{"title":"Zero width limit of the heat equation on moving thin domains","authors":"Tatsuya Miura","doi":"10.4171/IFB/376","DOIUrl":"https://doi.org/10.4171/IFB/376","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"278 1","pages":"31-77"},"PeriodicalIF":1.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72920152","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 a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar ...
{"title":"A free boundary problem with log term singularity","authors":"O. S. D. Queiroz, H. Shahgholian","doi":"10.4171/IFB/385","DOIUrl":"https://doi.org/10.4171/IFB/385","url":null,"abstract":"We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar ...","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"14 1","pages":"351-369"},"PeriodicalIF":1.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84808015","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":"Hysteresis in porous media: Modelling and analysis","authors":"B. Schweizer","doi":"10.4171/IFB/388","DOIUrl":"https://doi.org/10.4171/IFB/388","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"30 1","pages":"417-447"},"PeriodicalIF":1.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73625815","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":"Weak solutions to thin-film equations with contact-line friction","authors":"M. Chiricotto, L. Giacomelli","doi":"10.4171/IFB/382","DOIUrl":"https://doi.org/10.4171/IFB/382","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"32 1","pages":"243-271"},"PeriodicalIF":1.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81742946","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 and analyze in this paper a new derivation of a phase-field model to approximate inhomogeneous multiphase perimeters. It is based on suitable decompositions of perimeters under some embeddability condition which allows not only an explicit derivation of the model from the surface tensions, but also gives rise to a Γ-convergence result. We propose a simple and robust scheme to simulate the gradient flow of the approximating energy. We illustrate the efficiency of our approach with a series of numerical simulations in 2D and 3D, and we address in particular the dynamics of droplets evolving on a fixed solid.
{"title":"A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics","authors":"É. Bretin, S. Masnou","doi":"10.4171/IFB/379","DOIUrl":"https://doi.org/10.4171/IFB/379","url":null,"abstract":"We propose and analyze in this paper a new derivation of a phase-field model to approximate inhomogeneous multiphase perimeters. It is based on suitable decompositions of perimeters under some embeddability condition which allows not only an explicit derivation of the model from the surface tensions, but also gives rise to a Γ-convergence result. We propose a simple and robust scheme to simulate the gradient flow of the approximating energy. We illustrate the efficiency of our approach with a series of numerical simulations in 2D and 3D, and we address in particular the dynamics of droplets evolving on a fixed solid.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"47 1","pages":"141-182"},"PeriodicalIF":1.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74939977","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 the first part of this paper we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global $s$-minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e. local and global), showing that in bounded open sets with Lipschitz boundary they coincide. However, in general this is not true in unbounded open sets, where a global $s$-minimal set may fail to exist (we provide an example in the case of a cylinder $Omegatimesmathbb{R}$).
{"title":"Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global $s$-minimal surfaces","authors":"L. Lombardini","doi":"10.4171/IFB/402","DOIUrl":"https://doi.org/10.4171/IFB/402","url":null,"abstract":"In the first part of this paper we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global $s$-minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e. local and global), showing that in bounded open sets with Lipschitz boundary they coincide. However, in general this is not true in unbounded open sets, where a global $s$-minimal set may fail to exist (we provide an example in the case of a cylinder $Omegatimesmathbb{R}$).","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74604437","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":"Convergence of a threshold-type algorithm using the signed distance function","authors":"K. Ishii, M. Kimura","doi":"10.4171/IFB/371","DOIUrl":"https://doi.org/10.4171/IFB/371","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"56 1","pages":"479-522"},"PeriodicalIF":1.0,"publicationDate":"2016-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82852707","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":"On the Gamma-limit of joint image segmentation and registration functionals based on phase fields","authors":"B. Wirth","doi":"10.4171/IFB/370","DOIUrl":"https://doi.org/10.4171/IFB/370","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"172 1","pages":"441-477"},"PeriodicalIF":1.0,"publicationDate":"2016-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86462428","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 level-set method is used in many dierent applications to describe the propagation of shapes and domains. When scalar speed elds are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf-Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed elds without a xed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the FastMarching Method, and it provides a strong theoretical justication for a generalised Composite Fast-Marching method. Based on our Hopf-Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level-set in a measure-theoretic sense, derive a very general shape sensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the eect of perturbations in both the speed eld and the initial geometry.
{"title":"A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis","authors":"Daniel Kraft","doi":"10.4171/IFB/366","DOIUrl":"https://doi.org/10.4171/IFB/366","url":null,"abstract":"The level-set method is used in many dierent applications to describe the propagation of shapes and domains. When scalar speed elds are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf-Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed elds without a xed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the FastMarching Method, and it provides a strong theoretical justication for a generalised Composite Fast-Marching method. Based on our Hopf-Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level-set in a measure-theoretic sense, derive a very general shape sensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the eect of perturbations in both the speed eld and the initial geometry.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"26 1","pages":"317-353"},"PeriodicalIF":1.0,"publicationDate":"2016-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83249305","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":"The limit as $p to infty$ in a two-phase free boundary problem for the $p$-Laplacian","authors":"J. Rossi, Peiyong Wang","doi":"10.4171/IFB/359","DOIUrl":"https://doi.org/10.4171/IFB/359","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":"13 1","pages":"115-135"},"PeriodicalIF":1.0,"publicationDate":"2016-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82502903","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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