We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., it is valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known.
{"title":"Thin obstacle problem: Estimates of the distance to the exact solution","authors":"D. Apushkinskaya, S. Repin","doi":"10.4171/IFB/410","DOIUrl":"https://doi.org/10.4171/IFB/410","url":null,"abstract":"We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., it is valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75647433","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}
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $Bbb{R}^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.
{"title":"Incompressible fluid problems on embedded surfaces: Modeling and variational formulations","authors":"Thomas Jankuhn, M. Olshanskii, A. Reusken","doi":"10.4171/IFB/405","DOIUrl":"https://doi.org/10.4171/IFB/405","url":null,"abstract":"Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $Bbb{R}^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81041107","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":"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":null,"pages":null},"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":null,"pages":null},"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":"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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