Sobolev–Kantorovich inequalities under CD(0,∞) condition

We refine and generalize several interpolation inequalities bounding the [Formula: see text] norm of a probability density with respect to the reference measure [Formula: see text] by its Sobolev norm and the Kantorovich distance to [Formula: see text] on a smooth weighted Riemannian manifold satisfying [Formula: see text] condition.

A Minimizing Movement approach to a class of scalar reaction-diffusion equations

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction-diffusion equations of the form \partial_t u \ = \ \Lambda\cdot \mathrm{div}[u(\nabla F'(u) + \nabla V)] \ - \ \Sigma\cdot (F'(u) + V) u, \quad \text{ in } (0, +\infty)\times\Omega, with parameters $\Lambda, \Sigma > 0$ and no-flux boundary condition u(\nabla F'(u) + \nabla V)\cdot {\sf n} \ = \ 0, \quad \text{ on } (0, +\infty)\times\partial\Omega, which is built on their gradient-flow-like structure in the space $\mathcal{M}(\bar{\Omega})$ of finite nonnegative Radon measures on $\bar{\Omega}\subset\xR^d$, endowed with the recently introduced Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$. It is proved that, under natural general assumptions on $F: [0, +\infty)\to\xR$ and $V:\bar{\Omega}\to\xR$, the Minimizing Movement scheme \mu_\tau^0:=u_0\mathscr{L}^d \in\mathcal{M}(\bar{\Omega}), \quad \mu_\tau^n \text{ is a minimizer for } \mathcal{E}(\cdot)+\frac{1}{2\tau}\HK_{\Lambda, \Sigma}(\cdot, \mu_\tau^{n-1})^2, \ n\in\xN, for \mathcal{E}: \mathcal{M}(\bar{\Omega}) \to (-\infty, +\infty], \ \mathcal{E}(\mu):= \begin{cases} \int_\Omega{[F(u(x))+V(x)u(x)]\xdif x} &\text{ if } \mu=u\mathscr{L}^d, \\ +\infty &\text{ else}, \end{cases} yields weak solutions to the above equation as the discrete time step size $\tau\downarrow 0$. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$, which will play an important role in this context, is established in the general setting of a separable Hilbert space.