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We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $Omega$ that varies over all subdomains of a given bounded domain $D$ of ${bf R}^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.
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The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $mu$ describing the distribution of root hair cells, we seek to maximize a harvest functional $mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense.
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on the free boundary. We show the existence of an optimal domain under rather general assumptions and we study the cases when the optimal domains are open sets and have a finite perimeter.
The $Gamma $-limit of a family of functionals $umapsto int_{Omega }fleft( frac{x}{varepsilon },frac{x}{varepsilon ^{2}},D^{s}uright) dx$ is obtained for $s=1,2$ and when the integrand $f=fleft( y,z,vright) $ is a continous function, periodic in $y$ and $z$ and convex with respect to $v$ with nonstandard growth. The reiterated two-scale limits of second order derivative are characterized in this setting.
Optimal transportation theory is an area of mathematics with real-world applications in fields ranging from economics to optimal control to machine learning. We propose a new algorithm for solving discrete transport (network flow) problems, based on classical auction methods. Auction methods were originally developed as an alternative to the Hungarian method for the assignment problem, so the classic auction-based algorithms solve integer-valued optimal transport by converting such problems into assignment problems. The general transport auction method we propose works directly on real-valued transport problems. Our results prove termination, bound the transport error, and relate our algorithm to the classic algorithms of Bertsekas and Castanon.
We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized - the so-called variational discretization approach - and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests.
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