Suppose or has smooth boundary . If is smooth vector field on , satisfies

Then there exists a soleniodal vector field takes on the boundary. That is

Decompose , where . We can solve the Neumann problem

So if we can find takes on , is the solution.

**(1) **Suppose **.** We will construct , , then

Since is locally a curve in , we assume it has parameter representation , .

If we are forcing , then

since . So we can let and on . Such can be extended to the whole .

then is the vector we need. In particular case, if is simply connected, can be represented by a single .

**(2) **If . Then there exists a partition of unity, namely

each supports in a small domain . Let . We will construct a for each such that on . Then is the vector field we want to find. For one , for simplicity, ignore the subindex as . Change the coordinates such that can be represented as

Then . Verify that by the fact that where .

Refer to **Olga Aleksandrovna Ladyzhenskaya’s** book. The second part was taken from the lecture notes of Yanyan, Li.

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Reblogged this on Being simple and commented:

Good text, take your time.