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40
Eigenvalues of Euclidean wedge domains in higher dimensions ∗
, 2009
"... In this paper, we use a weighted isoperimetric inequality to give a lower bound for the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two d ..."
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In this paper, we use a weighted isoperimetric inequality to give a lower bound for the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two dimensions. 1
The ε− εβ property, the boundedness of isoperimetric sets in Rn with density, and some applications
 J. Reine Angew. Math. (Crelle
"... Abstract. We show that every isoperimetric set in RN with density is bounded if the density is continuous and bounded by above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we ..."
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Abstract. We show that every isoperimetric set in RN with density is bounded if the density is continuous and bounded by above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical “ε − ε ” property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an “ε − εβ ” version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the sole continuity is enough. We conclude by deriving some consequences of our result about the existence and regularity of isoperimetric sets. 1.
Isoperimetric regions in surfaces and in surfaces with density
, 2006
"... We study the isoperimetric problem, the leastperimeter way to enclose given area, in various surfaces. For example, in twodimensional Twisted Chimney space, a twodimensional analog of one of the ten flat, orientable models for the universe, we prove that isoperimetric regions are round discs or st ..."
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We study the isoperimetric problem, the leastperimeter way to enclose given area, in various surfaces. For example, in twodimensional Twisted Chimney space, a twodimensional analog of one of the ten flat, orientable models for the universe, we prove that isoperimetric regions are round discs or strips. In the Gauss plane, defined as the Euclidean plane with Gaussian density, we prove that in halfspaces y ≥ a vertical rays minimize perimeter. In R n with radial density and in certain products we provide partial results and conjectures. 1