Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or L1L^1 data

We investigate solutions to nonlinear elliptic Dirichlet problems of the type $$\left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right.$$ where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,\xi)$ is a Carath\'eodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,\xi)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For $L^1$-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces

On the monotonicity of non-local perimeter of convex bodies

Under mild assumptions on the kernel $K\ge0$, the non-local $K$-perimeter $P_K$ satisfies the monotonicity property on nested convex bodies, i.e., if $A\subset B\subset\mathbb{R}^n$ are two convex bodies, then $P_K(A)\le P_K(B)$. In this note, we prove quantitative lower bounds on the difference of the $K$-perimeters of $A$ and $B$ in terms of their Hausdorff distance, provided that $K$ satisfies suitable symmetry properties.Comment: 15 pages, 2 figure

On the convex components of a set in Rn\mathbb{R}^n

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, "Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019) and in M. La Civita and F. Leonetti, "Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008-2009) 71-78.Comment: 15 pages, 5 figure

Bi-Sobolev mappings with differential matrices in Orlicz Zygmund classes

AbstractBi-Sobolev mappings f:Ω⊂R2→ontoΩ′⊂R2 have been defined as those homeomorphisms such that f and f−1 belong to Wloc1,1. We deduce regularity properties of the distortion of f from the regularity of the differential matrix Df−1 and conversely

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements