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We develop the notion of higher Cheeger constants for a measurable set $Omega subset mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value [h_k(Omega) = inf max {h_1(E_1), dots, h_1(E_k)},] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $Omega$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional adjustment conditions and study their properties. A relation between $h_k(Omega)$ and spectral minimal $k$-partitions of $Omega$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of some planar domains.
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In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $Omega$. Here with $N$-Cluster we mean a family of $N$ sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any $N$-cluster attaining such a minimum a Cheeger $N$-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger $N$-clusters in a general ambient space dimension and we give a precise description of their structure in the the planar case. The last part is devoted to the relation between the functional introduced here (namely the $N$-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lins conjecture.
We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $Omega$. We show that this size is strongly related to the regularity of $partial Omega$ by providing bounds on the Hausdorff dimension of $partial Ecap partialOmega$. In particular we show that, if $partial Omega$ has $C^{1,alpha}$ regularity then $mathcal{H}^{d-2+alpha}(partial Ecap partialOmega)>0$. This shows that a sufficient condition to ensure that $mathcal{H}^{d-1}(partial Ecap partial Omega)>0$ is that $partial Omega$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $partial E$ as well, we obtain that $Omega$ convex, which yields $partial Ein C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.
The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space $H^{s}(mathbb{T})$ for any $s geq 0$. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.
Consider a locally Lipschitz function $u$ on the closure of a possibly unbounded open subset $Omega$ of $mathbb{R}^n$ with $C^{1,1}$ boundary. Suppose $u$ is semiconcave on $overline Omega$ with a fractional semiconcavity modulus. Is it possible to extend $u$ in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points.
We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $Omega subset mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $partial C cap Omega$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if $Omega$ is convex, then the free boundary of $C$ consists only of pieces of spheres and nodoids. This result remains valid for nonconvex domains when the generating curve of $C$ is closed, convex, and of class $mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of the fact that, for general nonconvex domains, pieces of unduloids or cylinders can also appear in the free boundary of $C$.
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