{"title":"A diffuse interface model of tumour evolution under a finite elastic confinement","authors":"Abramo Agosti, Riccardo Bardin, Pasquale Ciarletta, Maurizio Grasselli","doi":"10.4171/ifb/520","DOIUrl":"https://doi.org/10.4171/ifb/520","url":null,"abstract":"Diffuse interface models have gained a growing interest in cancer research for their ability to investigate the mechano-biological features during tumour progression and to provide simulation tools for personalised anti-cancer strategies at an affordable computational cost. Here we propose a diffuse interface model for tumour evolution which accounts for an interfacial structure mimicking a finite elastic confinement at the tumour boundary, possibly due either to a localised elastic stress induced by host tissue displacements, or collagen remodelling in the peritumoural area. This model consists of a partial differential equation of the Cahn–Hilliard type, with degenerate mobility, single-well potential, and an elastic non-local term acting as the effect of a membrane confinement in the chemical potential. Using mixture theory, we derive the corresponding governing equations from thermodynamic principles based on realistic physical and biological assumptions. First, we introduce a suitable regularised problem in order to deal with the degeneracy set of the mobility and the singularity of the potential. For this problem we find a weak solution and provide a regularity result. Then, we establish suitable a priori estimates which are uniform with respect to the regularisation parameters. Passing to the limit in the regularised problem, we prove existence results for different classes of weak solutions to the original problem. Finally, we propose a continuous Galerkin Finite-Element discretisation of the problem, where the positivity of the discrete solution is enforced through a variational inequality. Numerical simulations in a two-dimensional domain are also discussed in three test cases for illustrative purposes.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"27 1","pages":"27-64"},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ems.press/content/serial-article-files/47647","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147910517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Effective interface conditions for a porous medium type problem","authors":"Giorgia Ciavolella, Noemi David, Alexandre Poulain","doi":"10.4171/ifb/505","DOIUrl":"https://doi.org/10.4171/ifb/505","url":null,"abstract":"Motivated by biological applications on tumour invasion through thin membranes, we study a porous medium type equation where the density of the cell population evolves under Darcy’s law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem–Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator, which allows us to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"26 2","pages":"161-188"},"PeriodicalIF":0.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ems.press/content/serial-article-files/39265","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147896345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantitative convergence of the ``bulk'' free boundary in an oscillatory obstacle problem","authors":"Farhan Abedin, William M. Feldman","doi":"10.4171/ifb/501","DOIUrl":"https://doi.org/10.4171/ifb/501","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"48 24","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138952610","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}
{"title":"Embeddedness of liquid-vapour interfaces in stable equilibrium","authors":"Costante Bellettini","doi":"10.4171/ifb/490","DOIUrl":"https://doi.org/10.4171/ifb/490","url":null,"abstract":"We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g., water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g., air), it may come into contact with solid supports (e.g., a container), and it is subject to the action of an analytic potential field (e.g., gravity). The region occupied by the liquid is described as a set of locally finite perimeter (Caccioppoli set) in $R^3$; no a priori regularity assumption is made on its boundary. The (twofold) scope in this note is to propose a weakest possible set of mathematical assumptions that sensibly describe a condition of stable equilibrium for the liquid-vapour interface (the capillary surface), and to infer from those that this interface is a smoothly embedded analytic surface. (The liquid-solid-vapour junction, or free boundary, can be present but is not analysed here.) The result relies fundamentally on the recent varifold regularity theory developed by the author and Wickramasekera, and on the identification of a suitable formulation of the stability condition.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666551","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}
{"title":"Error estimate for classical solutions to the heat equation in a moving thin domain and its limit equation","authors":"Tatsu-Hiko Miura","doi":"10.4171/ifb/499","DOIUrl":"https://doi.org/10.4171/ifb/499","url":null,"abstract":"We consider the Neumann-type problem of the heat equation in a moving thin domain around a given closed moving hypersurface. The main result of this paper is an error estimate in the sup-norm for classical solutions to the thin domain problem and a limit equation on the moving hypersurface which appears in the thin-film limit of the heat equation. To prove the error estimate, we show a uniform a priori estimate for a classical solution to the thin domain problem based on the maximum principle. Moreover, we construct a suitable approximate solution to the thin domain problem from a classical solution to the limit equation based on an asymptotic expansion of the thin domain problem and apply the uniform a priori estimate to the difference of the approximate solution and a classical solution to the thin domain problem.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918853","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}
{"title":"A novel finite element approximation of anisotropic curve shortening flow","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.4171/ifb/500","DOIUrl":"https://doi.org/10.4171/ifb/500","url":null,"abstract":"We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal $H^1$-error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252054","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}
{"title":"Classification of global solutions of a free boundary problem in the plane","authors":"Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci","doi":"10.4171/ifb/494","DOIUrl":"https://doi.org/10.4171/ifb/494","url":null,"abstract":"We classify non-trivial, non-negative, positively homogeneous solutions of the equation $$ Delta u=gamma u^{gamma-1} $$ in the plane. The problem is motivated by the analysis of the classical Alt–Phillips free boundary problem, but considered here with negative exponents $gamma$. The proof relies on several bespoke results for ordinary differential equations.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690247","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}
{"title":"Instantaneous convexity breaking for the quasi-static droplet model","authors":"Albert Chau, B. Weinkove","doi":"10.4171/ifb/498","DOIUrl":"https://doi.org/10.4171/ifb/498","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87993911","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}
{"title":"Fully nonlinear free transmission problems","authors":"Edgard A. Pimentel, Makson S. Santos","doi":"10.4171/ifb/489","DOIUrl":"https://doi.org/10.4171/ifb/489","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"43 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79427418","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}