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

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 $\mathrm L^2(\mathbb{R}^d,\mathcal B^{2-m}\,|x|^{n-d}\,dx)$.