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1. Examples of non-Dini domains with large singular sets

   https://doi.org/10.1515/ans-2022-0058  

   Kenig, Carlos; Zhao, Zihui (January 2023, Advanced Nonlinear Studies)

   Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures. 

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2. Boundary unique continuation for the Laplace equation and the biharmonic operator

   S. Berhanu (January 2022, Communications in analysis and geometry)

   Kefeng Liu (Ed.)

   We establish results on unique continuation at the boundary for the solutions of ∆u = f, f harmonic, and the biharmonic equation ∆^2u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2]. 

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3. BOUNDARY UNIQUE CONTINUATION FOR THE LAPLACE EQUATION AND THE BIHARMONIC OPERATOR

   Berhanu, S. (January 2022, Communications in analysis and geometry)

   Kefeng Liu (Ed.)

   We establish results on unique continuation at the boundary for the solutions of ∆u = f, f harmonic, and the biharmonic equation ∆^2u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2]. 

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4. Projections onto L^p Bergman spaces of Reinhardt domains

   https://doi.org/10.1016/j.aim.2024.109790  

   Chakrabarti, Debraj; Edholm, Luke D (August 2024, Advances in Mathematics)

   For $$1<\infty$$, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of $L^p$-Bergman space. The construction gives an integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined by the kernel is shown to be an absolutely bounded projection on the $L^p$-Bergman space on a class of domains where the $L^p$-boundedness of the Bergman projection fails for certain $$p \neq 2$$. As an application, we identify the duals of these $L^p$-Bergman spaces with weighted Bergman spaces. 

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5. Resistance scaling on 4 N -carpets

   https://doi.org/10.1515/forum-2020-0330  

   Canner, Claire; Hayes, Christopher; Huang, Robin; Orwin, Michael; Rogers, Luke G. (December 2021, Forum Mathematicum)

   Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F . 

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