We consider variational problems with regular H{ö}lderian weight or boundary singularity, and Dirichlet condition. We prove the boundedness of the volume of the solutions to these equations on analytic domains.
We prove that functions that are homogeneous of degree $α\in (0,\,1)$ on $\mathbb{R}^n$ and have nowhere vanishing Hessian determinant cannot change sign.
We obtain local Lipschitz regularity for minima of autonomous integrals in the calculus of variations, assuming $q$-growth hypothesis and $W^{1,p}$-quasiconvexity only asymptotically, both in the sub-quadratic and the super-quadratic case.
As an orchestra or a rock star, archaeologists have their audience too. This paper wants to highlight an integrated approach between fieldwork, its account and its dissemination to the public in different ways, including social media. This potential integration has come to life in the 2011 excavation of the Roman mansio of Vignale (Italy) and it has been named “Excava(c)tion”. It doesn’t mean a new way of digging but another way of approaching the excavation, an approach integrated toward and with the public, both on site and on the social Web. “Excava(c)tion” conceives the site as a stage and digging as a performance, through a continuous dialogue between archaeologists and the public. Archaeologists share their work in the form of guided tours (live, theatrical-like performances), communicative diaries and videos (edited, motion-picture performances) and on a blog (www.uominiecoseavignale.it). They receive back comments and oral accounts from the local community about the main themes of common interest. “Excava(c)tion” means engagement both of archaeologists and the public in the pursuit of a global multivocality during archaeological excavation.
We prove the existence of non-trivial solutions for a fractional Schr$\ddot{o}$dinger-Poisson equation in $\mathbb{R}^{3}$. The proof is based on the perturbation method and the mountain pass theorem.
In this article, we explore convolutions of distributions with distributions given by (weighted) line integration. We also explore the scattering of singularities of such convolutions.
This paper is devoted to the characterization of the lack of compactness of the Sobolev embedding of $H^N(R^{2N})$ into the Orlicz space using Fourier analysis.
We give simple proofs of hypoelliptic estimates for some models of kinetic equations with a fractional order diffusion part. The proofs are based on energy estimates together with F. Bouchut and B. Perthame previous ideas.
We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.
We consider nonlinear elliptic equations which contains global coupling as a nonlinear term. We classify the existence of all possible positive solutions to this problem.
We compute the shape derivative of the first eigenvalue of the 1-Laplacian. As an application, we find that a ball is critical among all volume-preserving deformations.