On the isoperimetric problem in Euclidean space with density

By César Rosales, Antonio Cañete, Vincent Bayle, Frank Morgan. 2006

198 citations recorded by the source corpus.

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp$$(|x|^2)$$ by using symmetrization techniques.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e1ef324babc2128542fb08ee6871c51a9e4f3205

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In Applied Mathematics

Citations in the corpus

Sharp isoperimetric inequalities via the ABP method (2013, unknown): “For example, in R n with the Gaussian weight w(x) = e −| x | 2 all the minimizers are half-spaces [3, 16], and with w(x) = e | x | 2 all the minimizers are balls centered at the origin [46].”
Existence of isoperimetric regions in R n with density (2011, background): “gives our main existence results, and Section 8 collects some open problems. This paper focuses on Rnfor n2. In R1, most of our questions are trivial and much ner results are already known (see e.g. [15]). 2. Preliminaries Let us rst set some notation and list some known results. We conne attention ”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “Let us limit ourselves to mentioning that, besides those quoted above, recent contributions on isoperimetric problems in (domains in) R endowed with a measure ν include [18, 36, 43, 76].”
Some Isoperimetric Problems in Planes with Density (2009, methodology): “Proof The proof of this fact is essentially the same contained in [24]; in fact, we have that”
Stability properties and topology at infinity of $$f$$f-minimal hypersurfaces (2013, unknown): “[28– 30,41].”
Halfspace type theorems for self‐shrinkers (2014, background): “Moreover, we know (see [2]) that H ′ φ(0) = −Jφu −cu < 0, which means that the weighted mean curvature of Σs is strictly negative, that is, H φ(q) < 0 for all q ∈ Σs, for all s ∈ (− , ).”