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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 $d=2$.
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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 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.
We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,mu_X)$ satisfying a $2$-Poincare inequality. Given a bounded domain $Omegasubset X$ with $mu_X(XsetminusOmega)>0$, and a function $f$ in the Besov class $B^theta_{2,2}(X)cap L^2(X)$, we study the problem of finding a function $uin B^theta_{2,2}(X)$ such that $u=f$ in $XsetminusOmega$ and $mathcal{E}_theta(u,u)le mathcal{E}_theta(h,h)$ whenever $hin B^theta_{2,2}(X)$ with $h=f$ in $XsetminusOmega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Holder continuous on $Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source vertex to a target vertex. We consider a generalization of this classical problem in which the position of each vertex in the graph is a continuous decision variable, constrained to lie in a corresponding convex set. The length of an edge is then defined as a convex function of the positions of the vertices it connects. Problems of this form arise naturally in motion planning of autonomous vehicles, robot navigation, and even optimal control of hybrid dynamical systems. The price for such a wide applicability is the complexity of this problem, which is easily seen to be NP-hard. Our main contribution is a strong mixed-integer convex formulation based on perspective functions. This formulation has a very tight convex relaxation and makes it possible to efficiently find globally-optimal paths in large graphs and in high-dimensional spaces.
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