{"title":"Optimal Borel measure-valued controls to the viscous Cahn--Hilliard--Oberbeck--Boussinesq phase-field system on two-dimensional bounded domains","authors":"Gilbert Peralta","doi":"10.1051/cocv/2023025","DOIUrl":"https://doi.org/10.1051/cocv/2023025","url":null,"abstract":"We consider an optimal control problem for the two-dimensional viscous Cahn--Hilliard--Oberbeck--Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equation governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"52 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74861678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emerson Abreu, Leandro G. Fernandes, Joel Cruz Ramirez
{"title":"The $varepsilon$ - $varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights\u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 ","authors":"Emerson Abreu, Leandro G. Fernandes, Joel Cruz Ramirez","doi":"10.1051/cocv/2023023","DOIUrl":"https://doi.org/10.1051/cocv/2023023","url":null,"abstract":"We consider a class of monomial weights $x^{A}=vert x_{1}vert^{a_{1}}ldotsvert x_{N}vert^{a_{N}}$ in $mathbb{R}^{N}$, where $a_{i}$ is a nonnegative real number for each $iin{1,ldots,N}$, and we establish the $varepsilon-varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for $N=2$, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"6 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81844355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}