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Upper and lower bounds for the optimal constant in the extended Sobolev inequality. Derivation and numerical results. (English) Zbl 1428.26047

Summary: We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant \(k_0 = k_0(n, \alpha)\) in the inequality \[\|u\|_{2n/(n-2\alpha)} \leqslant k_0 \|\nabla u\|^\alpha_2 \|u\|^{1-\alpha}_2,\quad u \in H^1(\mathbb R^n),\] for \(n = 1\), \(0 < \alpha \leqslant 1/2\), and \(n \geqslant 2\), \(0 < \alpha < 1\).
This constant \(k_0\) is the reciprocal of the infimum \(\lambda_{n, \alpha}\) for \(u \in H^1(\mathbb R^n)\) of the functional \[\Lambda_{n,\alpha} = \frac{\|\nabla u\|^\alpha_2 \|u\|^{1-\alpha}_2}{\|u\|_{2n/(n-2\alpha)}},\quad u \in H^1(\mathbb R^n),\] where for \(n = 1\), \(0 < \alpha \leqslant 1/2\), and for \(n \geqslant 2\), \(0 < \alpha < 1\).
The lowest point in the point spectrum of the Schrödinger operator \(\tau = -\Delta + q\) on \(\mathbb R^n\) with the real-valued potential \(q\) can be expressed in \(\lambda_{n, \alpha}\) for all \(q_-= \max(0,-q) \in L^p(\mathbb R^n)\), for \(n = 1\), \(1 \leqslant p < \infty\), and \(n\geqslant 2\), \(n/2 < p < \infty\), and the norm \(\|q_-\|_p\).

MSC:

26D15 Inequalities for sums, series and integrals
41A44 Best constants in approximation theory

Software:

zeroin; DLMF
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