{"title":"Free boundary regularity for a degenerate problem with right hand side","authors":"R. Leitão, G. Ricarte","doi":"10.4171/IFB/413","DOIUrl":"https://doi.org/10.4171/IFB/413","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83082861","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 develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional Fλ which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter λ. We prove existence of a solution for λ large enough as well as convergence to a solution of the free boundary problem as λ tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis may be largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this one-dimensional problem and prove an error estimate which includes an order of convergence with respect to the grid size.
{"title":"Approximation of minimal surfaces with free boundaries","authors":"U. Dierkes, Tristan Jenschke, Paola Pozzi","doi":"10.4171/IFB/412","DOIUrl":"https://doi.org/10.4171/IFB/412","url":null,"abstract":"In this paper we develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional Fλ which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter λ. We prove existence of a solution for λ large enough as well as convergence to a solution of the free boundary problem as λ tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis may be largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this one-dimensional problem and prove an error estimate which includes an order of convergence with respect to the grid size.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89282039","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 consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization argument.
{"title":"An obstacle problem for elastic curves: Existence results","authors":"Marius Muller","doi":"10.4171/IFB/418","DOIUrl":"https://doi.org/10.4171/IFB/418","url":null,"abstract":"We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization argument.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81609820","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":"Schwarz P surfaces and a non local perturbation of the perimeter","authors":"M. Rizzi","doi":"10.4171/IFB/404","DOIUrl":"https://doi.org/10.4171/IFB/404","url":null,"abstract":"","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79727727","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 work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of p−Laplacian type. The problem in its variational form is as follows: min v ∫ Ω∩{v>0} ( 1 p |∇v|p +λ p +(x)+ f+(x)v ) dx+ ∫ Ω∩{v≤0} ( 1 q |∇v|q +λ q −(x)+ f−(x)v ) dx . Here we minimize among all admissible functions v in an appropriate Sobolev space with a prescribed boundary datum v = g on ∂Ω. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where p and q go to infinity, obtaining a limiting free boundary problem governed by the ∞−Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.
本文研究了一个两相的最小化问题，其中在每个相域中，问题由一个p -拉普拉斯型的拟线性椭圆算子来控制。问题的变分形式如下:min v∫Ω∩v >{0}(1页| |∇v p +λp + f (x) + + (x) v) dx +∫Ω∩{v≤0}(1 q |∇v | q +λq−−(x) + f (x) v) dx。在这里，我们在一个适当的Sobolev空间中最小化所有可容许的函数v，在∂Ω上有一个规定的边界基准v = g。首先，我们证明了最小化器的存在性，证明了一些性质，并给出了一个非唯一性的例子。此外，我们分析了p和q趋于无穷时的极限情况，得到了一个由∞-拉普拉斯算子支配的极限自由边界问题。得到了任意极限解的Lipschitz正则性。最后，我们建立了解的一些弱几何性质。
{"title":"A limit case in non-isotropic two-phase minimization problems driven by $p$-Laplacians","authors":"J. V. Silva, J. Rossi","doi":"10.4171/IFB/406","DOIUrl":"https://doi.org/10.4171/IFB/406","url":null,"abstract":"In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of p−Laplacian type. The problem in its variational form is as follows: min v ∫ Ω∩{v>0} ( 1 p |∇v|p +λ p +(x)+ f+(x)v ) dx+ ∫ Ω∩{v≤0} ( 1 q |∇v|q +λ q −(x)+ f−(x)v ) dx . Here we minimize among all admissible functions v in an appropriate Sobolev space with a prescribed boundary datum v = g on ∂Ω. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where p and q go to infinity, obtaining a limiting free boundary problem governed by the ∞−Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87145981","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 diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn-Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in the sharp interface limit, which is derived from first principles. Results from numerical simulations support the theoretical findings. The main novelties are centred around the conditions in the triple junctions where three fluids meet. Specifically the case of local chemical equilibrium with respect to the surfactant is considered, which allows for interfacial surfactant flow through the triple junctions.
{"title":"Phase field modelling of surfactants in multi-phase flow","authors":"Oliver R. A. Dunbar, K. F. Lam, B. Stinner","doi":"10.4171/ifb/429","DOIUrl":"https://doi.org/10.4171/ifb/429","url":null,"abstract":"A diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn-Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in the sharp interface limit, which is derived from first principles. Results from numerical simulations support the theoretical findings. The main novelties are centred around the conditions in the triple junctions where three fluids meet. Specifically the case of local chemical equilibrium with respect to the surfactant is considered, which allows for interfacial surfactant flow through the triple junctions.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88351746","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}
Non-transversal intersection of the free and fixed boundary is shown to hold and a classification of blow-up solutions is given for obstacle problems generated by fully nonlinear uniformly elliptic operators in two dimensions which appear in the mean-field theory of superconducting vortices.
{"title":"Non-transversal intersection of the free and fixed boundary in the mean-field theory of superconductivity","authors":"E. Indrei","doi":"10.4171/IFB/423","DOIUrl":"https://doi.org/10.4171/IFB/423","url":null,"abstract":"Non-transversal intersection of the free and fixed boundary is shown to hold and a classification of blow-up solutions is given for obstacle problems generated by fully nonlinear uniformly elliptic operators in two dimensions which appear in the mean-field theory of superconducting vortices.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/IFB/423","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72395231","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 a global existence result for weak solutions to a one-dimensional free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a priori unknown interface never disappears. As main ingredients of the global existence proof, we rely on a local weak solvability result for our problem, uniform estimates of the solution, integral estimates on quantities defined at the free boundary, as well as a fine pointwise lower bound for the position of the moving boundary. Some of the estimates are time-independent. They allow us to explore the large time behavior of the position of the moving boundary. The approach is specific to one-dimensional settings.
{"title":"Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces","authors":"K. Kumazaki, A. Muntean","doi":"10.4171/ifb/431","DOIUrl":"https://doi.org/10.4171/ifb/431","url":null,"abstract":"We prove a global existence result for weak solutions to a one-dimensional free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a priori unknown interface never disappears. As main ingredients of the global existence proof, we rely on a local weak solvability result for our problem, uniform estimates of the solution, integral estimates on quantities defined at the free boundary, as well as a fine pointwise lower bound for the position of the moving boundary. Some of the estimates are time-independent. They allow us to explore the large time behavior of the position of the moving boundary. The approach is specific to one-dimensional settings.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89475397","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 describe a convex relaxation for the Gilbert-Steiner problem both in $R^d$ and on manifolds, extending the framework proposed in [9], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.
{"title":"A convex approach to the Gilbert–Steiner problem","authors":"M. Bonafini, 'Edouard Oudet","doi":"10.4171/ifb/436","DOIUrl":"https://doi.org/10.4171/ifb/436","url":null,"abstract":"We describe a convex relaxation for the Gilbert-Steiner problem both in $R^d$ and on manifolds, extending the framework proposed in [9], and we discuss its sharpness by means of calibration type arguments. The minimization of the resulting problem is then tackled numerically and we present results for an extensive set of examples. In particular we are able to address the Steiner tree problem on surfaces.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84743562","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 extend the results of [1] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.
{"title":"Long-time behaviour of solutions to a singular heat equation with an application to hydrodynamics","authors":"G. Kitavtsev, R. Taranets","doi":"10.4171/ifb/437","DOIUrl":"https://doi.org/10.4171/ifb/437","url":null,"abstract":"In this paper, we extend the results of [1] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80418972","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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