Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have $||u||_4^4\leq C||u||_2||\nabla u||_2^3$, where the constant $C$ is depending on $Q$.

Now if our domain is $\Omega=\mathbb{R}^N\setminus B_1(0)$, i.e., the exterior domain outside the unit ball, then we ask the question, what's the corresponding best constant? I.e., find the best $C_1$ s.t. $\|u\|_{4,\Omega}^4 \le $ $C_1$ $\|u\|_{2,\Omega}$

$\|\nabla u\|_{2,\Omega}^3$.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.