The fractional Laplacian in power-weighted LpL^p spaces: integration-by-parts formulas and self-adjointness

We consider the fractional Laplacian operator $(-\Delta)^s$ (let $s \in (0,1)$) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $L^2(\mathbb{R}^d)$ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space $\dot{H}^s(\mathbb{R}^d)$. More precisely, we focus on functions belonging to some weighted $L^2$ space whose fractional Laplacian belongs to another weighted $L^2$ space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight $\rho(x)$ at infinity. The latter is assumed to be like a power both near the origin and at infinity (the two powers being possibly different). Our results have direct consequences for the self-adjointness of the linear operator formally given by $\rho^{-1}(-\Delta)^s$. The generality of the techniques developed allows us to deal with weighted $L^p$ spaces as well

Porous medium equations on manifolds with critical negative curvature: unbounded initial data

We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan-Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity

Radial Fast Diffusion on the Hyperbolic Space

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.Comment: To appear in Proc. London Math. So

Smoothing effects for the filtration equation with different powers

We study the nonlinear diffusion equation $u_t=\Delta\phi(u)$ on general Euclidean domains, with homogeneous Neumann boundary conditions. We assume that $\phi^\prime(u)$ is bounded from below by $|u|^{m_1-1}$ for small $|u|$ and by $|u|^{m_2-1}$ for large $|u|$, the two exponents $m_1,m_2$ being possibly different and larger than one. The equality case corresponds to the well-known porous medium equation. We establish sharp short- and long-time $L^{q_0}$-$L^\infty$ smoothing estimates: similar issues have widely been investigated in the literature in the last few years, but the Neumann problem with different powers had not been addressed yet. This work extends some previous results in many directions

Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods

This paper is the second part of the study. In Part~I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part~II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights will be emphasized

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

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-Poincar{\'e} inequality which is interpreted as 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

Conditions at infinity for the inhomogeneous filtration equation

We investigate existence and uniqueness of solutions to the filtration equation with an inhomogeneous density in ${\mathbb R}^N$, approaching at infinity a given continuous datum of Dirichlet type.Comment: To appear in Annales de l'Institut Henri Poincar\'e (C) Analyse Non Lin\'eair