{"title":"Révolution et contre-révolution fr","authors":"Leonore Bazinek","doi":"10.4000/variations.2339","DOIUrl":"https://doi.org/10.4000/variations.2339","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136014446","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}
{"title":"La Théorie critique en tant que révolution permanente","authors":"Oskar Negt","doi":"10.4000/variations.2340","DOIUrl":"https://doi.org/10.4000/variations.2340","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136014571","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}
Antonin Chambolle, Daniele de Gennaro, Massimiliano Morini
Abstract In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus–Sturzenhecker to such flows, the latter result holding in low dimension and conditionally to the convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows one to simplify many arguments.
{"title":"Minimizing movements for anisotropic and inhomogeneous mean curvature flows","authors":"Antonin Chambolle, Daniele de Gennaro, Massimiliano Morini","doi":"10.1515/acv-2022-0102","DOIUrl":"https://doi.org/10.1515/acv-2022-0102","url":null,"abstract":"Abstract In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus–Sturzenhecker to such flows, the latter result holding in low dimension and conditionally to the convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows one to simplify many arguments.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547527","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}
Iwona Chlebicka, Flavia Giannetti, Anna Zatorska-Goldstein
Abstract We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 𝒜 {{mathcal{A}}} -superharmonic functions with nonlinear operator 𝒜:Ω×ℝn→ℝn {{mathcal{A}}:Omegatimes{mathbb{R}^{n}}to{mathbb{R}^{n}}} having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space.
{"title":"Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data","authors":"Iwona Chlebicka, Flavia Giannetti, Anna Zatorska-Goldstein","doi":"10.1515/acv-2023-0005","DOIUrl":"https://doi.org/10.1515/acv-2023-0005","url":null,"abstract":"Abstract We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒜</m:mi> </m:math> {{mathcal{A}}} -superharmonic functions with nonlinear operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒜</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {{mathcal{A}}:Omegatimes{mathbb{R}^{n}}to{mathbb{R}^{n}}} having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549134","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}
Pub Date : 2023-10-01DOI: 10.1515/acv-2023-frontmatter4
{"title":"Frontmatter","authors":"","doi":"10.1515/acv-2023-frontmatter4","DOIUrl":"https://doi.org/10.1515/acv-2023-frontmatter4","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134935446","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}
Abstract By using the monotonicity of the log Sobolev functionals, we prove a no breathers theorem for noncompact harmonic Ricci flows under conditions on infimum of log Sobolev functionals and curvatures. As an application, we obtain a no breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature.
{"title":"No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature","authors":"Jiarui Chen, Qun Chen","doi":"10.1515/acv-2023-0003","DOIUrl":"https://doi.org/10.1515/acv-2023-0003","url":null,"abstract":"Abstract By using the monotonicity of the log Sobolev functionals, we prove a no breathers theorem for noncompact harmonic Ricci flows under conditions on infimum of log Sobolev functionals and curvatures. As an application, we obtain a no breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46817241","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}
Pub Date : 2023-07-01DOI: 10.1515/acv-2023-frontmatter3
{"title":"Frontmatter","authors":"","doi":"10.1515/acv-2023-frontmatter3","DOIUrl":"https://doi.org/10.1515/acv-2023-frontmatter3","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135364084","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}
Abstract The Ambrosio–Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford–Shah image segmentation functional depending on the weight ω d x {omega,dx} , where ω is a special function of bounded variation, and on its values at the jumps.
{"title":"Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters","authors":"Irene Fonseca, Pan Liu, Xin Yang Lu","doi":"10.1515/acv-2021-0071","DOIUrl":"https://doi.org/10.1515/acv-2021-0071","url":null,"abstract":"Abstract The Ambrosio–Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford–Shah image segmentation functional depending on the weight ω d x {omega,dx} , where ω is a special function of bounded variation, and on its values at the jumps.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42236082","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}
Abstract We study minimisation problems in L ∞ {L^{infty}} for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p {L^{p}} approximations as p → ∞ {ptoinfty} , we illustrate the existence of a special L ∞ {L^{infty}} minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ {L^{infty}} variational problem.
{"title":"On isosupremic vectorial minimisation problems in L ∞ with general nonlinear constraints","authors":"E. Clark, Nikos Katzourakis","doi":"10.1515/acv-2022-0068","DOIUrl":"https://doi.org/10.1515/acv-2022-0068","url":null,"abstract":"Abstract We study minimisation problems in L ∞ {L^{infty}} for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p {L^{p}} approximations as p → ∞ {ptoinfty} , we illustrate the existence of a special L ∞ {L^{infty}} minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ {L^{infty}} variational problem.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47685576","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}
Abstract In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ( θ ) ϕ ( ρ ) α − 2 ϕ ( ρ ) 2 + | ∇ ¯ ρ | 2 , K=frac{tilde{f}(theta)}{phi(rho)^{alpha-2}sqrt{phi(rho)^{2}+lvertoverline{nabla}rhorvert^{2}}}, a generalization of the Alexandrov problem ( α = n + 1 alpha=n+1 ) in hyperbolic space, where f ~ tilde{f} is a smooth positive function on S n mathbb{S}^{n} , 𝜌 is the radial function of the hypersurface, ϕ ( ρ ) = sinh ρ phi(rho)=sinhrho and 𝐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 alphageq n+1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 mathbb{H}^{n+1} . We also consider the cases 2 < α ≤ n + 1 2
{"title":"A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1","authors":"Haizhong Li, Ruijia Zhang","doi":"10.1515/acv-2022-0033","DOIUrl":"https://doi.org/10.1515/acv-2022-0033","url":null,"abstract":"Abstract In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ( θ ) ϕ ( ρ ) α − 2 ϕ ( ρ ) 2 + | ∇ ¯ ρ | 2 , K=frac{tilde{f}(theta)}{phi(rho)^{alpha-2}sqrt{phi(rho)^{2}+lvertoverline{nabla}rhorvert^{2}}}, a generalization of the Alexandrov problem ( α = n + 1 alpha=n+1 ) in hyperbolic space, where f ~ tilde{f} is a smooth positive function on S n mathbb{S}^{n} , 𝜌 is the radial function of the hypersurface, ϕ ( ρ ) = sinh ρ phi(rho)=sinhrho and 𝐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 alphageq n+1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 mathbb{H}^{n+1} . We also consider the cases 2 < α ≤ n + 1 2<alphaleq n+1 under the evenness assumption of f ~ tilde{f} and prove the existence of solutions to the above equations.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48480192","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}
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