Volume 14, Issue 2

Using the optimal mass transport method and a suitable quasi-conformal mapping, we study the sharp weighted isoperimetric, Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. The class of weight functions under consideration includes all nonnegative homogeneous weights satisfying a concavity condition that is equivalent to a usual curvature-dimension bound and the nonnegativity of a Bakry–Émery Ricci tensor. Though our densities are not radial in general, the optimizers are radially symmetric.

In the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of BV{\mathrm{BV}} functions with zero boundary values. In particular, we show that the class is the closure of compactly supported BV{\mathrm{BV}} functions in the BV{\mathrm{BV}} norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and BV{\mathrm{BV}} analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

We consider a weakly coupled system of discounted Hamilton–Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to 0. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton–Jacobi systems and on suitable random representation formulae for the discounted solutions.

We define a class of Sobolev W1,p⁢(Ω,ℝn){W^{1,p}(\Omega,\mathbb{R}^{n})} functions, with p>n-1{p>n-1}, such that its trace on ∂⁡Ω{\partial\Omega} is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.

In this article we study two classical problems in convex geometry associated to 𝒜{\mathcal{A}}-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p -Laplace equation. Let p be fixed with 2≤n≤p<∞{2\leq n\leq p<\infty}. For a convex compact set E in ℝn{\mathbb{R}^{n}}, we define and then prove the existence and uniqueness of the so-called 𝒜{\mathcal{A}}-harmonic Green’s function for the complement of E with pole at infinity. We then define a quantity C𝒜⁢(E){\mathrm{C}_{\mathcal{A}}(E)} which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that C𝒜⁢(⋅){\mathrm{C}_{\mathcal{A}}(\,\cdot\,)} satisfies the following Brunn–Minkowski-type inequality: [C𝒜⁢(λ⁢E1+(1-λ)⁢E2)]1p-n≥λ⁢[C𝒜⁢(E1)]1p-n+(1-λ)⁢[C𝒜⁢(E2)]1p-n[\mathrm{C}_{\mathcal{A}}(\lambda E_{1}+(1-\lambda)E_{2})]^{\frac{1}{p-n}}\geq% \lambda[\mathrm{C}_{\mathcal{A}}(E_{1})]^{\frac{1}{p-n}}+(1-\lambda)[\mathrm{C% }_{\mathcal{A}}(E_{2})]^{\frac{1}{p-n}} when n<p<∞{n<p<\infty}, 0≤λ≤1{0\leq\lambda\leq 1}, and E1,E2{E_{1},E_{2}} are nonempty convex compact sets in ℝn{\mathbb{R}^{n}}. While p=n{p=n} then C𝒜⁢(λ⁢E1+(1-λ)⁢E2)≥λ⁢C𝒜⁢(E1)+(1-λ)⁢C𝒜⁢(E2),\mathrm{C}_{\mathcal{A}}(\lambda E_{1}+(1-\lambda)E_{2})\geq\lambda\mathrm{C}_% {\mathcal{A}}(E_{1})+(1-\lambda)\mathrm{C}_{\mathcal{A}}(E_{2}), where 0≤λ≤1{0\leq\lambda\leq 1} and E1,E2{E_{1},E_{2}} are convex compact sets in ℝn{\mathbb{R}^{n}} containing at least two points. Moreover, if equality holds in the either of the above inequalities for some E1{E_{1}} and E2{E_{2}}, then under certain regularity and structural assumptions on 𝒜{\mathcal{A}} we show that these two sets are homothetic. The classical Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere 𝕊n-1{\mathbb{S}^{n-1}} to be the surface area measure of a convex compact set in ℝn{\mathbb{R}^{n}} under the Gauss mapping for the boundary of this convex set. In the second part of this article we study a Minkowski-type problem for a measure associated to the 𝒜{\mathcal{A}}-harmonic Green’s function for the complement of a convex compact set E when n≤p<∞{n\leq p<\infty}. If μE{\mu_{E}} denotes this measure, then we show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem. Using the Brunn–Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation.