Refined convex Sobolev inequalities (Q2566086)

This paper deals with several formulations of a refined convex Sobolev inequality in the context of entropy methods [see: \textit{A. Arnold} et al., Monatsh. Math. 142, No. 1--2, 35--43 (2004; Zbl 1063.35109)]. Let \(\psi_p(s)=s^p-1-p(s-1)\), \(s\in\mathbb R_0^+\), \(p\in(1,2]\) and \(D(x)\) be a positive scalar diffusion coefficient which belongs to \(W^{2,\infty} _{\text{loc}}(\mathbb R^n)\). The main result is as follows: if \(\rho_{\infty}\in L^1_+(\mathbb R^n),\) with \(\|\rho_{\infty}\|_{L^1}=1\), and \(D(x)>0\) are such that the usual Poincaré constant \(\Lambda_2\) is stricly positive, the Sobolev inequality \[ k \Biggl(\int_{\mathbb R^n} \psi_p\biggl(\frac{\rho}{\rho_{\infty}}\biggr) \rho_{\infty} (x)\,dx\Biggr)\leqslant \frac{1}{2\lambda} \int_{\mathbb R^n }\psi_p'' \biggl(\frac{\rho}{\rho_{\infty}}\biggr)D(x) \biggl| \nabla \biggl(\frac{\rho}{\rho_{\infty}}\biggr)\biggr| ^2 \rho_{\infty}(x)\,dx\tag{1} \] holds for some \(\lambda >0\), for any \(\rho\in L^1_+(\mathbb R^n)\) such that \(\| \rho\| _{L^1}=1\), and \(p\in(1,2]\), where \(k(e)\) is the solution of the differential problem \(k'(e)=1+\kappa\frac{k(e)}{1+e}, k(0)=0,\) with \(\kappa=\frac{2-p}{p}.\) In (1) the optimal constant \(\Lambda_p\) satisfies the inequality \[ 4\biggl(\frac{p-1}{p}\biggr)^2\Lambda_2\leqslant\Lambda_p\leqslant\Lambda_2. \] It is noticed that, for any \(e>0\), \(k(e)>e\), for \(1<p<2.\) The main step in the proof of (1) is an a priori estimate which is sharper than in \textit{A. Arnold} et al. [Commun. Partial Differ. Equations 26, No. 1--2, 43--100 (2001; Zbl 0982.35113)]. Some different formulations of (1) are given. Possible generalizations of the above results to Holley-Strook-type perturbations are investigated.