A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters.
{"title":"Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations","authors":"Sébastien Court, K. Kunisch, Laurent Pfeiffer","doi":"10.4171/ifb/424","DOIUrl":"https://doi.org/10.4171/ifb/424","url":null,"abstract":"A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74003843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free-boundary. Moreover we show that at such points the only admissible frequencies are $2m-1+s$ and $2m$, $mgeq 1$.
{"title":"Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem","authors":"Maria Colombo, L. Spolaor, B. Velichkov","doi":"10.4171/ifb/452","DOIUrl":"https://doi.org/10.4171/ifb/452","url":null,"abstract":"We answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimizers $u$ of the lower dimensional obstacle problem is unique at generic point of the free-boundary. Moreover we show that at such points the only admissible frequencies are $2m-1+s$ and $2m$, $mgeq 1$.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83105986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a `kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions.
{"title":"A tractable mathematical model for tissue growth","authors":"J. Eyles, John King, V. Styles","doi":"10.4171/ifb/428","DOIUrl":"https://doi.org/10.4171/ifb/428","url":null,"abstract":"Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a `kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78721031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having $n$ microvoids of radius $varepsilon$. The constraint is added that the cavities should reach at least certain minimum areas $v_{1},...,v_{n}$ after the deformation takes place. They can be thought of as the current areas of the cavities during a quasistatic loading, the variational problem being the way to determine the state to be attained by the elastic body in a subsequent time step. It is proved that if each $v_{i}$ is smaller than the area of a disk having a certain well defined radius, which is comparable to the distance, in the reference configuration, to either the boundary of the domain or the nearest cavity (whichever is closer), then there exists a range of external loads for which the cavities opened in the body are circular in the $varepsilon rightarrow 0$ limit.
{"title":"A lower bound for the void coalescence load in nonlinearly elastic solids","authors":"Victor Canulef-Aguilar, Duvan Henao","doi":"10.4171/ifb/427","DOIUrl":"https://doi.org/10.4171/ifb/427","url":null,"abstract":"The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having $n$ microvoids of radius $varepsilon$. The constraint is added that the cavities should reach at least certain minimum areas $v_{1},...,v_{n}$ after the deformation takes place. They can be thought of as the current areas of the cavities during a quasistatic loading, the variational problem being the way to determine the state to be attained by the elastic body in a subsequent time step. It is proved that if each $v_{i}$ is smaller than the area of a disk having a certain well defined radius, which is comparable to the distance, in the reference configuration, to either the boundary of the domain or the nearest cavity (whichever is closer), then there exists a range of external loads for which the cavities opened in the body are circular in the $varepsilon rightarrow 0$ limit.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85124013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A general model covering a large variety of the so-called adhesive or cohesive, possibly also frictional, contact interfaces between visco-elastic bodies with inertia considered in a thermodynamical context is presented. A semi-implicit time discretisation which conserves energy, is numerically stable and convergent, and which advantageously decouples the system is devised. An extension to porous media with adhesive contact influenced by a diffusant concentration is devised, too.
{"title":"A general thermodynamical model for adhesive frictional contacts between viscoelastic or poro-viscoelastic bodies at small strains","authors":"Tom'avs Roub'ivcek","doi":"10.4171/IFB/420","DOIUrl":"https://doi.org/10.4171/IFB/420","url":null,"abstract":"A general model covering a large variety of the so-called adhesive or cohesive, possibly also frictional, contact interfaces between visco-elastic bodies with inertia considered in a thermodynamical context is presented. A semi-implicit time discretisation which conserves energy, is numerically stable and convergent, and which advantageously decouples the system is devised. An extension to porous media with adhesive contact influenced by a diffusant concentration is devised, too.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88078479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive a comparison principle for a degenerate elliptic partial differential equation without boundary conditions which arises naturally in optimal learning strategies. Our argument is direct and exploits the degeneracy of the differential operator to construct (logarithmically) diverging barriers.
{"title":"Well-posedness for degenerate elliptic PDE arising in optimal learning strategies","authors":"Tim Laux, J. M. Villas-Boas","doi":"10.4171/ifb/434","DOIUrl":"https://doi.org/10.4171/ifb/434","url":null,"abstract":"We derive a comparison principle for a degenerate elliptic partial differential equation without boundary conditions which arises naturally in optimal learning strategies. Our argument is direct and exploits the degeneracy of the differential operator to construct (logarithmically) diverging barriers.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76242432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 derive a model for heat diffusion in a composi te medium in which the different components are separated by thermally active interf ac s. The previous result is obtained via a concentrated capacity procedure and leads to a non-sta ntard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For suc h a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady s tate is proved. 2010 Mathematics Subject Classification: Primary 35K20; Secondary 35K90, 35B40.
{"title":"Existence, uniqueness and concentration for a system of PDEs involving the Laplace–Beltrami operator","authors":"M. Amar, R. Gianni","doi":"10.4171/IFB/416","DOIUrl":"https://doi.org/10.4171/IFB/416","url":null,"abstract":"In this paper we derive a model for heat diffusion in a composi te medium in which the different components are separated by thermally active interf ac s. The previous result is obtained via a concentrated capacity procedure and leads to a non-sta ntard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For suc h a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady s tate is proved. 2010 Mathematics Subject Classification: Primary 35K20; Secondary 35K90, 35B40.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88597897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson-Kendall-Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis.
{"title":"From individual-based mechanical models of multicellular systems to free-boundary problems","authors":"T. Lorenzi, P. Murray, M. Ptashnyk","doi":"10.4171/ifb/439","DOIUrl":"https://doi.org/10.4171/ifb/439","url":null,"abstract":"In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson-Kendall-Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85127945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable scaling limits steady states converge to solutions of some obstacle type problems. For these limiting problems we prove the onset of polarization if the total mass of protein is sufficiently small. For some variants we can even characterize precisely the critical mass for which polarization occurs.
{"title":"A bulk-surface reaction-diffusion system for cell polarization","authors":"B. Niethammer, M. Roger, J. Vel'azquez","doi":"10.4171/ifb/433","DOIUrl":"https://doi.org/10.4171/ifb/433","url":null,"abstract":"We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable scaling limits steady states converge to solutions of some obstacle type problems. For these limiting problems we prove the onset of polarization if the total mass of protein is sufficiently small. For some variants we can even characterize precisely the critical mass for which polarization occurs.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90742685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 define the entropy of Radon measures, especially the measures associated to the parabolic Allen-Cahn equation. We show that when the entropy of the initial data is small enough (less than twice of the energy of the one dimensional standing wave), the limit measure of the parabolic Allen-Cahn equation has unit density.
{"title":"On the entropy of parabolic Allen–Cahn equation","authors":"Ao Sun","doi":"10.4171/ifb/460","DOIUrl":"https://doi.org/10.4171/ifb/460","url":null,"abstract":"In this paper we define the entropy of Radon measures, especially the measures associated to the parabolic Allen-Cahn equation. We show that when the entropy of the initial data is small enough (less than twice of the energy of the one dimensional standing wave), the limit measure of the parabolic Allen-Cahn equation has unit density.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72476446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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