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Uniqueness of the Cheeger set of a convex body
 Pacific J. Math
"... Abstract We prove that if C ⊂ IR N is a an open bounded convex set, then there is only one Cheeger set inside C and it is convex. The Cheeger set of C is the set which minimizes for sets inside C the ratio perimeter over volume. ..."
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Abstract We prove that if C ⊂ IR N is a an open bounded convex set, then there is only one Cheeger set inside C and it is convex. The Cheeger set of C is the set which minimizes for sets inside C the ratio perimeter over volume.
Maximum flows and minimum cuts in the plane
 Journal of Global Optimization
"... A continuous maximum flow problem finds the largest t such that div v = t F (x, y) is possible with a capacity constraint �(v1, v2) � ≤ c(x, y). The dual problem finds a minimum cut ∂S which is filled to capacity by the flow through it. This model problem has found increasing application in medical ..."
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A continuous maximum flow problem finds the largest t such that div v = t F (x, y) is possible with a capacity constraint �(v1, v2) � ≤ c(x, y). The dual problem finds a minimum cut ∂S which is filled to capacity by the flow through it. This model problem has found increasing application in medical imaging, and the theory continues to develop (along with new algorithms). Remaining difficulties include explicit streamlines for the maximum flow, and constraints that are analogous to a directed graph.
Maximum Area with Minkowski Measures of Perimeter
"... The oldest competition for an optimal shape (areamaximizing) was won by the circle. But if the fixed perimeter is measured by the line integral of dx  + dy, a square would win. Or if the boundary integral of max(dx, dy) is given, a diamond has maximum area. For any norm in R 2, we show that ..."
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The oldest competition for an optimal shape (areamaximizing) was won by the circle. But if the fixed perimeter is measured by the line integral of dx  + dy, a square would win. Or if the boundary integral of max(dx, dy) is given, a diamond has maximum area. For any norm in R 2, we show that when the integral of �(dx, dy) � around the boundary is prescribed, the area inside is maximized by a ball in the dual norm (rotated by π/2). This “isoperimetrix ” was found by Busemann. For polyhedra it was described by Wulff in the theory of crystals. In our approach, the EulerLagrange equation for the support function of S has a particularly nice form. This has application to computing minimum cuts and maximum flows in a plane domain. 1.
Ito diffusions, modified capacity, and harmonic measure. Applications to Schrödinger operators
, 2013
"... We observe that some special Itô diffusions are related to scattering properties of a Schrödinger operator on R d, d ≥ 2. We introduce FeynmanKac type formulae for these stochastic processes which lead us to results on the preservation of the a.c. spectrum of the Schrödinger operator. To better un ..."
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We observe that some special Itô diffusions are related to scattering properties of a Schrödinger operator on R d, d ≥ 2. We introduce FeynmanKac type formulae for these stochastic processes which lead us to results on the preservation of the a.c. spectrum of the Schrödinger operator. To better understand the analytic properties of the processes, we construct and study a special version of the potential theory. The modified capacity and harmonic measure play an important role in these considerations. Various applications to Schrödinger operators are also given. For example, we relate the presence of the absolutely continuous spectrum to the geometric properties of the support of the potential.
Bounds on the principal frequency of the pLaplacian
 Arxiv:1304.5131v2
"... Abstract. This paper is concerned with the lower bounds for the principal frequency of the pLaplacian on ndimensional Euclidean domains. For p> n, we obtain a lower bound for the first eigenvalue of the pLaplacian on a domain in terms of its inradius, without any assumptions on the topology. M ..."
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Abstract. This paper is concerned with the lower bounds for the principal frequency of the pLaplacian on ndimensional Euclidean domains. For p> n, we obtain a lower bound for the first eigenvalue of the pLaplacian on a domain in terms of its inradius, without any assumptions on the topology. Moreover, we show that a similar lower bound can be obtained if p> n − 1 assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains. Extensions of some known results for the first eigenvalue of the Laplacian to the case p 6 = 2 are also discussed. 1. Introduction and
The Cheeger constant of curved strips
"... Abstract We study the Cheeger constant and Cheeger set for domains obtained as striplike neighbourhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the ha ..."
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Abstract We study the Cheeger constant and Cheeger set for domains obtained as striplike neighbourhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the halfwidth of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about noncomplete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.
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, 2013
"... Abstract. We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann’s one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. ( ..."
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Abstract. We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann’s one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. (SIAM J Control Optim 47: 1330–1350, 2008), which in turn is the continuous reformulation of Wardrop’s model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show
NONLOCAL PERIMETER, CURVATURE AND MINIMAL SURFACES FOR MEASURABLE SETS
"... Abstract. In this paper, we study the nonlocal perimeter associated with a nonnegative radial kernel J: RN → R, compactly supported, verifying ∫RN J(z)dz = 1. The nonlocal perimeter studied here is given by the interactions (measured in terms of the kernel J) of particles from the outside of a measu ..."
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Abstract. In this paper, we study the nonlocal perimeter associated with a nonnegative radial kernel J: RN → R, compactly supported, verifying ∫RN J(z)dz = 1. The nonlocal perimeter studied here is given by the interactions (measured in terms of the kernel J) of particles from the outside of a measurable set E with particles from the inside, that is, PJ (E):=