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 identityon the level set

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

Next we prove the following

The second identityon the level set

Combining the above two identities, we can prove

The third identityon the level set

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.

Assume that is a non-negative classical solution to the following equation

in smooth, bounded domain together with the following boundary conditions

on . Then we have the following Pohozaev type identity

**1.** **Estimate of on **. Denote

Since is constant along , the gradient and the unit outer normal have the same direction. Hence

Since , we deduce that

which gives the desired identity.

**2. Estimate of on **. We also denote

We shall show that the second identity actually follows from the first identity. By direction computation we have

which implies

Hence

since on . However, because

clearly is the zero level set of . Hence

which implies that

Thus, we have shown that

Note that, in the same fashion of the first identity, we further have

Hence, we arrive at

This completes the proof.

**3. Estimate of on **. As routine, we denote

On one hand, because , we easily get

Now because is the zero level set of , we know that

on ; hence yielding

on . Now as in the proof of the second identity (or by the first identity for ) we obtain

This gives us the desired identity.

**4. A Pohozaev type identity.** Multiplying both sides of the equation by and integrating the resulting equation over give

Integration by parts gives

Using the boundary condition, we know that

For the remaining boundary term we use the second identity to get

Of course, in view of the third identity, we do not distinguish the two terms and on . Finally, the last term can be estimated by using to get

However, integration by parts gives

Hence

Thus

Hence, we obtain the following Pohozaev type identity

See also:

- The Pohozaev identity: Toda systems and a priori estimates
- The Pohozaev identity: Elliptic problem with biharmonic operator
- The Pohozaev identity: Integral equation with exponential nonlinearity
- The Pohozaev identity: Semilinear elliptic problem with polygonal nonlinear
- The Pohozaev identity: Semilinear elliptic problem with exponential nonlinearity

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