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Regularity of stable solutions up to dimension 7 in domains of double revolution
 Comm. Partial Differential Equations
, 2013
"... Abstract. We consider the class of semistable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex ..."
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Cited by 8 (4 self)
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Abstract. We consider the class of semistable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori Lp and H10 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of −∆u = λf(u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 3 for any domain Ω, in dimension n = 4 when the domain is convex, and in dimensions 5 ≤ n ≤ 9 in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities f in dimensions 5 ≤ n ≤ 9. 1. Introduction and
Sharp isoperimetric inequalities via the ABP method
"... We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of Rn. Our result ..."
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Cited by 7 (2 self)
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We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of Rn. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial —except for the constant ones. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.
Euclidean balls solve some isoperimetric problems with nonradial weights
"... Abstract. In this note we present the solution of some isoperimetric problems in open convex cones of Rn in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperim ..."
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Cited by 5 (2 self)
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Abstract. In this note we present the solution of some isoperimetric problems in open convex cones of Rn in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990. Résumé. Dans cette note, nous présentons la solution de certains problèmes isopérimétriques dans des cônes convexes de Rn ou ̀ le périmètre et le volume sont mesurés par rapport a ̀ certains poids non radiaux. Contrairement a ̀ ce que l’on pourrait penser, les boules euclidiennes centrées a ̀ l’origine (intersectées avec le cône) minimisent le quotient isopérimétrique. Notre résultat s’applique aux poids strictement positifs, homogènes et satisfaisant une condition de concavite ́ dans le cône. Lorsque le poids est constant, le résultat a éte ́ établi par Lions et Pacella en 1990. 1.
Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities
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Regularity of minimizers up to dimension 7 in domains of double revolution
 Comm. Partial Differential Equations
"... Abstract. We consider the class of semistable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n − 2. When the domain is conve ..."
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Cited by 2 (1 self)
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Abstract. We consider the class of semistable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori Lp and H10 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of −∆u = λf(u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 3 for any domain Ω, in dimensions n ≤ 4 when the domain is convex, and in dimensions n ≤ 9 in the radial case. In dimensions 5 ≤ n ≤ 9 it remains an open question. 1. Introduction and
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities
"... on halfspaces via mass transport and consequences ..."
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