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1. Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

   https://doi.org/10.1515/ans-2022-0055  

   De Silva, Daniela ; Savin, Ovidiu ( January 2023 , Advanced Nonlinear Studies)

   Abstract We investigate the rigidity of global minimizers u ≥ 0 u\ge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , \mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2), when the exponent γ \gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{\mathbb{R}}}^{n} are one-dimensional if γ \gamma is close to 2 and n ≤ 7 n\le 7 , or if γ \gamma is close to 0 and n ≤ 4 n\le 4 . 

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2. On big pieces approximations of parabolic hypersurfaces

   https://doi.org/10.54330/afm.115417  

   Bortz, Simon ; Hoffman, John ; Hofmann, Steve ; Luna-Garcia, Jose Luis ; Nyström, Kaj ( November 2021 , Annales Fennici Mathematici)

   Let \(\Sigma\) be a closed subset of \(\mathbb{R}^{n+1}\) which is parabolic Ahlfors-David regular and assume that \(\Sigma\) satisfies a 2-sided corkscrew condition. Assume, in addition, that \(\Sigma\) is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that \(\Sigma\) satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that \(\Sigma\) contain suniform big pieces of Lip(1,1/2) graphs. When \(\Sigma\) is parabolic uniformly rectifiable the construction can be refined and in this case we prove that \(\Sigma\) contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if \(\Omega\subset\mathbb{R}^{n+1}\) is a connected component of \(\mathbb{R}^{n+1}\setminus\Sigma\) and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if \(\Omega\) is a one-sided parabolic chord arc domain, and if \(\Sigma\) is parabolic uniformly rectifiable, then \(\Omega\) is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition. 

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3. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie ; Malinnikova, Eugenia ( June 2021 , Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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4. $L^2(I;H^1(\Omega)^d)$ and $L^2(I;L^2(\Omega)^d)$ best approximation type error estimates for Galerkin solutions of transient Stokes problems

   Leykekhman, Dmitriy ; Vexler, Boris ( October 2023 , Calcolo)

   In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in $L^2(I;L^2(\Omega)^d)$ and $L^2(I;H^1(\Omega)^d)$ norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type $L^2(I;H^1(\Omega))$ error estimate are new even for scalar parabolic problems. 

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5. Random quantum graphs

   https://doi.org/10.1090/tran/8584  

   Chirvasitu, Alexandru ; Wasilewski, Mateusz ( April 2022 , Transactions of the American Mathematical Society)

   We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples ( X 1 , ⋯ , X d ) (X_1,\cdots ,X_d) of traceless self-adjoint operators in the n × n n\times n matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: 2 ≤ d ≤ n 2 − 3 2\le d\le n^2-3 . Moreover, the automorphism group is generically abelian in the larger parameter range 1 ≤ d ≤ n 2 − 2 1\le d\le n^2-2 . This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of X i X_i ’s (mimicking the Erdős-Rényi G ( n , p ) G(n,p) model) has trivial/abelian automorphism group almost surely. 

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