Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Amp\`ere equation $\det D^2 u(z)=(1/\psi(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=\varphi$, for $\psi$ a smooth positive function and $\varphi$ a bounded lower semicontinuous function. We then prove that a domain of dependence $D$ contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function $\varphi$ is in the Zygmund class. Moreover in this case the surface of constant curvature $K$ contained in $D$ has bounded principal curvatures, for every $K<0$. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of $\partial \mathbb{D}$. Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature $K$, as $K$ varies in $(-\infty,0)$.Comment: 45 pages, 17 figures. Final version, improved presentation and details of some proof

On semigroup maximal operators associated with divergence-form operators with complex coefficients

Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $\Omega\subseteq\mathbb{R}^{d}$. We prove that the maximal operator ${\mathscr M}^{A} f=\sup_{t>0}|\exp(-tL_{A})f|$ is bounded in $L^{p}(\Omega)$, whenever $A$ is $p$-elliptic in the sense of [10]. The relevance of this result is that, in general, the semigroup generated by $-L_{A}$ is neither contractive in $L^{\infty}$ nor positive, therefore neither the Hopf--Dunford--Schwartz maximal ergodic theorem [15, Chap.~VIII] nor Akcoglu's maximal ergodic theorem [1] can be used. We also show that if $d\geq 3$ and the domain of the sesquilinear form associated with $L_{A}$ embeds into $L^{2^{*}}(\Omega)$ with $2^{*}=2d/(d-2)$, then the range of $L^{p}$-boundedness of ${\mathscr M}^{A}$ improves into the interval $(rd/((r-1)d+2),rd/(d-2))$, where $r\geq 2$ is such that $A$ is $r$-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator $\sup_{s,t>0}|T^{A_{1}}_{s}T^{A_{2}}_{t}f|$.Comment: 15 page

Planelike interfaces in long-range ising models and connections with nonlocal minimal surfaces

This paper contains three types of results:the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane,the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces.In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit resultPostprint (author's final draft

Nonlinear differential equations having non-sign-definite weights

In the present PhD thesis we deal with the study of the existence, multiplicity and complex behaviors of solutions for some classes of boundary value problems associated with second order nonlinear ordinary differential equations of the form $u''+f(u)u'+g(t,u)=s,$ or $u''+g(t,u)=0,$ $t\in I$, where $I$ is a bounded interval, $f\colon\mathbb{R}\to\mathbb{R}$ is continuous, $s\in\mathbb{R}$ and $g: I\times \mathbb{R}\to\mathbb{R}$ is a perturbation term characterizing the problems. The results carried out in this dissertation are mainly based on dynamical and topological approaches. The issues we address have arisen in the field of partial differential equations. For this reason, we do not treat only the case of ordinary differential equations, but also we take advantage of some results achieved in the one dimensional setting to give applications to nonlinear boundary value problems associated with partial differential equations. In the first part of the thesis, we are interested on a problem suggested by Antonio Ambrosetti in ``Observations on global inversion theorems'' (2011). In more detail, we deal with a periodic boundary value problem associated with the first differential equation where the perturbation term is given by $g(t,u):=a(t)\phi(u)-p(t)$. We assume that $a,$ $p\in L^{\infty}(I)$ and $\phi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying $\lim_{|\xi|\to\infty}\phi(\xi)=+\infty$. In this context, if the weight term $a(t)$ is such that $a(t)\geq 0$ for a.e. $t\in I$ and $\int_{I}a(t)\,dt>0$, we generalize the result of multiplicity of solutions given by Fabry, Mawhin and Nakashama in ``A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations'' (1986). We extend this kind of improvement also to more general nonlinear terms under local coercivity conditions. In this framework, we also treat in the same spirit Neumann problems associated with second order ordinary differential equations and periodic problems associated with first order ones. Furthermore, we face the classical case of a periodic Ambrosetti-Prodi problem with a weight term $a(t)$ which is constant and positive. Here, considering in the second differential equation a nonlinearity $g(t,u):=\phi(u)-h(t)$, we provide several conditions on the nonlinearity and the perturbative term that ensure the presence of complex behaviors for the solutions of the associated $T$-periodic problem. We also compare these outcomes with the result of stability carried out by Ortega in ``Stability of a periodic problem of Ambrosetti-Prodi type'' (1990). The case with damping term is discussed as well. In the second part of this work, we solve a conjecture by Yuan Lou and Thomas Nagylaki stated in ``A semilinear parabolic system for migration and selection in population genetics'' (2002). The problem refers to the number of positive solutions for Neumann boundary value problems associated with the second differential equation when the perturbation term is given by $g(t,u):=\lambda w(t)\psi(u)$ with $\lambda>0$, $w\in L^{\infty}(I)$ a sign-changing weight term such that $\int_{I}w(t)\,dt<0$ and $\psi\colon[0,1]\to[0,\infty[$ a non-concave continuous function satisfying $\psi(0)=0=\psi(1)$ and such that the map $\xi\mapsto \psi(\xi)/\xi$ is monotone decreasing. In addition to this outcome, other new results of multiplicity of positive solutions are presented as well, for both Neumann or Dirichlet boundary value problems, by means of a particular choice of indefinite weight terms $w(t)$ and different positive nonlinear terms $\psi(u)$ defined on the interval $[0,1]$ or on the positive real semi-axis $[0,+\infty[$

Sublinear Rigidity of Lattices in Semisimple Lie Groups

Let $G$ be a real centre-free semisimple Lie group without compact factors. I prove that irreducible lattices in $G$ are rigid under two types of sublinear distortions. I show that if $\Lambda\leq G$ is a discrete subgroup that sublinearly covers a lattice, then $\Lambda$ is itself a lattice. I use this result to prove that the class of lattices in groups that do not admit $\mathbb{R}$-rank 1 factors is SBE complete: if $\Lambda$ is an abstract finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice in $G$, then $\Lambda$ can be homomorphically mapped into $G$ with finite kernel and image a lattice in $G$. This generalizes the well known quasi-isometric completeness of lattices in semisimple Lie groups

Multiplicity results for a differential inclusion problem with non-standard growth

AbstractIn this paper we examine the multiplicity of solutions of a differential inclusion problem involving p(x)-Laplacian of the type[Pλ]{−Δp(x)u+V(x)|u|p(x)−2u∈∂j1(x,u(x))+λ∂j2(x,u(x)),in Ω,∂u∂n=0,on ∂Ω. By using the nonsmooth version of Ricceri variational principle we get three critical points of the corresponding energy Motreanu–Panagiotopoulos type functional, which are the solutions of (Pλ)

Real Analysis, Harmonic Analysis and Applications

[no abstract available