An equivalent path functional formulation of branched transportation problems
Lorenzo Brasco Filippo Santambrogio
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 845-871 doi: 10.3934/dcds.2011.29.845
We consider two models for branched transport: the one introduced in Bernot et al. (Publ Mat 49:417-451, 2005), which makes use of a functional defined on measures over the space of Lipschitz paths, and the path functional model presented in Brancolini et al. (J Eur Math Soc 8:415-434, 2006), where one minimizes some suitable action functional defined over the space of measure-valued Lipschitz curves, getting sort of a Riemannian metric on the space of probabilities, favouring atomic measures, with a cost depending on the masses of each of their atoms. We prove that modifying the latter model according to Brasco (Ann Mat Pura Appl 189:95-125, 2010), then the two models turn out to be equivalent.
keywords: Branched transport; irrigation patterns; path functionals.
On fractional Hardy inequalities in convex sets
Lorenzo Brasco Eleonora Cinti
Discrete & Continuous Dynamical Systems - A 2018, 38(8): 4019-4040 doi: 10.3934/dcds.2018175

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiĭ spaces of order $(s, p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<∞$ and zhongwenzy $0<s<1$, with a constant which is stable as $s$ goes to 1.

keywords: Hardy inequality nonlocal operators fractional Sobolev spaces
Global compactness results for nonlocal problems
Lorenzo Brasco Marco Squassina Yang Yang
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 391-424 doi: 10.3934/dcdss.2018022

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.

keywords: Fractional $p$-Laplacian variational methods global compactness

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