We consider the minimization of an average distance functional defined on a two-dimensional domain Ω with an Euler elastica penalization associated with ∂Ω, the boundary of Ω. The average distance is given by ∫ Ω dist(x, ∂Ω) dx where p ≥ 1 is a given parameter, and dist(x, ∂Ω) is the Hausdorff distance between {x} and ∂Ω. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ∂Ω, which is proportional to the integrated squared curvature defined on ∂Ω, as given by λ ∫ ∂Ω κ∂Ω dH x∂Ω, where κ∂Ω denotes the (signed) curvature of ∂Ω and λ > 0 denotes a penalty constant. The domain Ω is allowed to vary among compact, convex sets of R2 with Hausdorff dimension equal to 2. Under no a priori assumptions on the regularity of the boundary ∂Ω, we prove the existence of minimizers of Ep,λ. Moreover, we establish the C1,1-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.
{"title":"The average-distance problem with an Euler elastica penalization","authors":"Q. Du, Xinran Lu, Chongzeng Wang","doi":"10.4171/ifb/470","DOIUrl":"https://doi.org/10.4171/ifb/470","url":null,"abstract":"We consider the minimization of an average distance functional defined on a two-dimensional domain Ω with an Euler elastica penalization associated with ∂Ω, the boundary of Ω. The average distance is given by ∫ Ω dist(x, ∂Ω) dx where p ≥ 1 is a given parameter, and dist(x, ∂Ω) is the Hausdorff distance between {x} and ∂Ω. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ∂Ω, which is proportional to the integrated squared curvature defined on ∂Ω, as given by λ ∫ ∂Ω κ∂Ω dH x∂Ω, where κ∂Ω denotes the (signed) curvature of ∂Ω and λ > 0 denotes a penalty constant. The domain Ω is allowed to vary among compact, convex sets of R2 with Hausdorff dimension equal to 2. Under no a priori assumptions on the regularity of the boundary ∂Ω, we prove the existence of minimizers of Ep,λ. Moreover, we establish the C1,1-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85465820","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}
This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimization steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points.
{"title":"Longest minimal length partitions","authors":"Beniamin Bogosel, Édouard Oudet","doi":"10.4171/ifb/468","DOIUrl":"https://doi.org/10.4171/ifb/468","url":null,"abstract":"This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimization steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532054","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 study the equations of overdamped motion of an inextensible triod with three fixed ends and a free junction under the action of gravity. The problem can be expressed as a system of PDE that involves unknown Lagrange multipliers and non-standard boundary conditions related to the freely moving junction. It can also be formally interpreted as a gradient flow of the potential energy on a certain submanifold of the Otto-Wasserstein space of probability measures. We prove global existence of generalized solutions to this problem.
{"title":"Overdamped dynamics of a falling inextensible network: Existence of solutions","authors":"Ayk Telciyan, D. Vorotnikov","doi":"10.4171/ifb/492","DOIUrl":"https://doi.org/10.4171/ifb/492","url":null,"abstract":"We study the equations of overdamped motion of an inextensible triod with three fixed ends and a free junction under the action of gravity. The problem can be expressed as a system of PDE that involves unknown Lagrange multipliers and non-standard boundary conditions related to the freely moving junction. It can also be formally interpreted as a gradient flow of the potential energy on a certain submanifold of the Otto-Wasserstein space of probability measures. We prove global existence of generalized solutions to this problem.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78340465","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 formulate, analyse and numerically simulate what are arguably the two simplest Stokes-flow free boundary problems relevant to tissue growth, extending the classical Stokes free boundary problem by incorporating (i) a volumetric source (the nutrient-rich case) and (ii) a volumetric sink, a surface source and surface compression (the nutrient-poor case). Both two- and three-dimensional cases are considered. A number of phenomena are identified and characterised thereby, most notably a buckling-associated instability in case (ii).
{"title":"Free boundary problems for Stokes flow, with applications to the growth of biological tissues","authors":"John King, C. Venkataraman","doi":"10.4171/ifb/459","DOIUrl":"https://doi.org/10.4171/ifb/459","url":null,"abstract":"We formulate, analyse and numerically simulate what are arguably the two simplest Stokes-flow free boundary problems relevant to tissue growth, extending the classical Stokes free boundary problem by incorporating (i) a volumetric source (the nutrient-rich case) and (ii) a volumetric sink, a surface source and surface compression (the nutrient-poor case). Both two- and three-dimensional cases are considered. A number of phenomena are identified and characterised thereby, most notably a buckling-associated instability in case (ii).","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74012290","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 weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478v2] for any such cluster. This extends the two-dimensional construction to the threedimensional case of surfaces meeting along triple junctions.
导出了三维三线团簇多相平均曲率流的BV解的弱-强唯一性原理。我们的证明是基于Fischer et al. [arXiv: 2003.054778 v2]最近工作意义上的梯度流校准的显式构造。这将二维结构扩展到沿三重结的曲面相遇的三维情况。
{"title":"Weak-strong uniqueness for the mean curvature flow of double bubbles","authors":"S. Hensel, Tim Laux","doi":"10.4171/ifb/484","DOIUrl":"https://doi.org/10.4171/ifb/484","url":null,"abstract":"We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478v2] for any such cluster. This extends the two-dimensional construction to the threedimensional case of surfaces meeting along triple junctions.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87817045","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}
Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulae. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical experiments in codimension 2 illustrate and complement our theoretical results.
{"title":"A convergent finite element algorithm for mean curvature flow in arbitrary codimension","authors":"Tim Binz, Bal'azs Kov'acs","doi":"10.4171/ifb/493","DOIUrl":"https://doi.org/10.4171/ifb/493","url":null,"abstract":"Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulae. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical experiments in codimension 2 illustrate and complement our theoretical results.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79509581","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 diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces. Introduction Tumor growth is nowadays one of the most active area of scientific research, especially due to the impact on the quality of life for cancer patients. Starting with the seminal work of Burton [10] and Greenspan [35], many mathematical models have been proposed to describe the complex biological and chemical processes that occur in tumor growth, with the aim of better understanding and ultimately controlling the behavior of cancer cells. In recent years, there has been a growing interest in the mathematical modelling of cancer, see for example [1, 2, 5, 9, 16, 20, 22]. Mathematical models for tumor growth may have different analytical features: in the present work, we are focusing on the subclass of continuum models, namely diffuse interface models. There are various ways to model the interaction between the tumor and the surrounding host tissue. A classical approach is to represent the interfaces between the tumor and healthy tissues as idealized surfaces of zero thickness, leading to a sharp interface description that differentiates the tumor and the surrounding host tissue cell-bycell. These sharp interface models are often difficult to analyze mathematically, and may fail when the interface undergoes a topological change. Metastasis, which is the spreading of cancer to other parts of the body, is one important example of a change of topology. In such an event, the interface can no longer be represented as a mathematical surface, and thus the sharp interface models do no longer properly describe the reality. On the other hand, diffuse interface models consider the interface between the tumor and the healthy tissues as a layer of non-infinitesimal thickness in which tumor and healthy cells can coexist. The main advantage of this approach is that the mathematical description is less sensitive to topological changes. This is the reason why recent efforts in the mathematical modeling of tumor growth have been mostly focused on diffuse interface models, see for example [15, 16, 21, 30, 33, 36, 43, 50], and their numerical simulations demonstrating complex changes in tumor morphologies due to mechanical stresses and interactions with chemical species such as nutrients or toxic agents. Regarding the recent literature on the ma
{"title":"Analysis of a tumor model as a multicomponent deformable porous medium","authors":"P. Krejčí, E. Rocca, J. Sprekels","doi":"10.4171/ifb/472","DOIUrl":"https://doi.org/10.4171/ifb/472","url":null,"abstract":"We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces. Introduction Tumor growth is nowadays one of the most active area of scientific research, especially due to the impact on the quality of life for cancer patients. Starting with the seminal work of Burton [10] and Greenspan [35], many mathematical models have been proposed to describe the complex biological and chemical processes that occur in tumor growth, with the aim of better understanding and ultimately controlling the behavior of cancer cells. In recent years, there has been a growing interest in the mathematical modelling of cancer, see for example [1, 2, 5, 9, 16, 20, 22]. Mathematical models for tumor growth may have different analytical features: in the present work, we are focusing on the subclass of continuum models, namely diffuse interface models. There are various ways to model the interaction between the tumor and the surrounding host tissue. A classical approach is to represent the interfaces between the tumor and healthy tissues as idealized surfaces of zero thickness, leading to a sharp interface description that differentiates the tumor and the surrounding host tissue cell-bycell. These sharp interface models are often difficult to analyze mathematically, and may fail when the interface undergoes a topological change. Metastasis, which is the spreading of cancer to other parts of the body, is one important example of a change of topology. In such an event, the interface can no longer be represented as a mathematical surface, and thus the sharp interface models do no longer properly describe the reality. On the other hand, diffuse interface models consider the interface between the tumor and the healthy tissues as a layer of non-infinitesimal thickness in which tumor and healthy cells can coexist. The main advantage of this approach is that the mathematical description is less sensitive to topological changes. This is the reason why recent efforts in the mathematical modeling of tumor growth have been mostly focused on diffuse interface models, see for example [15, 16, 21, 30, 33, 36, 43, 50], and their numerical simulations demonstrating complex changes in tumor morphologies due to mechanical stresses and interactions with chemical species such as nutrients or toxic agents. Regarding the recent literature on the ma","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81233740","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}
Les problemes de partition minimale consistent a determiner une partition d’un domaine en un nombre donne de composantes de maniere a minimiser un critere geometrique. Dans les champs d’application tels que le traitement d’images et la mecanique des milieux continus, il est courant d’incorporer dans cet objectif une energie d’interface qui prend en compte les longueurs des interfaces entre composantes. Ce travail est focalise sur le traitement theorique et numerique de problemes de partition minimale avec energie d’interface. L’approche consideree est basee sur une approximation par Gamma-convergence et des techniques de dualite.
{"title":"Variational approximation of interface energies and applications","authors":"Mohammed S. M. Zabiba","doi":"10.4171/IFB/450","DOIUrl":"https://doi.org/10.4171/IFB/450","url":null,"abstract":"Les problemes de partition minimale consistent a determiner une partition d’un domaine en un nombre donne de composantes de maniere a minimiser un critere geometrique. Dans les champs d’application tels que le traitement d’images et la mecanique des milieux continus, il est courant d’incorporer dans cet objectif une energie d’interface qui prend en compte les longueurs des interfaces entre composantes. Ce travail est focalise sur le traitement theorique et numerique de problemes de partition minimale avec energie d’interface. L’approche consideree est basee sur une approximation par Gamma-convergence et des techniques de dualite.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81933612","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}
Spatial segregation occurs in population dynamics when k species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competitiondiffusion system of k differential equations −∆ui(x) = −μui(x) ∑ j 6=i uj(x) i = 1, ..., k in a domain D with appropriate boundary conditions. Any ui represents a population density and the parameter μ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as μ −→ +∞ on a planar domain for any number of species. If k is even we show that some limiting configurations are strictly connected to the solution of a Dirichlet problem for the Laplace equation. 2010 Mathematics Subject Classification: Primary 35Bxx, 35J47; Secondary 92D25.
在种群动态中，当k个物种以高度竞争的方式相互作用时，就会发生空间分离。作为研究这一现象的模型，我们考虑了k微分方程的竞争扩散系统-∆ui(x) = - μui(x)∑j 6=i uj(x) i = 1，…，在具有适当边界条件的定义域D中的k。ui表示种群密度，参数μ决定种群之间的相互作用强度。本文的目的是研究平面上任意数量物种的极限位形μ−→+∞的几何性质。如果k是偶数，我们证明了一些极限构型与拉普拉斯方程的狄利克雷问题的解是严格相关的。2010数学学科分类:初级35Bxx、35J47;二次92 d25。
{"title":"Some remarks on segregation of $k$ species in strongly competing systems","authors":"F. Lanzara, Eugenio Montefusco","doi":"10.4171/ifb/458","DOIUrl":"https://doi.org/10.4171/ifb/458","url":null,"abstract":"Spatial segregation occurs in population dynamics when k species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competitiondiffusion system of k differential equations −∆ui(x) = −μui(x) ∑ j 6=i uj(x) i = 1, ..., k in a domain D with appropriate boundary conditions. Any ui represents a population density and the parameter μ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as μ −→ +∞ on a planar domain for any number of species. If k is even we show that some limiting configurations are strictly connected to the solution of a Dirichlet problem for the Laplace equation. 2010 Mathematics Subject Classification: Primary 35Bxx, 35J47; Secondary 92D25.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80551184","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 prove the higher integrability of the gradient for local minimizers of the thermal insulation problem, an analogue of De Giorgi’s conjecture for the Mumford-Shah functional. We deduce that the singular part of the free boundary has Hausdorff dimension strictly less than n− 1. AMS Subject Classifications: 35R35, 35J20, 49N60, 49Q20.
{"title":"Higher integrability of the gradient for the thermal insulation problem","authors":"Camille Labourie, Emmanouil Milakis","doi":"10.4171/ifb/481","DOIUrl":"https://doi.org/10.4171/ifb/481","url":null,"abstract":"We prove the higher integrability of the gradient for local minimizers of the thermal insulation problem, an analogue of De Giorgi’s conjecture for the Mumford-Shah functional. We deduce that the singular part of the free boundary has Hausdorff dimension strictly less than n− 1. AMS Subject Classifications: 35R35, 35J20, 49N60, 49Q20.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89881352","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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