \big)=(\partial_\nu^2 f) (x \cdot \nu)

on the level set

\displaystyle \big\{ x \in \mathbf R^n : \partial_1 f(x) = \cdots = \partial_n f(x) = 0 \big\}.

Combining the above two identities, we can prove

The third identity

\displaystyle \partial_\nu^2 f=\Delta f

on the level set

\displaystyle \big\{ x \in \mathbf R^n : \partial_1 f(x) = \cdots = \partial_n f(x) = 0 \big\}

which basically tells us how to compute the restriction of Laplacian on level sets. This note is devoted to a rigorous proof of the above facts together with a simple application of all these identities.

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