Pub Date : 2020-01-01DOI: 10.4310/maa.2020.v27.n4.a1
J. Benamou, W. IJzerman, G. Rukhaia
The point source far field reflector design problem is one of the main classic optimal transport problems with a non-euclidean displacement cost [Wang, 2004] [Glimm and Oliker, 2003]. This work describes the use of Entropic Optimal Transport and the associated Sinkhorn algorithm [Cuturi, 2013] to solve it numerically. As the reflector modelling is based on the Kantorovich potentials , several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of [Berman, 2017] and in particular the imposed discretization requirements therein. Secondly, the correction of the Entropic bias induced by the Entropic OT, as discussed in particular in [Ramdas et al., 2017] [Genevay et al., 2018] [Feydy et al., 2018], is another important tool to achieve reasonable results. The paper reviews the necessary mathematical and numerical tools needed to produce and discuss the obtained numerical results. We find that Sinkhorn algorithm may be adapted, at least in simple academic cases, to the resolution of the far field reflector problem. Sinkhorn canonical extension to continuous potentials is needed to generate continuous reflector approximations. The use of Sinkhorn divergences [Feydy et al., 2018] is useful to mitigate the entropic bias.
点源远场反射器设计问题是非欧氏位移代价的经典最优输运问题之一[Wang, 2004] [Glimm and Oliker, 2003]。这项工作描述了使用熵最优传输和相关的Sinkhorn算法[Cuturi, 2013]来数值解决它。由于反射器建模是基于坎托罗维奇势，因此产生了几个问题。首先，关于离散熵近似的收敛性，这里我们遵循[Berman, 2017]最近的工作，特别是其中施加的离散化要求。其次，对Entropic OT引起的熵偏的校正，如[Ramdas et al.， 2017] [Genevay et al.， 2018] [Feydy et al.， 2018]中所讨论的，是获得合理结果的另一个重要工具。本文回顾了产生所获得的数值结果所需的必要的数学和数值工具，并对其进行了讨论。我们发现，至少在简单的学术案例中，Sinkhorn算法可以适用于解决远场反射器问题。为了得到连续反射器近似，需要对连续势进行Sinkhorn正则扩展。使用Sinkhorn散度[Feydy等人，2018]有助于减轻熵偏差。
{"title":"An entropic optimal transport numerical approach to the reflector problem","authors":"J. Benamou, W. IJzerman, G. Rukhaia","doi":"10.4310/maa.2020.v27.n4.a1","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n4.a1","url":null,"abstract":"The point source far field reflector design problem is one of the main classic optimal transport problems with a non-euclidean displacement cost [Wang, 2004] [Glimm and Oliker, 2003]. This work describes the use of Entropic Optimal Transport and the associated Sinkhorn algorithm [Cuturi, 2013] to solve it numerically. As the reflector modelling is based on the Kantorovich potentials , several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of [Berman, 2017] and in particular the imposed discretization requirements therein. Secondly, the correction of the Entropic bias induced by the Entropic OT, as discussed in particular in [Ramdas et al., 2017] [Genevay et al., 2018] [Feydy et al., 2018], is another important tool to achieve reasonable results. The paper reviews the necessary mathematical and numerical tools needed to produce and discuss the obtained numerical results. We find that Sinkhorn algorithm may be adapted, at least in simple academic cases, to the resolution of the far field reflector problem. Sinkhorn canonical extension to continuous potentials is needed to generate continuous reflector approximations. The use of Sinkhorn divergences [Feydy et al., 2018] is useful to mitigate the entropic bias.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489472","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n3.a3
Qilin Yang, D. Zaffran
{"title":"On Fano threefolds with semi-free $mathbb{C}^{ast}$-actions, I","authors":"Qilin Yang, D. Zaffran","doi":"10.4310/maa.2020.v27.n3.a3","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n3.a3","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489402","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n2.a2
J. Tervo, M. Herty
{"title":"On approximation of a hyper-singular transport operator and existence of solutions","authors":"J. Tervo, M. Herty","doi":"10.4310/maa.2020.v27.n2.a2","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n2.a2","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489060","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n1.a2
Mohamad Houda, P. Lescot
We determine the isovectors of the backward heat equation with quadratic potential term in the space variable. This generalize the calculations of Lescot-Zambrini (cf.[4]). These results first appeared in the first author’s PhD thesis (Rouen, 2013). 1. Backward Equation The fundamental equation of Euclidean Quantum Mechanics of Zambrini is the backward heat equation with potential V (t, q): θ ∂η u ∂t = − θ 2 ∂η u ∂q2 + V (t, q)η V u (C (V ) 1 ) η u (0, q) = u(q) where t represents the time variable, q the space variable, and θ is a real parameter, strictly positive (in physics, θ = √ ~). This equation is not well–posed in general, but existence and uniqueness of a solution are insured whenever the initial condition u belongs to the set of analytic vectors for the operator appearing on the right–hand side of the equation (see e.g. [1], Lemma 4, p. 429). We shall denote by ηu(t, q) = η 0 u(t, q) the solution of the backward heat equation with null potential : θ 2 ∂ηu ∂t = − θ 2 ∂ηu ∂q2 (C (0) 1 ) ηu(0, q) = η 0 u(0, q) = u(q). The authors wish to thank the referees for constructive comments. 1.1. Generalization of results due to Lescot-Zambrini (cf.[4]). Theorem 1.1. Let a(t), b(t) and c(t) be continuous functions and V (t, q) = a(t)q+ b(t)q + c(t). For all initial conditions u, the solution η u (t, q) of: θ ∂η u ∂t = − 4 2 ∂η u ∂q2 + V (t, q)η u (C (V ) 1 ) such that η u (0, q) = u(q) is given by: η u (t, q) = φ1(t, q) ηu (φ2(t, q), φ3(t, q)) , (1.1) Date: 01 November 2019. 1 2 MOHAMAD HOUDA & PAUL LESCOT where: ηu(t, q) = η 0 u(t, q) and φ1(t, q), φ2(t, q), φ3(t, q) only depend on a(t), b(t) and c(t) (via formulas (1.7), (1.8), (1.9), (1.11), (1.12), (1.13) and (1.14)). Proof. We shall make formula 1.1 explicit with the following initial conditions: φ1(0, q) = 1 φ2(0, q) = 0 φ3(0, q) = q. In this case : η u (0, q) = ηu(0, q) = u(q). We now differentiate formula 1.1 with respect to t and q; ηu and its derivatives will be taken in (φ2(t, q), φ3(t, q)). ∂η u ∂t = ∂φ1 ∂t ηu + φ1( ∂ηu ∂t ∂φ2 ∂t + ∂ηu ∂q ∂φ3 ∂t ) , ∂η u ∂q = ∂φ1 ∂q ηu + φ1( ∂ηu ∂t ∂φ2 ∂q + ∂ηu ∂q ∂φ3 ∂q ) , ∂η u ∂q2 = ∂φ1 ∂q2 ηu + 2 ∂φ1 ∂q ( ∂ηu ∂t ∂φ2 ∂q + ∂ηu ∂q ∂φ3 ∂q ) + φ1 [ ∂ηu ∂t2 ( ∂φ2 ∂q ) 2 + ∂ηu ∂q2 ( ∂φ3 ∂q ) 2 + 2 ∂ηu ∂t∂q ∂φ2 ∂q ∂φ3 ∂q + ∂ηu ∂t ∂φ2 ∂q2 + ∂ηu ∂q ∂φ3 ∂q2 ] and V (t, q)η u = a(t)q 2 φ1 ηu + b(t)q φ1 ηu + c(t) φ1 ηu. So, it’s enough to have: θφ1 ∂φ2 ∂t − θφ1( ∂φ3 ∂q ) 2 + θ 2 (2 ∂φ1 ∂q ∂φ2 ∂q + ∂φ2 ∂q2 ) = 0 (1.2) θ 2 φ1( ∂φ2 ∂q ) 2 = 0 (1.3) θφ1 ∂φ3 ∂t + θ 2 (2 ∂φ1 ∂q ∂φ3 ∂q + φ1 ∂φ3 ∂q2 ) = 0 (1.4) θ ∂φ2 ∂q ∂φ3 ∂q = 0 (1.5) θ ∂φ1 ∂t + θ 2 ∂φ1 ∂q2 − φ1 ( a(t)q + b(t)q + c(t) ) = 0 . (1.6) Two calculates concerned the isovectors of backward heat equation with potential term 3 As φ1(0, q) = 1, the equation (1.3) gives us: (2 ∂q ) 2 = 0, then φ2 = φ2(t) and (1.5) is then automatically satisfied. So, the equation (1.2) implies: φ1(t, q)( ∂φ2 ∂t − (3 ∂q )) = 0, then for all (t, q) : ∂φ2 ∂t = ( ∂φ3 ∂q ) 2
{"title":"Two computations concerning the isovectors of the backward heat equation with quadratic potential","authors":"Mohamad Houda, P. Lescot","doi":"10.4310/maa.2020.v27.n1.a2","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n1.a2","url":null,"abstract":"We determine the isovectors of the backward heat equation with quadratic potential term in the space variable. This generalize the calculations of Lescot-Zambrini (cf.[4]). These results first appeared in the first author’s PhD thesis (Rouen, 2013). 1. Backward Equation The fundamental equation of Euclidean Quantum Mechanics of Zambrini is the backward heat equation with potential V (t, q): θ ∂η u ∂t = − θ 2 ∂η u ∂q2 + V (t, q)η V u (C (V ) 1 ) η u (0, q) = u(q) where t represents the time variable, q the space variable, and θ is a real parameter, strictly positive (in physics, θ = √ ~). This equation is not well–posed in general, but existence and uniqueness of a solution are insured whenever the initial condition u belongs to the set of analytic vectors for the operator appearing on the right–hand side of the equation (see e.g. [1], Lemma 4, p. 429). We shall denote by ηu(t, q) = η 0 u(t, q) the solution of the backward heat equation with null potential : θ 2 ∂ηu ∂t = − θ 2 ∂ηu ∂q2 (C (0) 1 ) ηu(0, q) = η 0 u(0, q) = u(q). The authors wish to thank the referees for constructive comments. 1.1. Generalization of results due to Lescot-Zambrini (cf.[4]). Theorem 1.1. Let a(t), b(t) and c(t) be continuous functions and V (t, q) = a(t)q+ b(t)q + c(t). For all initial conditions u, the solution η u (t, q) of: θ ∂η u ∂t = − 4 2 ∂η u ∂q2 + V (t, q)η u (C (V ) 1 ) such that η u (0, q) = u(q) is given by: η u (t, q) = φ1(t, q) ηu (φ2(t, q), φ3(t, q)) , (1.1) Date: 01 November 2019. 1 2 MOHAMAD HOUDA & PAUL LESCOT where: ηu(t, q) = η 0 u(t, q) and φ1(t, q), φ2(t, q), φ3(t, q) only depend on a(t), b(t) and c(t) (via formulas (1.7), (1.8), (1.9), (1.11), (1.12), (1.13) and (1.14)). Proof. We shall make formula 1.1 explicit with the following initial conditions: φ1(0, q) = 1 φ2(0, q) = 0 φ3(0, q) = q. In this case : η u (0, q) = ηu(0, q) = u(q). We now differentiate formula 1.1 with respect to t and q; ηu and its derivatives will be taken in (φ2(t, q), φ3(t, q)). ∂η u ∂t = ∂φ1 ∂t ηu + φ1( ∂ηu ∂t ∂φ2 ∂t + ∂ηu ∂q ∂φ3 ∂t ) , ∂η u ∂q = ∂φ1 ∂q ηu + φ1( ∂ηu ∂t ∂φ2 ∂q + ∂ηu ∂q ∂φ3 ∂q ) , ∂η u ∂q2 = ∂φ1 ∂q2 ηu + 2 ∂φ1 ∂q ( ∂ηu ∂t ∂φ2 ∂q + ∂ηu ∂q ∂φ3 ∂q ) + φ1 [ ∂ηu ∂t2 ( ∂φ2 ∂q ) 2 + ∂ηu ∂q2 ( ∂φ3 ∂q ) 2 + 2 ∂ηu ∂t∂q ∂φ2 ∂q ∂φ3 ∂q + ∂ηu ∂t ∂φ2 ∂q2 + ∂ηu ∂q ∂φ3 ∂q2 ] and V (t, q)η u = a(t)q 2 φ1 ηu + b(t)q φ1 ηu + c(t) φ1 ηu. So, it’s enough to have: θφ1 ∂φ2 ∂t − θφ1( ∂φ3 ∂q ) 2 + θ 2 (2 ∂φ1 ∂q ∂φ2 ∂q + ∂φ2 ∂q2 ) = 0 (1.2) θ 2 φ1( ∂φ2 ∂q ) 2 = 0 (1.3) θφ1 ∂φ3 ∂t + θ 2 (2 ∂φ1 ∂q ∂φ3 ∂q + φ1 ∂φ3 ∂q2 ) = 0 (1.4) θ ∂φ2 ∂q ∂φ3 ∂q = 0 (1.5) θ ∂φ1 ∂t + θ 2 ∂φ1 ∂q2 − φ1 ( a(t)q + b(t)q + c(t) ) = 0 . (1.6) Two calculates concerned the isovectors of backward heat equation with potential term 3 As φ1(0, q) = 1, the equation (1.3) gives us: (2 ∂q ) 2 = 0, then φ2 = φ2(t) and (1.5) is then automatically satisfied. So, the equation (1.2) implies: φ1(t, q)( ∂φ2 ∂t − (3 ∂q )) = 0, then for all (t, q) : ∂φ2 ∂t = ( ∂φ3 ∂q ) 2","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488984","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n2.a1
Tujin Kim, D. Cao
{"title":"Mixed boundary value problems of the system for steady flow of heat-conducting incompressible viscous fluids with dissipative heating","authors":"Tujin Kim, D. Cao","doi":"10.4310/maa.2020.v27.n2.a1","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n2.a1","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489047","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n4.a3
Jianchun Chu, Wenshuai Jiang
. In this note, we study nodal sets of harmonic functions on Riemannian manifolds with lower Ricci curvature bound and the noncollapsing lower volume bound. Both upper and lower bound measure estimates of nodal sets are established.
{"title":"A note on nodal sets on manifolds with lower Ricci bound","authors":"Jianchun Chu, Wenshuai Jiang","doi":"10.4310/maa.2020.v27.n4.a3","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n4.a3","url":null,"abstract":". In this note, we study nodal sets of harmonic functions on Riemannian manifolds with lower Ricci curvature bound and the noncollapsing lower volume bound. Both upper and lower bound measure estimates of nodal sets are established.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489487","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n3.a2
Chunhua Wang, Jing Zhou
. We study a nonlinearly coupled Schr¨odinger equations in R N (2 ≤ N < 6) . Assume that the potentials in the system are continuous functions satisfying some suitable decay assumptions but without any symmetric properties, and the parameters in the system satisfy some restrictions. Applying the Liapunov-Schmidt reduction methods twice and combining localized energy method, we prove that the problem has infinitely many positive synchronized solutions.
{"title":"Infinitely many synchronized solutions to a nonlinearly coupled Schrödinger equations with non-symmetric potentials","authors":"Chunhua Wang, Jing Zhou","doi":"10.4310/maa.2020.v27.n3.a2","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n3.a2","url":null,"abstract":". We study a nonlinearly coupled Schr¨odinger equations in R N (2 ≤ N < 6) . Assume that the potentials in the system are continuous functions satisfying some suitable decay assumptions but without any symmetric properties, and the parameters in the system satisfy some restrictions. Applying the Liapunov-Schmidt reduction methods twice and combining localized energy method, we prove that the problem has infinitely many positive synchronized solutions.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488693","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 : 2020-01-01DOI: 10.4310/maa.2020.v27.n4.a4
M. Cullen, M. Feldman, A. Tudorascu
We prove that if the initial data is well prepared, then certain solutions to the Euler system converge to a solution of the Semi-Geostrophic system with constant Coriolis force. The main assumptions on the strong solution are the boundedness of the velocity field as well as the uniform convexity of the Legendre-Fenchel transform of the modified pressure.
{"title":"From Euler to the semi-geostrophic system: convergence under uniform convexity","authors":"M. Cullen, M. Feldman, A. Tudorascu","doi":"10.4310/maa.2020.v27.n4.a4","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n4.a4","url":null,"abstract":"We prove that if the initial data is well prepared, then certain solutions to the Euler system converge to a solution of the Semi-Geostrophic system with constant Coriolis force. The main assumptions on the strong solution are the boundedness of the velocity field as well as the uniform convexity of the Legendre-Fenchel transform of the modified pressure.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489547","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 : 2019-12-30DOI: 10.4310/maa.2022.v29.n4.a1
Moritz Reintjes, B. Temple
We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $Gamma$, with components $Gamma in L^{2p}$ and components of its Riemann curvature ${rm Riem}(Gamma)$ in $L^p$, in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, $Gamma in W^{1,p}$ (one derivative smoother than the curvature), $p>max{n/2,2}$, dimension $ngeq 2$. For Lorentzian metrics in General Relativity this implies that shock wave solutions of the Einstein-Euler equations are non-singular -- geodesic curves, locally inertial coordinates and the Newtonian limit, all exist in a classical sense, and the Einstein equations hold in the strong sense. The proof is based on an $L^p$ existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in $L^infty$, with curvature in $L^{p}$, $p>n$, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a ``geometric'' improvement of the generalized Div-Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another -- what one could take to be the ``starting assumption of geometry''.
{"title":"On the optimal regularity implied by the assumptions of geometry, I: connections on tangent bundles","authors":"Moritz Reintjes, B. Temple","doi":"10.4310/maa.2022.v29.n4.a1","DOIUrl":"https://doi.org/10.4310/maa.2022.v29.n4.a1","url":null,"abstract":"We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $Gamma$, with components $Gamma in L^{2p}$ and components of its Riemann curvature ${rm Riem}(Gamma)$ in $L^p$, in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, $Gamma in W^{1,p}$ (one derivative smoother than the curvature), $p>max{n/2,2}$, dimension $ngeq 2$. For Lorentzian metrics in General Relativity this implies that shock wave solutions of the Einstein-Euler equations are non-singular -- geodesic curves, locally inertial coordinates and the Newtonian limit, all exist in a classical sense, and the Einstein equations hold in the strong sense. The proof is based on an $L^p$ existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in $L^infty$, with curvature in $L^{p}$, $p>n$, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a ``geometric'' improvement of the generalized Div-Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another -- what one could take to be the ``starting assumption of geometry''.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42464386","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 : 2019-08-18DOI: 10.4310/maa.2021.v28.n3.a8
H. Jian, You Li, Xushan Tu
In this paper we shall prove the existence, uniqueness and global H$ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Ampere type equations which may be degenerate and singular on the boundary of convex domains. We will establish a relation of the H$ddot{o}$lder exponent for the solutions with the convexity for the domains.
{"title":"On a class of degenerate and singular Monge–Ampère equations","authors":"H. Jian, You Li, Xushan Tu","doi":"10.4310/maa.2021.v28.n3.a8","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n3.a8","url":null,"abstract":"In this paper we shall prove the existence, uniqueness and global H$ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Ampere type equations which may be degenerate and singular on the boundary of convex domains. \u0000We will establish a relation of the H$ddot{o}$lder exponent for the solutions with the convexity for the domains.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42725118","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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