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1. 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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2. Fast Regression for Structured Inputs

   Meyer, Raphael ; Musco, Cameron ; Musco, Christopher ; Woodruff, David P. ; Zhou, Samson ( April 2022 , International Conference on Learning Representations (ICLR))

   We study the $\ell_p$ regression problem, which requires finding $\mathbf{x}\in\mathbb R^{d}$ that minimizes $\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$ for a matrix $\mathbf{A}\in\mathbb R^{n \times d}$ and response vector $\mathbf{b}\in\mathbb R^{n}$. There has been recent interest in developing subsampling methods for this problem that can outperform standard techniques when $n$ is very large. However, all known subsampling approaches have run time that depends exponentially on $p$, typically, $d^{\mathcal{O}(p)}$, which can be prohibitively expensive. We improve on this work by showing that for a large class of common \emph{structured matrices}, such as combinations of low-rank matrices, sparse matrices, and Vandermonde matrices, there are subsampling based methods for $\ell_p$ regression that depend polynomially on $p$. For example, we give an algorithm for $\ell_p$ regression on Vandermonde matrices that runs in time $\mathcal{O}(n\log^3 n+(dp^2)^{0.5+\omega}\cdot\text{polylog}\,n)$, where $\omega$ is the exponent of matrix multiplication. The polynomial dependence on $p$ crucially allows our algorithms to extend naturally to efficient algorithms for $\ell_\infty$ regression, via approximation of $\ell_\infty$ by $\ell_{\mathcal{O}(\log n)}$. Of practical interest, we also develop a new subsampling algorithm for $\ell_p$ regression for arbitrary matrices, which is simpler than previous approaches for $p \ge 4$. 

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3. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A ; Jauslin, Ian ; Lieb, Elliott H ; Loss, Michael P ( January 2021 , International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $f \geqslant f\star f$ for real functions in $L^1({\mathbb{R}}^d)$ where $f\star f$ denotes the convolution of $f$ with itself. We show that all such functions $f$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $1 < p \leqslant 2$, for which the convolution is defined. We also show that all solutions in $L^1({\mathbb{R}}^d)$ satisfy $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$. Moreover, if $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$, then $f$ must decay fairly slowly: $\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $, and this is sharp since for all $r< 1$, there are solutions with $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$ and $\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $. However, if $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$, the decay at infinity can be much more rapid: we show that for all $a<\tfrac 12$, there are solutions such that for some $\varepsilon>0$, $\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $. 

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4. Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

   Pauthier, Antoine ; Poláčik, Peter ( August 2022 , Partial Differential Equations and Applications)

   Abstract This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation $$u_t=u_{xx}+f(u)$$ u t = u xx + f ( u ) on the real line whose initial data $$u_0=u(\cdot ,0)$$ u 0 = u ( · , 0 ) have finite limits $$\theta ^\pm $$ θ ± as $$x\rightarrow \pm \infty $$ x → ± ∞ . We assume that f is a locally Lipschitz function on $$\mathbb {R}$$ R satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u ( x ,  t ) as $$t\rightarrow \infty $$ t → ∞ . In the first two parts of this series we mainly considered the cases where either $$\theta ^-\ne \theta ^+$$ θ - ≠ θ + ; or $$\theta ^\pm =\theta _0$$ θ ± = θ 0 and $$f(\theta _0)\ne 0$$ f ( θ 0 ) ≠ 0 ; or else $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is a stable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of $$u(\cdot ,t)$$ u ( · , t ) as $$t\rightarrow \infty $$ t → ∞ are steady states. The limit profiles, or accumulation points, are taken in $$L^\infty _{loc}(\mathbb {R})$$ L loc ∞ ( R ) . In the present paper, we take on the case that $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is an unstable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on $$u(\cdot ,t)$$ u ( · , t ) is that it is nonoscillatory (has only finitely many critical points) at some $$t\ge 0$$ t ≥ 0 . Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal. 

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5. Singularity formation in the harmonic map flow with free boundary

   https://doi.org/10.1353/ajm.2023.a902958  

   Sire, Yannick ; Wei, Juncheng ; Zheng, Youquan ( August 2023 , American Journal of Mathematics)

   abstract: In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary $$ (0.1)\hskip77pt\cases{u_t=\Delta u& in $\Bbb{R}^2_+\times (0,T)$,\cr u(x,0,t)\in\Bbb{S}^1& for all $(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$,\cr {du\over dy}(x,0,t)\perp T_{u(x,0,t)}\Bbb{S}^1& for all $(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$,\cr u(\cdot, 0)=u_0& in $\Bbb{R}^2_+$} $$ for a function $u:\Bbb{R}^2_+\times [0,T)\to\Bbb{R}^2$. Here $u_0 :\Bbb{R}^2_+\to\Bbb{R}^2$ is a given smooth map and $\perp$ stands for orthogonality. We prove the existence of initial data $u_0$ such that (0.1) blows up at finite time with a profile being the half-harmonic map. This answers a question raised by Chen and Lin.

    

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