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Kondrachov embedding theorem

On a compact manifold with boundary, the Kondrachov embedding theorem states that if andthen the Sobolev embedding

:

is completely continuous (compact). Note that the condition is just as in the first part of the Sobolev embedding theorem, with the equality replaced by an inequality, thus requiring a more regular space .

Gagliardo-Nirenberg-Sobolev inequality

Assume that is a continuously differentiable

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