arXiv subjects
Publications and source records attributed to Bruno Volzone.
We prove a Brezis-Bourgain-Mironescu type formula for a class of nonlocal magnetic spaces, which builds a bridge between a fractional magnetic operator recently introduced and the classical theory.
We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $Ω$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $Ω$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.
Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data.
We obtain a priori estimates with best constants for the solutions of the fractional fast diffusion equation $u_t+(-Δ)^{σ/2}u^m=0$, posed in the whole space with $0<σ<2$, $0<m\le 1$. The estimates are expressed in terms of convenient norms of the initial data, the preferred norms being the $L^1$-norm and the Marcinkiewicz norm. The estimates contain exact exponents and best constants. We also obtain optimal estimates for the extinction time of the solutions in the range $m$ near 0 where solutions may vanish completely in finite time. Actually, our results apply to equations with a more general nonlinearity. Our main tools are symmetrization techniques and comparison of concentrations. Classical results for $σ=2$ are recovered in the limit.
We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-Δ)^{σ/2}u=f$ and itselliptic counterpart $h v +(-Δ)^{σ/2}v=f$, $h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, $\partial_t u+(-Δ)^{σ/2}A(u)=f$, but only when $A:\re_+\to\re_+$ is a concave function. In the elliptic case, complete symmetrization results are proved for $\,B(v)+(-Δ)^{σ/2}v=f$ \ when $B(v)$ is a convex nonnegative function for $v>0$ with $B(0)=0$, and partial results when $B$ is concave. Remarkable counterexamples are constructed for the parabolic equation when $A$ is convex, resp. for the elliptic equation when $B$ is concave. Such counterexamples do not exist in the standard diffusion case $σ=2$.
In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian.
Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy-Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.
In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of $u$ or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.