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Sharp constant in Nash’s inequality. (English) Zbl 0822.35018
The Nash inequality states that $\biggl( \int_{\mathbb{R}^ n} | f(x) |^ 2 d^ n x\biggr)^{1+ 2/n}\leq C_ n \int_{\mathbb{R}^ n} |\nabla f(x) |^ 2 d^ n x \biggl(\int_{\mathbb{R}^ n} | f(x)| d^ n x\biggr)^{4/n} \tag{1}$ for a constant $$C_ n$$ depending only on $$n$$. This inequality is a particular case of the Gagliardo-Nirenberg inequalities for which numerous applications have been found. In this note we compute the sharp constant in (1) and determine all of the cases of equality. A particularly striking feature of the result is that all of the extremals have compact support.
Theorem. Let $$f$$ be an integrable function on $$\mathbb{R}^ n$$ such that the distributional gradient of $$f$$, $$\nabla f$$, is a square integrable function. Then (1) holds with $C_ n= {{2((n+ 2)/2 )^{(n+ 2)/n}} \over {n\lambda_ 1^ N (B^ n) | B^ n |^{2/n}}},$ where $$| B^ n |$$ denotes the volume of the unit ball $$B^ n$$ in $$\mathbb{R}^ n$$, and where $$\lambda^ N_ 1 (B^ n)$$ denotes the first nonzero Neumann eigenvalue of the Laplacian $$(-\nabla \cdot \nabla)$$ on radial functions on $$B^ n$$. There is equality if and only if, after a possible translation, scaling, and normalization, $f_ n (x)= \begin{cases} u_ n(| x|)- u_ n(1), \quad &\text{for } | x|\leq 1\\ 0, &\text{for }| x|\geq 1 \end{cases}$ where $$u(| x|)$$ is the normalized eigenfunction of the Neumann Laplacian in $$B^ n$$ with the eigenvalue $$\lambda^ N_ 1 (B^ n)$$.

##### MSC:
 35B45 A priori estimates in context of PDEs 26D10 Inequalities involving derivatives and differential and integral operators
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