Abstract. Let Ω ⊂ R n+1 , n ≥ 2, be 1-sided NTA domain also known as uniform domain), i.e., a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that ∂Ω is n-dimensional Ahlfors-David regular. We characterize the rectifiability of ∂Ω in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that ∂Ω can be covered H n -a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of Ω and to the fact that ∂Ω possesses exterior corkscrew points in a qualitative way H n -a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.
In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, Cap A , where A-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the p-Laplace equation and whose solutions in an open set are called A-harmonic.In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity:
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 ℝ n {\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 C 𝒜 ( 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 C 𝒜 ( ⋅ ) {\mathrm{C}_{\mathcal{A}}(\,\cdot\,)} satisfies the following Brunn–Minkowski-type inequality: [ C 𝒜 ( λ E 1 + ( 1 - λ ) E 2 ) ] 1 p - n ≥ λ [ C 𝒜 ( E 1 ) ] 1 p - n + ( 1 - λ ) [ C 𝒜 ( E 2 ) ] 1 p - 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 E 1 , E 2 {E_{1},E_{2}} are nonempty convex compact sets in ℝ n {\mathbb{R}^{n}} . While p = n {p=n} then C 𝒜 ( λ E 1 + ( 1 - λ ) E 2 ) ≥ λ C 𝒜 ( E 1 ) + ( 1 - λ ) C 𝒜 ( E 2 ) , \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 E 1 , E 2 {E_{1},E_{2}} are convex compact sets in ℝ n {\mathbb{R}^{n}} containing at least two points. Moreover, if equality holds in the either of the above inequalities for some E 1 {E_{1}} and E 2 {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 𝕊 n - 1 {\mathbb{S}^{n-1}} to be the surface area measure of a convex compact set in ℝ n {\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 μ E {\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.
We discuss what is known about homogeneous solutions u to the p− Laplace equation, p fixed, 1 < p < ∞, when (A) u is an entire p− harmonic function on Euclidean n space, R n , or (B) u > 0 is p− harmonic in the cone, K(α) = {x = (x 1 ,. .. , x n) : x 1 > cos α |x|} ⊂ R n , n ≥ 2, with continuous boundary value zero on ∂K(α) \ {0} when α ∈ (0, π]. We also outline a proof of our new result concerning the exact value, λ = 1 − (n − 1)/p, for an eigenvalue problem in an ODE associated with u when u is pharmonic in K(π) and p > n − 1. Generalizations of this result are stated. Our result complements work of Krol'-Maz'ya for 1 < p ≤ n − 1.
Let f be a smooth convex homogeneous function of degree p, 1 < p < ∞, on C \ {0}. We show that if u is a minimizer for the functional whose integrand is f (∇v), v in a certain subclass of the Sobolev space W 1, p ( ), and ∇u = 0 at z ∈ , then in a neighborhood of z, log f (∇u) is a sub, super, or solution (depending on whether p > 2, p < 2, or p = 2) to L wherewe then indicate the importance of this fact in previous work of the authors when f (η) = |η| p and indicate possible future generalizations of this work in which this fact will play a fundamental role.
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