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Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

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  • In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy -entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN).

    We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy -entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincaré inequality which is interpretedas a linearized entropy -entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods.Consequences for the (WFD) flow will be studied in Part Ⅱ of this work.

    Mathematics Subject Classification: Primary: 35K55, 46E35, 49K30; Secondary: 26D10, 35B06, 49K20, 35J20.


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  • Figure 1.  We consider the admissible range for the $(\beta,\gamma)$ parameters. The grey area is the area of validity of (2) and it is given by $\gamma-2 < \beta < \tfrac{d-2}d\,\gamma$ if $\gamma < d$ and $\tfrac{d-2}d\,\gamma < \beta < \gamma-2$ if $\gamma>d$: the cones corresponding to $\gamma < d$ and $\gamma>d$ are in one-to-one correspondance by an inversion symmetry: see details in Section 2.1 Notice that the case $\gamma>d$ has been excluded in (3) in order to simplify the statements. The hyperbola defined by the Felli & Schneider curve determines a region (dark grey area) of symmetry breaking which is valid for any $p\in(1,p_\star)$, and independent of $p$. However, since $p_\star$ depends on $\beta$ and $\gamma$, this induces an additional restriction on the admissible range of $(\beta,\gamma)$, which depends on $p$: see Fig.2 and Fig.3. Here we consider the special case $d=5$.

    Figure 2.  With $d=3$, the left figure is essentially an enlargement of Fig.1 and represents the symmetry breaking region, while on the right figure, we choose $p=2$, so that the admissible range of parameters $(\beta,\gamma)$ is restricted by the condition $p\le p_\star(\beta,\gamma)$, i.e., $\beta\ge d-2-(d-\gamma)/p$. This lower bound corresponds to the line determined by the points $(\beta,\gamma)=(d-2,d)$ and $(\beta,\gamma)$ given by the condition $\Lambda_\star=\Lambda_{0,1}=\Lambda_{1,0}$. The curve $\beta=\sigma(\gamma,p)$ in Theorem 3 is represented by a dotted curve. To $\beta\ge\sigma(\gamma,p)$ corresponds the case $\Lambda=\Lambda_{0,1}\le\Lambda_{1,0}$, while $\beta\le\sigma(\gamma,p)$ corresponds to the case $\Lambda_{0,1}\ge\Lambda_{1,0}=\Lambda$, when $\gamma\in(-\infty,d)$.

    Figure 3.  Enlargement of Fig.2 in a neighborhood of $(\beta,\gamma)=(0,0)$. On the right, the equality case $\Lambda_{0,1}=\Lambda_{1,0}$ determines the dotted curve $\beta=\sigma(\gamma,p)$. Notice that the symmetry breaking region is contained in the region in which the spectral gap is $\Lambda=\Lambda_{0,1}$.

    Figure 4.  In the dark grey region, symmetry breaking occurs. The plot is done for $p=2$ and $d=3$. See Appendix B for a more detailed description of the properties of the lowest eigenvalues.

    Figure 5.  The spectral gap and the lowest eigenvalues of $\mathcal L$ for $n=3$. The parabola represents $\Lambda_{\rm ess}$ as a function of $\delta$, $\Lambda_{1,0}$ is tangent to the parabola and $\Lambda_{0,1}$ is shown for $\eta=3.5$ (left), $\eta=1.4$ (center) and $\eta=0.35$ (right). The eigenvalues $\Lambda_{1,0}$ and $\Lambda_{0,1}$ are represented by a plain line only if the corresponding eigenvalues are in the space $\mathrm L^2(\mathbb{R}^d,\mathcal B^{2-m}\,|x|^{n-d}\,dx)$.

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