arXiv subjects
Publications and source records attributed to Stefano Nardulli.
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological and homological invariants of the underlying domain $Ω$.
In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,α}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allard's regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov.
We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with strong bounded geometry, as in Definition $2.3$, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_0\in\mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of $\mathbb{M}^n_{k_0}$ the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of [Dru02b] in which the same result was proved in the case where $(M^n, g)$ is assumed to be just compact. As a consequence of our result we give an asymptotic expansion in Puiseux's series up to the second nontrivial term of the isoperimetric profile function for small volumes. Finally, as a corollary of our isoperimetric comparison result, it is shown, in the special case of manifolds with strong bounded geometry, and $S_g<n(n-1)k_0$ that for small volumes the Aubin-Cartan-Hadamard's Conjecture in any dimension $n$ is true.
For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-Hölder continuous function and so in particular it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume. Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown.
For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. Then under an extra hypothesis on the geometry of $M$, we apply this result to prove some differentiability property of $I_M$ and a differential inequality satisfied by $I_M$, extending in this way well known results for compact manifolds, to this class of noncompact complete Riemannian manifolds with bounded geometry.
We study the problem of existence of isoperimetric regions for large volumes, in $C^0$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named $C^0$-strongly asymptotic Schwarzschild, extending results of [BE13]. Such results are of interest in the field of mathematical general relativity.
The first known example of a complete Riemannian manifold whose isoperimetric profile is discontinuous is given.
For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extends previous results contained in [Nar14a], in such a way that the generalized existence theorem, Theorem 1 of [Nar14a] is actually a generalized compactness theorem. The suitable modifications to the arguments and statements of the results in [Nar14a] are non-trivial. As a consequence we give a multipointed version of Theorem 1.1 of [LW11], and a simple proof of the continuity of the isoperimetric profile function.
Erdős introduced the noncommuting graph, in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph is becoming relevant in the last years for various reasons. Here we deal with a numerical aspect, showing for the first time an isoperimetric inequality and an analytic condition in terms of Sobolev inequalities. This last result holds in the more general context of weighted locally finite graphs.
For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to such noncompact manifolds.
In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.
We prove existence of isoperimetric regions for every volume in non-compact Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq (n-1) k_0 g$ and being locally asymptotic to the simply connected space form of constant sectional curvature $k_0$; moreover in case $k_0=0$ we show that the isoperimetric regions are indecomposable. We also discuss some physically and geometrically relevant examples. Finally, under assumptions on the scalar curvature we prove existence of isoperimetric regions of small volume.
We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumption, this can happen only on topological spheres.
It is shown that, in dimensions $<8$, isoperimetric profiles of compact real analytic Riemannian manifolds are semi-analytic.
We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifold, there is a unique family concentric pseudo balls which contains all the pseudo balls $C^{2,α}$-close to small spheres. This allows us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.