Sharp isoperimetric inequalities via the ABP method

By X. Cabré, Xavier Ros-Oton, J. Serra. 2013

124 citations recorded by the source corpus.

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 $\mathbb{R}^n$. 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/68e149fa0c5907cec6ee01e0f0187c755a82e404

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In Applied Mathematics

Citations in the corpus

Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “The first one is highlighted by the works of Cabré and Ros-Oton [9] and Cabré, Ros-Oton and Serra [10] which is based on PDE techniques, notably on Aleksandrov-Bakelman-Pucci estimates; this is the so called ABP-method.”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “4, this is actually known to hold for arbitrary Borel sets K (see [15, 54]).”
Symmetry results for critical anisotropic p-Laplacian equations in convex cones (2019, methodology): “Sobolev inequalities have been studied for more general norms as well as in convex cones (see [4, 8, 19, 20, 27, 28]), where they take the form ‖u‖Lp∗ (Σ) ≤ SΣ,H‖H(∇u)‖Lp(Σ) , (1.”
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences (2013, methodology): “Their method provides a proof for isoperimetric inequalities with monominal weights (see [9]), which can be generalized to inequalities on open convex cone with general weights (see [10]), which includes the half-space case discussed here (Proposition 1.”
Sobolev Inequalities in Manifolds with Nonnegative Curvature (2020, unknown): “[5], [13], [14], [15].”
Overdetermined problems and constant mean curvature surfaces in cones (2018, background): “Alternative proofs and generalizations of this theorem in convex cones are provided in [7], [10], [25].”
Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures (2013, methodology): “As described in [12], one of the present authors has noted that the case when p = 1/N and N is an integer may be reduced to the p = ∞ case.”