Area-minimizing regions with small volume in Riemannian manifolds with boundary

Fall, Mouhamed Moustapha

doi:10.2140/pjm.2010.244.235

Cited by 22 publications

(13 citation statements)

References 19 publications

(23 reference statements)

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“…, N , g ij = ∂F M ∂y i , ∂F M ∂y j ; the quantity |g| is the determinant of g and g ij is the component of the inverse of the matrix (g ij ) 2≤i,j≤N . Since g ij = δ ij + O(y 1 ) + O(|y| 2 ) (see [19]), we have the following Taylor expansion…”

Section: Fermi Coordinatesmentioning

confidence: 99%

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

Fall

2013

Journal of Functional Analysis

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Section: Fermi Coordinatesmentioning

confidence: 99%

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

Fall

2013

Journal of Functional Analysis

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“…For r > 0 small, we introduce the following system of coordinates centered at 0 (see [9]) via the mapping F : B + r → Ω given by…”

Section: Lemma 21mentioning

confidence: 99%

A note on Hardy’s inequalities with boundary singularities

Fall

2012

Nonlinear Analysis: Theory, Methods & Applications

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“…More precisely the estimate from above is inspired by computations performed in [32]. On the other hand, the estimate from below exploits sharp relative isoperimetric inequalities proved by Fall in [22]. As explained in such paper, asymptotic spherical optimal shapes for isoperimetric inequalities with small volume constraints have been object of large interest in the last years.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic spherical shapes in some spectral optimization problems

Mazzoleni

Pellacci

Verzini

2020

Journal de Mathématiques Pures et Appliquées

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We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box Ω ⊂ R N , which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is led to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Using α-symmetrization techniques on cones, we prove that, for suitable choices of the box Ω, the optimal shapes for this second problem are indeed spherical. Moreover, for general Ω, we show that small volume spectral drops are asymptotically spherical, centered near points of ∂Ω having largest mean curvature. RésuméOn etudie l'optimisation de la valeur propre principale positive d'un problème avec un poids indéfini, associé au Laplacien avec conditions au bord de Neumann dans une boîte Ω ⊂ R N , qui apparaît dans l'étude naturellement comme valeur limite de survie en dynamique des populations. En essayant de minimiser une telle valeur propre par rapport au poids, on est amenéà considérer un problème d'optimisation de forme, qui n'admet pas de formes sphériques comme solutions (chose qui contredit certaines conjecturesénoncées précédemment). Nousétudions si des formes sphériques peuventêtre recouvrées comme solutions de certaines perturbations singulières du problème. Notamment, on démontre que, si la partie négative du poids diverge, le problème d'optimisation de forme se rapprocheà la limiteà un problème de "goutte spectrale", cetteà dire un problème de minimisation de la première valeur propre du Laplacien avec conditions mixtes Dirichletet-Neumann au bord. En utilisant techniques de α-symétrisation sur les cônes, on démontre que, pour des choix opportunes de la boîte Ω, les formes optimales pour ce second problème sont bien sphériques. De plus, pour Ω générique, on démontre que les gouttes spectrales s'approchent asymptotiquement -lorsque leur volume devient infinitésimal-à des gouttes sphériques, centrées près des points de ∂Ω ayant courbure moyenne maximale.

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Area-minimizing regions with small volume in Riemannian manifolds with boundary

Abstract: Given a domain of a Riemannian manifold, we prove that regions minimizing the area (relative to) are nearly the maxima of the mean curvature of ∂ when their volume tends to zero. We deduce some sharp local relative isoperimetric inequalities involving mean curvature comparisons.

Cited by 22 publications

References 19 publications

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

A note on Hardy’s inequalities with boundary singularities

Asymptotic spherical shapes in some spectral optimization problems

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