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1. Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

   https://doi.org/10.1515/crelle-2022-0029  

   Xiao, Ming (June 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F = ( F 1 , … , F m ) {F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹 n {\mathbb{B}^{n}} , n ≥ 2 {n\geq 2} , to a bounded symmetric domain Ω = Ω 1 × ⋯ × Ω m {\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ω i ′ {\Omega_{i}^{\prime}} s are irreducible factors of Ω. We prove every non-constant component F i {F_{i}} must map generic boundary points of 𝔹 n {\mathbb{B}^{n}} to the boundary of Ω i {\Omega_{i}} . In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to aproduct of unit balls and Lie balls. 

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2. Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

   https://doi.org/https://doi.org/10.1090/tran/6927  

   Akman, Murat; Badger, Matthew; Hofmann, Steve; Martell, Jose Maria (July 2017, Transactions of the American Mathematical Society)

   Let $$ \Omega \subset \mathbb{R}^{n+1}$$, $$ n\geq 2$$, be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that $$ \partial \Omega $$ is $ n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $$ \partial \Omega $$ 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 $$ \partial \Omega $$ can be covered $$ \mathcal {H}^n$$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $$ \Omega $$ and to the fact that $$ \partial \Omega $$ possesses exterior corkscrew points in a qualitative way $$ \mathcal {H}^n$$-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. 

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3. Finite element approximation of scalar curvature in arbitrary dimension

   https://doi.org/10.1090/mcom/4038  

   Gawlik, Evan S; Neunteufel, Michael (November 2024, Mathematics of Computation)

   We analyze finite element discretizations of scalar curvature in dimension $$N \ge 2$$. Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric $$g$$ on a simplicial triangulation of a polyhedral domain $$\Omega \subset \mathbb{R}^N$$ having maximum element diameter $$h$$. We show that if such an interpolant $$g_h$$ has polynomial degree $$r \ge 0$$ and possesses single-valued tangential-tangential components on codimension-1 simplices, then it admits a natural notion of (densitized) scalar curvature that converges in the $$H^{-2}(\Omega)$$-norm to the (densitized) scalar curvature of $$g$$ at a rate of $$O(h^{r+1})$$ as $$h \to 0$$, provided that either $N = 2$ or $$r \ge 1$$. As a special case, our result implies the convergence in $$H^{-2}(\Omega)$$ of the widely used ``angle defect'' approximation of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on the interpolated metric $$g_h$$. We present numerical experiments that indicate that our analytical estimates are sharp. 

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4. Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

   https://doi.org/10.1515/acv-2021-0053  

   Akman, Murat; Hofmann, Steve; Martell, José María; Toro, Tatiana (May 2022, Advances in Calculus of Variations)

   Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u )   and   L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-%\operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion[M. Akman, S. Hofmann, J. M. Martell and T. Toro,Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition,preprint 2021, https://arxiv.org/abs/1901.08261v3 ]is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaininga square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work ofDahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above. 

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5. An inequality for the normal derivative of the Lane–Emden ground state

   https://doi.org/10.1515/acv-2022-0005  

   Frank, Rupert L.; Larson, Simon (May 2022, Advances in Calculus of Variations)

   Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions.Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality.Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity. 

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