The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $pneq 2$, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$.
{"title":"L p positivity preservation and self-adjointness of Schrödinger operators on incomplete Riemannian manifolds","authors":"Andrea Bisterzo, Giona Veronelli","doi":"10.1017/prm.2024.64","DOIUrl":"https://doi.org/10.1017/prm.2024.64","url":null,"abstract":"The aim of this paper is to prove a qualitative property, namely the <jats:italic>preservation of positivity</jats:italic>, for Schrödinger-type operators acting on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000647_inline2.png\"/> </jats:alternatives> </jats:inline-formula> functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case <jats:inline-formula> <jats:alternatives> <jats:tex-math>$pneq 2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000647_inline3.png\"/> </jats:alternatives> </jats:inline-formula>, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000647_inline4.png\"/> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"48 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.
{"title":"Distribution in homology classes and discrete fractal dimension","authors":"James Everitt, Richard Sharp","doi":"10.1017/prm.2024.67","DOIUrl":"https://doi.org/10.1017/prm.2024.67","url":null,"abstract":"In this note, we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.
{"title":"Spectral asymptotics for linear elasticity: the case of mixed boundary conditions","authors":"Matteo Capoferri, Isabel Mann","doi":"10.1017/prm.2024.65","DOIUrl":"https://doi.org/10.1017/prm.2024.65","url":null,"abstract":"We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"195 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(operatorname {div};Omega )cap H_0(operatorname {curl};Omega )$ into $H^1(Omega )$.
{"title":"A global higher regularity result for the static relaxed micromorphic model on smooth domains","authors":"Dorothee Knees, Sebastian Owczarek, Patrizio Neff","doi":"10.1017/prm.2024.63","DOIUrl":"https://doi.org/10.1017/prm.2024.63","url":null,"abstract":"We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$H(operatorname {div};Omega )cap H_0(operatorname {curl};Omega )$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000635_inline1.png\"/> </jats:alternatives> </jats:inline-formula> into <jats:inline-formula> <jats:alternatives> <jats:tex-math>$H^1(Omega )$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000635_inline2.png\"/> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"32 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.
{"title":"Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence","authors":"Yan-Xia Feng, Wan-Tong Li, Fei-Ying Yang","doi":"10.1017/prm.2024.62","DOIUrl":"https://doi.org/10.1017/prm.2024.62","url":null,"abstract":"<p>Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (<span>J. Differ. Equ.</span> 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"41 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a flat model structure on the category ${_{mathcal {Q},,R}mathsf {Mod}}$ of additive functors from a small preadditive category $mathcal {Q}$ satisfying certain conditions to the module category ${_{R}mathsf {Mod}}$ over an associative ring $R$, whose homotopy category is the $mathcal {Q}$-shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in ${_{mathcal {Q},,R}mathsf {Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $mathcal {Q}$, and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.
{"title":"Flat model structures and Gorenstein objects in functor categories","authors":"Zhenxing Di, Liping Li, Li Liang, Yajun Ma","doi":"10.1017/prm.2024.60","DOIUrl":"https://doi.org/10.1017/prm.2024.60","url":null,"abstract":"<p>We construct a flat model structure on the category <span><span><span data-mathjax-type=\"texmath\"><span>${_{mathcal {Q},,R}mathsf {Mod}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline1.png\"/></span></span> of additive functors from a small preadditive category <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {Q}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline2.png\"/></span></span> satisfying certain conditions to the module category <span><span><span data-mathjax-type=\"texmath\"><span>${_{R}mathsf {Mod}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline3.png\"/></span></span> over an associative ring <span><span><span data-mathjax-type=\"texmath\"><span>$R$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline4.png\"/></span></span>, whose homotopy category is the <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {Q}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline5.png\"/></span></span>-shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring <span><span><span data-mathjax-type=\"texmath\"><span>$R$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline6.png\"/></span></span>, an object in <span><span><span data-mathjax-type=\"texmath\"><span>${_{mathcal {Q},,R}mathsf {Mod}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline7.png\"/></span></span> is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {Q}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513183242284-0041:S030821052400060X:S030821052400060X_inline8.png\"/></span></span>, and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"123 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $mathcal {L}_s u=u^{frac {Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $mathcal {L}_s$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.
{"title":"Optimal decay for solutions of nonlocal semilinear equations with critical exponent in homogeneous groups","authors":"Nicola Garofalo, Annunziata Loiudice, Dimiter Vassilev","doi":"10.1017/prm.2024.58","DOIUrl":"https://doi.org/10.1017/prm.2024.58","url":null,"abstract":"<p>In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {L}_s u=u^{frac {Q+2s}{Q-2s}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513091116353-0352:S0308210524000581:S0308210524000581_inline1.png\"/></span></span> in a homogeneous Lie group, where <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {L}_s$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240513091116353-0352:S0308210524000581:S0308210524000581_inline2.png\"/></span></span> represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"206 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$-dimensional unit sphere $mathbb {S}^{2n+1}$ admitting a Sasakian structure $(varphi,,xi,,eta,,g)$ for $nge 3$, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-varphi h$ is semi-parallel, which is introduced as a natural extension of $C$-parallel second fundamental form $h$. Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.
{"title":"On conformally flat minimal Legendrian submanifolds in the unit sphere","authors":"Cece Li, Cheng Xing, Jiabin Yin","doi":"10.1017/prm.2024.57","DOIUrl":"https://doi.org/10.1017/prm.2024.57","url":null,"abstract":"This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(2n+1)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline1.png\"/> </jats:alternatives> </jats:inline-formula>-dimensional unit sphere <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathbb {S}^{2n+1}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline2.png\"/> </jats:alternatives> </jats:inline-formula> admitting a Sasakian structure <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(varphi,,xi,,eta,,g)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline3.png\"/> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$nge 3$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline4.png\"/> </jats:alternatives> </jats:inline-formula>, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor <jats:inline-formula> <jats:alternatives> <jats:tex-math>$K:=-varphi h$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline5.png\"/> </jats:alternatives> </jats:inline-formula> is semi-parallel, which is introduced as a natural extension of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$C$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline6.png\"/> </jats:alternatives> </jats:inline-formula>-parallel second fundamental form <jats:inline-formula> <jats:alternatives> <jats:tex-math>$h$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052400057X_inline7.png\"/> </jats:alternatives> </jats:inline-formula>. Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"66 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.
{"title":"Catenaries and minimal surfaces of revolution in hyperbolic space","authors":"Luiz C. B. da Silva, Rafael López","doi":"10.1017/prm.2024.56","DOIUrl":"https://doi.org/10.1017/prm.2024.56","url":null,"abstract":"We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant [ left{begin{array}{@{}ll} u_t=Delta u^{m}-nablacdot(unabla v),quad & xin Omega,quad t>0, v_t=Delta v-uv,quad & xin Omega,quad t>0 end{array}right. ]in a bounded domain $Omega subset mathbb {R}^N(N=3,,4,,5)$ with smooth boundary $partial Omega$. It is shown that if $m>max {1,,frac {3N-2}{2N+2}}$, for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(bar {u}_0,,0)$ in an appropriate sense as $trightarrow infty$, where $bar {u}_0=frac {1}{|Omega |}int _Omega u_0$. This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m>frac {3N-2}{2N}$
{"title":"Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant","authors":"Minghua Zhang, Chunlai Mu, Hongying Yang","doi":"10.1017/prm.2024.54","DOIUrl":"https://doi.org/10.1017/prm.2024.54","url":null,"abstract":"This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant <jats:disp-formula> <jats:alternatives> <jats:tex-math>[ left{begin{array}{@{}ll} u_t=Delta u^{m}-nablacdot(unabla v),quad & xin Omega,quad t>0, v_t=Delta v-uv,quad & xin Omega,quad t>0 end{array}right. ]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000544_eqnU1.png\"/> </jats:alternatives> </jats:disp-formula>in a bounded domain <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Omega subset mathbb {R}^N(N=3,,4,,5)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline1.png\"/> </jats:alternatives> </jats:inline-formula> with smooth boundary <jats:inline-formula> <jats:alternatives> <jats:tex-math>$partial Omega$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline2.png\"/> </jats:alternatives> </jats:inline-formula>. It is shown that if <jats:inline-formula> <jats:alternatives> <jats:tex-math>$m>max {1,,frac {3N-2}{2N+2}}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline3.png\"/> </jats:alternatives> </jats:inline-formula>, for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium <jats:inline-formula> <jats:alternatives> <jats:tex-math>$(bar {u}_0,,0)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline4.png\"/> </jats:alternatives> </jats:inline-formula> in an appropriate sense as <jats:inline-formula> <jats:alternatives> <jats:tex-math>$trightarrow infty$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline5.png\"/> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:tex-math>$bar {u}_0=frac {1}{|Omega |}int _Omega u_0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline6.png\"/> </jats:alternatives> </jats:inline-formula>. This result not only partly extends the previous global boundedness result in Fan and Jin (<jats:italic>J. Math. Phys.</jats:italic>58 (2017), 011503) and Wang and Xiang (<jats:italic>Z. Angew. Math. Phys.</jats:italic>66 (2015), 3159–3179) to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$m>frac {3N-2}{2N}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000544_inline7.png\"/> </jats:altern","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"77 3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
of additive functors from a small preadditive category $mathcal {Q}$
satisfying certain conditions to the module category ${_{R}mathsf {Mod}}$
over an associative ring $R$
, whose homotopy category is the $mathcal {Q}$
-shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$
, an object in ${_{mathcal {Q},,R}mathsf {Mod}}$
is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $mathcal {Q}$
, and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.
in a homogeneous Lie group, where $mathcal {L}_s$
represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.
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