Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userArea Minimizing Sets Subject to a Volume Constraint in a Convex Set
107
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we consider the problem of minimizing area subject to a volume constraint in a given convex set.
rate research
Read More
We consider minimizing harmonic maps $u$ from $Omega subset mathbb{R}^n$ into a closed Riemannian manifold $mathcal{N}$ and prove: (1) an extension to $n geq 4$ of Almgren and Liebs linear law. That is, if the fundamental group of the target manifold $mathcal{N}$ is finite, we have [ mathcal{H}^{n-3}(textrm{sing } u) le C int_{partial Omega} | abla_T u|^{n-1} ,d mathcal{H}^{n-1}; ] (2) an extension of Hardt and Lins stability theorem. Namely, assuming that the target manifold is $mathcal{N}=mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s in (frac{1}{2},1]$ and $p in [2,infty)$ satisfying $sp geq 2$. We also discuss sharpness.
We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n geq 4$. For minimizing harmonic maps $uin W^{1,2}(Omega,mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Liebs linear law, namely [mathcal{H}^{n-3}(textrm{sing} u) le C int_{partial Omega} | abla_T u|^{n-1} ,dmathcal{H}^{n-1};] (2) An extension of Hardt and Lins stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $Gamma$ that, similarly to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset $C$ whose translations by elements in $Gamma$ dont intersect $Gamma$. However, a priori the cone $C$ may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second type.
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.
Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a $5.83$ approximation and runs in $O(n log n)$ time, i.e., at least a factor $n$ faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a $9$-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
