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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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