We study the classical Hénon family
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight
 Award ID(s):
 1800527
 NSFPAR ID:
 10223849
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Calculus of Variations and Partial Differential Equations
 Volume:
 60
 Issue:
 3
 ISSN:
 09442669
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract ,$$f_{a,b}:(x,y)\mapsto (1ax^2+y,bx)$$ ${f}_{a,b}:(x,y)\mapsto (1a{x}^{2}+y,bx)$ ,$$0$0<a<2$ , and prove that given an integer$$0$0<b<1$ , there is a set of parameters$$k\ge 1$$ $k\ge 1$ of positive twodimensional Lebesgue measure so that$$E_k$$ ${E}_{k}$ , for$$f_{a,b}$$ ${f}_{a,b}$ , has at least$$(a,b)\in E_k$$ $(a,b)\in {E}_{k}$k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73–169, 1991). A corresponding statement also holds for the Hénonlike families of Mora and Viana (Acta Math 171:1–71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original Hénon maps. The final main result of the paper is the existence, within the classical Hénon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors. 
Abstract The elliptic flow
of$$(v_2)$$ $\left({v}_{2}\right)$ mesons from beautyhadron decays (nonprompt$${\textrm{D}}^{0}$$ ${\text{D}}^{0}$ was measured in midcentral (30–50%) Pb–Pb collisions at a centreofmass energy per nucleon pair$${\textrm{D}}^{0})$$ ${\text{D}}^{0})$ TeV with the ALICE detector at the LHC. The$$\sqrt{s_{\textrm{NN}}} = 5.02$$ $\sqrt{{s}_{\text{NN}}}=5.02$ mesons were reconstructed at midrapidity$${\textrm{D}}^{0}$$ ${\text{D}}^{0}$ from their hadronic decay$$(y<0.8)$$ $\left(\righty<0.8)$ , in the transverse momentum interval$$\mathrm {D^0 \rightarrow K^\uppi ^+}$$ ${D}^{0}\to {K}^{}{\pi}^{+}$ GeV/$$2< p_{\textrm{T}} < 12$$ $2<{p}_{\text{T}}<12$c . The result indicates a positive for nonprompt$$v_2$$ ${v}_{2}$ mesons with a significance of 2.7$${{\textrm{D}}^{0}}$$ ${\text{D}}^{0}$ . The nonprompt$$\sigma $$ $\sigma $ meson$${{\textrm{D}}^{0}}$$ ${\text{D}}^{0}$ is lower than that of prompt nonstrange D mesons with 3.2$$v_2$$ ${v}_{2}$ significance in$$\sigma $$ $\sigma $ , and compatible with the$$2< p_\textrm{T} < 8~\textrm{GeV}/c$$ $2<{p}_{\text{T}}<8\phantom{\rule{0ex}{0ex}}\text{GeV}/c$ of beautydecay electrons. Theoretical calculations of beautyquark transport in a hydrodynamically expanding medium describe the measurement within uncertainties.$$v_2$$ ${v}_{2}$ 
Abstract Let
denote the standard Haar system on [0, 1], indexed by$$(h_I)$$ $\left({h}_{I}\right)$ , the set of dyadic intervals and$$I\in \mathcal {D}$$ $I\in D$ denote the tensor product$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ ,$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto {h}_{I}\left(s\right){h}_{J}\left(t\right)$ . We consider a class of twoparameter function spaces which are completions of the linear span$$I,J\in \mathcal {D}$$ $I,J\in D$ of$$\mathcal {V}(\delta ^2)$$ $V\left({\delta}^{2}\right)$ ,$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ . This class contains all the spaces of the form$$I,J\in \mathcal {D}$$ $I,J\in D$X (Y ), whereX andY are either the Lebesgue spaces or the Hardy spaces$$L^p[0,1]$$ ${L}^{p}[0,1]$ ,$$H^p[0,1]$$ ${H}^{p}[0,1]$ . We say that$$1\le p < \infty $$ $1\le p<\infty $ is a Haar multiplier if$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ , where$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D({h}_{I}\otimes {h}_{J})={d}_{I,J}{h}_{I}\otimes {h}_{J}$ , and ask which more elementary operators factor through$$d_{I,J}\in \mathbb {R}$$ ${d}_{I,J}\in R$D . A decisive role is played by theCapon projection given by$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta}^{2}\right)\to V\left({\delta}^{2}\right)$ if$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_{I}\otimes {h}_{J}={h}_{I}\otimes {h}_{J}$ , and$$I\le J$$ $\leftI\right\le \leftJ\right$ if$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_{I}\otimes {h}_{J}=0$ , as our main result highlights: Given any bounded Haar multiplier$$I > J$$ $\leftI\right>\leftJ\right$ , there exist$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ such that$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$ i.e., for all$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})\text { approximately 1projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)\phantom{\rule{0ex}{0ex}}\text{approximately 1projectionally factors through}\phantom{\rule{0ex}{0ex}}D,\end{array}$ , there exist bounded operators$$\eta > 0$$ $\eta >0$A ,B so thatAB is the identity operator ,$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$ and$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert \xb7\Vert B\Vert =1$ . Additionally, if$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})  ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)ADB\Vert <\eta $ is unbounded on$$\mathcal {C}$$ $C$X (Y ), then and then$$\lambda = \mu $$ $\lambda =\mu $ either factors through$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$D or .$${{\,\textrm{Id}\,}}D$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}D$ 
Abstract A wellknown open problem of Meir and Moser asks if the squares of sidelength 1/
n for can be packed perfectly into a rectangle of area$$n\ge 2$$ $n\ge 2$ . In this paper we show that for any$$\sum _{n=2}^\infty n^{2}=\pi ^2/61$$ ${\sum}_{n=2}^{\infty}{n}^{2}={\pi}^{2}/61$ , and any$$1/2 $1/2<t<1$ that is sufficiently large depending on$$n_0$$ ${n}_{0}$t , the squares of sidelength for$$n^{t}$$ ${n}^{t}$ can be packed perfectly into a square of area$$n\ge n_0$$ $n\ge {n}_{0}$ . This was previously known (if one packs a rectangle instead of a square) for$$\sum _{n=n_0}^\infty n^{2t}$$ ${\sum}_{n={n}_{0}}^{\infty}{n}^{2t}$ (in which case one can take$$1/2 $1/2<t\le 2/3$ ).$$n_0=1$$ ${n}_{0}=1$ 
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma n} \nabla $$ ${L}_{\beta ,\gamma}=\text{div}{D}^{d+1+\gamma n}\nabla $ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset {R}^{n}$ of dimension$$\Gamma $$ $\Gamma $ , the now usual distance to the boundary$$d < n1$$ $d<n1$ given by$$D = D_\beta $$ $D={D}_{\beta}$ for$$D_\beta (X)^{\beta } = \int _{\Gamma } Xy^{d\beta } d\sigma (y)$$ ${D}_{\beta}{\left(X\right)}^{\beta}={\int}_{\Gamma}{Xy}^{d\beta}d\sigma \left(y\right)$ , where$$X \in \Omega $$ $X\in \Omega $ and$$\beta >0$$ $\beta >0$ . In this paper we show that the Green function$$\gamma \in (1,1)$$ $\gamma \in (1,1)$G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ ${L}_{\beta ,\gamma}$ , in the sense that the function$$D^{1\gamma }$$ ${D}^{1\gamma}$ satisfies a Carleson measure estimate on$$\big  D\nabla \big (\ln \big ( \frac{G}{D^{1\gamma }} \big )\big )\big ^2$$ $D\nabla (ln(\frac{G}{{D}^{1\gamma}})){}^{2}$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).$$\Omega $$ $\Omega $