We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d ge 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^∗$ which exists for all d $d ge 7$. For $d = 9$, we study the stability of $u^∗$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$ . In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d = 7$ and $d = 9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.
{"title":"On blowup for the supercritical quadratic wave equation","authors":"E. Csobo, Irfan Glogi'c, Birgit Schorkhuber","doi":"10.5445/IR/1000138775","DOIUrl":"https://doi.org/10.5445/IR/1000138775","url":null,"abstract":"We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d ge 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^∗$ which exists for all d $d ge 7$. For $d = 9$, we study the stability of $u^∗$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$ . In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d = 7$ and $d = 9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80145171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show global wellposedness for the defocusing cubic nonlinear Schrodinger equation (NLS) in $H^1(mathbb{R}) + H^{3/2+}(mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(mathbb{R}) + H^{5/2+}(mathbb{T})$. This complements local results for the cubic NLS [6] and global results for the quadratic NLS [8] in this hybrid setting.
{"title":"Global wellposedness of NLS in $H^1(mathbb{R})+H^s(mathbb{T})$","authors":"Friedrich Klaus, P. Kunstmann","doi":"10.5445/IR/1000137946","DOIUrl":"https://doi.org/10.5445/IR/1000137946","url":null,"abstract":"We show global wellposedness for the defocusing cubic nonlinear Schrodinger equation (NLS) in $H^1(mathbb{R}) + H^{3/2+}(mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(mathbb{R}) + H^{5/2+}(mathbb{T})$. This complements local results for the cubic NLS [6] and global results for the quadratic NLS \u0000[8] in this hybrid setting.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84939408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show new local well-posedness results for quasilinear Maxwell equations in three spatial dimensions with an emphasis on the Kerr nonlinearity. For this purpose, new Strichartz estimates are proved for solutions with rough permittivity by conjugation to half-wave equations. We use the Strichartz estimates in a known combination with energy estimates to derive the new well-posedness results.
{"title":"Well-posedness for Maxwell equations with Kerr nonlinearity in three dimensions via Strichartz estimates","authors":"R. Schippa","doi":"10.5445/IR/1000136611","DOIUrl":"https://doi.org/10.5445/IR/1000136611","url":null,"abstract":"We show new local well-posedness results for quasilinear Maxwell equations in three spatial dimensions with an emphasis on the Kerr nonlinearity. For this purpose, new Strichartz estimates are proved for solutions with rough permittivity by conjugation to half-wave equations. We use the Strichartz estimates in a known combination with energy estimates to derive the new well-posedness results.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73876678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the true signal, to which the observations correspond, in all higher-order Sobolev topologies. This work complements previous results in the literature where conditions were identified under which synchronization is guaranteed either with respect to only the $H^1$--topology, in the case of general observables, or to the analytic Gevrey topology, in the case of spectral observables. To accommodate the property of synchronization in the stronger topologies, the framework of general interpolant observable operators, originally introduced by Azouani, Olson, and Titi, is expanded to a far richer class of operators. A significant effort is dedicated to the development of this more expanded framework, specifically, their basic approximation properties, the identification of subclasses of such operators relevant to obtaining synchronization, as well as the detailed relation between the structure of these operators and the system regarding the syncrhonization property. One of the main features of this framework is its "mesh-free" aspect, which allows the observational data itself to dictate the subdivision of the domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in all higher-order Sobolev norms are obtained, thus properly generalizing previously known bounds; such estimates are required for establishing the synchronization property of the algorithm in the higher-order topologies.
{"title":"Higher-order synchronization of a nudging-based algorithm for data assimilation for the 2D NSE: a refined paradigm for global interpolant observables","authors":"A. Biswas, K. Brown, V. Martinez","doi":"10.13016/M2FYO1-BNGF","DOIUrl":"https://doi.org/10.13016/M2FYO1-BNGF","url":null,"abstract":"This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the true signal, to which the observations correspond, in all higher-order Sobolev topologies. This work complements previous results in the literature where conditions were identified under which synchronization is guaranteed either with respect to only the $H^1$--topology, in the case of general observables, or to the analytic Gevrey topology, in the case of spectral observables. To accommodate the property of synchronization in the stronger topologies, the framework of general interpolant observable operators, originally introduced by Azouani, Olson, and Titi, is expanded to a far richer class of operators. A significant effort is dedicated to the development of this more expanded framework, specifically, their basic approximation properties, the identification of subclasses of such operators relevant to obtaining synchronization, as well as the detailed relation between the structure of these operators and the system regarding the syncrhonization property. One of the main features of this framework is its \"mesh-free\" aspect, which allows the observational data itself to dictate the subdivision of the domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in all higher-order Sobolev norms are obtained, thus properly generalizing previously known bounds; such estimates are required for establishing the synchronization property of the algorithm in the higher-order topologies.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79366587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schrodinger operators on the cubic lattice.
{"title":"Fourier transform of surface-carried measures of two-dimensional generic surfaces and applications","authors":"Jean-Claude Cuenin, R. Schippa","doi":"10.5445/IR/1000136612","DOIUrl":"https://doi.org/10.5445/IR/1000136612","url":null,"abstract":"We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schrodinger operators on the cubic lattice.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88487449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinearequation $$Delta^2u=g(x,u)qquadtext{ in }mathbb{R}^N$$ with a Caratheodory function $g:mathbb{R}^Ntimesmathbb{R}tomathbb{R}$, $Nge5$. The regularity results give rise to the existence of ground state solutions provided that g has a general subcritical growth at infinity. We also conceive a newbiharmonic logarithmic Sobolev inequality$$int_{mathbb{R}^N}|u|^2log|u|,dx le frac{N}{8}logleft(Cint_{mathbb{R}^N}|Delta u|^2,dxright), quadtext{ for } uin H^2(mathbb{R}^N), int_{mathbb{R}^N}u^2,dx=1,$$for a constant $0
我们证明了具有卡拉多函数$g:mathbb{R}^Ntimesmathbb{R}tomathbb{R}$, $Nge5$的双调和非线性方程$$Delta^2u=g(x,u)qquadtext{ in }mathbb{R}^N$$弱解的一个brezis - kato型正则性结果。如果g在无穷远处具有一般的亚临界增长，则正则性结果可以得到基态解的存在性。对于常数$0
{"title":"Biharmonic nonlinear scalar field equations","authors":"Jarosław Mederski, Jakub Siemianowski","doi":"10.5445/IR/1000135513","DOIUrl":"https://doi.org/10.5445/IR/1000135513","url":null,"abstract":"We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinearequation $$Delta^2u=g(x,u)qquadtext{ in }mathbb{R}^N$$ with a Caratheodory function $g:mathbb{R}^Ntimesmathbb{R}tomathbb{R}$, $Nge5$. The regularity results give rise to the existence of ground state solutions provided that g has a general subcritical growth at infinity. We also conceive a newbiharmonic logarithmic Sobolev inequality$$int_{mathbb{R}^N}|u|^2log|u|,dx le frac{N}{8}logleft(Cint_{mathbb{R}^N}|Delta u|^2,dxright), quadtext{ for } uin H^2(mathbb{R}^N), int_{mathbb{R}^N}u^2,dx=1,$$for a constant $0<C<left(frac{2}{pi e N}right)^2$ and we characterize its minimizers.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73285665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-29DOI: 10.22541/AU.162227013.31240680/V1
Jingning He
We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono’s type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when t>0. In particular, with the aid of the viscous term, we prove the so-called instantaneous separation property of the phase variable such that it stays away from the pure states ±1 as long as t>0. Furthermore, we study long-time behavior of the system, by proving the existence of a global attractor and characterizing its ω-limit set.
{"title":"On the Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential","authors":"Jingning He","doi":"10.22541/AU.162227013.31240680/V1","DOIUrl":"https://doi.org/10.22541/AU.162227013.31240680/V1","url":null,"abstract":"We analyze a diffuse interface model that couples a viscous\u0000Cahn-Hilliard equation for the phase variable with a diffusion-reaction\u0000equation for the nutrient concentration. The system under consideration\u0000also takes into account some important mechanisms like chemotaxis,\u0000active transport as well as nonlocal interaction of Oono’s type. When\u0000the spatial dimension is three, we prove the existence and uniqueness of\u0000global weak solutions to the model with singular potentials including\u0000the physically relevant logarithmic potential. Then we obtain some\u0000regularity properties of the weak solutions when t>0. In\u0000particular, with the aid of the viscous term, we prove the so-called\u0000instantaneous separation property of the phase variable such that it\u0000stays away from the pure states ±1 as long as t>0.\u0000Furthermore, we study long-time behavior of the system, by proving the\u0000existence of a global attractor and characterizing its ω-limit set.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89963886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show new local $L^p$-smoothing estimates for the Schrodinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and $L^p$-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in $L^p(mathbb{R}^d)$ for $d ge 1$ and $p>2$, which was previously known for $d ge 2$. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schrodinger equation on the line. Lastly, we show $ell^2$ -decoupling inequalities for variable-coefficient versions of elliptic and non-elliptic Schrodinger phase functions.
我们通过解耦不等式给出了调制空间中具有初始数据的薛定谔方程的新的局部L^p$平滑估计。进一步，我们通过knapp类型的例子探讨了在调制和L^p$-空间中具有初始数据的解的时空估计的必要条件。实例显示了平滑估计在一定范围内达到端点规则性的清晰度。此外，这些例子排除了初始数据在$L^p(mathbb{R}^d)$中对于$d ge 1$和$p>2$的全局Strichartz估计，它以前被称为$d ge 2$。利用这些估计给出了三次非线性薛定谔方程在直线上的新的局部和全局适定性结果。最后，我们给出了椭圆型和非椭圆型薛定谔相函数变系数版本的$ well ^2$ -解耦不等式。
{"title":"On smoothing estimates in modulation spaces and the NLS with slowly decaying initial data","authors":"R. Schippa","doi":"10.5445/IR/1000132421","DOIUrl":"https://doi.org/10.5445/IR/1000132421","url":null,"abstract":"We show new local $L^p$-smoothing estimates for the Schrodinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and $L^p$-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in $L^p(mathbb{R}^d)$ for $d ge 1$ and $p>2$, which was previously known for $d ge 2$. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schrodinger equation on the line. Lastly, we show $ell^2$ -decoupling inequalities for variable-coefficient versions of elliptic and non-elliptic Schrodinger phase functions.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88179520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simon Kohler, Wolfgang Reichel Institute for Analysis, Karlsruhe Institute of Technology, D. Karlsruhe, H Germany
We consider the $(1 + 1)$-dimensional quasilinear wave equation $g(x)w_{tt} − w_{xx} + h(x)(w^3_t)_t = 0$ on $mathbb{R}timesmathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $gin L^{infty}(mathbb{R})$ is even with $gnotequiv 0$ and $h(x) = gammadelta_0(x)$ with $gammainmathbb{R}backslash{0}$ and $delta_0$ the delta distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k = frac{d^2}{dx^2}-k^2omega^2g$ on $L^2(mathbb{R})$ for all $kin 2mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitely given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.
{"title":"Breather solutions for a quasilinear (1+1)-dimensional wave equation","authors":"Simon Kohler, Wolfgang Reichel Institute for Analysis, Karlsruhe Institute of Technology, D. Karlsruhe, H Germany","doi":"10.5445/IR/1000132263","DOIUrl":"https://doi.org/10.5445/IR/1000132263","url":null,"abstract":"We consider the $(1 + 1)$-dimensional quasilinear wave equation $g(x)w_{tt} − w_{xx} + h(x)(w^3_t)_t = 0$ on $mathbb{R}timesmathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $gin L^{infty}(mathbb{R})$ is even with $gnotequiv 0$ and $h(x) = gammadelta_0(x)$ with $gammainmathbb{R}backslash{0}$ and $delta_0$ the delta distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k = frac{d^2}{dx^2}-k^2omega^2g$ on $L^2(mathbb{R})$ for all $kin 2mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitely given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78098018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $$dot{B}^{2,1}_{frac{d}{2}}(mathbb{R}^d) times dot{B}^{2,1}_{frac{d}{2}-2}(mathbb{R}^d)$$ for $ d geq 3 $. Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.
我们证明了$ d geq 3 $在标度临界空间$$dot{B}^{2,1}_{frac{d}{2}}(mathbb{R}^d) times dot{B}^{2,1}_{frac{d}{2}-2}(mathbb{R}^d)$$上具有非一般二次非线性和小初始数据的导数双谐波方程的整体存在性。由于解具有较高的初始数据正则性，我们得到了一类目标流形(包括球面)双调和波映射方程的小数据全局正则性结果。
{"title":"Global results for a Cauchy problem related to biharmonic wave maps","authors":"Tobias Schmid","doi":"10.5445/IR/1000130150","DOIUrl":"https://doi.org/10.5445/IR/1000130150","url":null,"abstract":"We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $$dot{B}^{2,1}_{frac{d}{2}}(mathbb{R}^d) times dot{B}^{2,1}_{frac{d}{2}-2}(mathbb{R}^d)$$ for $ d geq 3 $. Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83261233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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