Francesco Della Pietra, C. Nitsch, Francescantonio Oliva, C. Trombetti
Abstract In this paper, we study the Γ-limit, as p → 1 {pto 1} , of the functional J p ( u ) = ∫ Ω | ∇ u | p + β ∫ ∂ Ω | u | p ∫ Ω | u | p , J_{p}(u)=frac{int_{Omega}lvertnabla urvert^{p}+betaint_{partialOmega% }lvert urvert^{p}}{int_{Omega}lvert urvert^{p}}, where Ω is a smooth bounded open set in ℝ N {mathbb{R}^{N}} , p > 1 {p>1} and β is a real number. Among our results, for β > - 1 {beta>-1} , we derive an isoperimetric inequality for Λ ( Ω , β ) = inf u ∈ BV ( Ω ) , u ≢ 0 | D u | ( Ω ) + min ( β , 1 ) ∫ ∂ Ω | u | ∫ Ω | u | Lambda(Omega,beta)=inf_{uinoperatorname{BV}(Omega),,unotequiv 0}% frac{lvert Durvert(Omega)+min(beta,1)int_{partialOmega}lvert urvert% }{int_{Omega}lvert urvert} which is the limit as p → 1 + {pto 1^{+}} of λ ( Ω , p , β ) = min u ∈ W 1 , p ( Ω ) J p ( u ) {lambda(Omega,p,beta)=min_{uin W^{1,p}(Omega)}J_{p}(u)} . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ( Ω , β ) {Lambda(Omega,beta)} when β ∈ ( - 1 , 0 ) {betain(-1,0)} and minimizes Λ ( Ω , β ) {Lambda(Omega,beta)} when β ∈ [ 0 , ∞ ) {betain[0,infty)} .
文摘本文研究了Γ限制,如p→1 p {1},功能性的J p(u) =∫Ω|∇u | p +β∫∂Ω| | u p∫Ω| | u p, J_ {p} (u) = 压裂{ int_{ω} lvert 微分算符u rvert ^ {p} +β int_{ω部分 %} lvert u rvert ^ {p}} { int_{ω} lvert u rvert ^ {p}},Ω是一个光滑的有界开集在ℝN { mathbb {R} ^ {N}}, p > 1 {p > 1},β是一个实数。我们的结果,对β> - 1{β> 1},我们获得的等周不等式Λ(Ω,β)=正u∈BV(Ω),u≢0| Du |(Ω)+分钟(β1)∫∂Ω| |你∫Ω| | uλ(ω β)= inf_ {u operatorname {BV}(ω),u不 枚0}% 压裂{ lvert Du rvert(ω)+ min(β1) int_{ω部分} lvert u rvert %} { int_{ω} lvert u rvert}这是p的极限→1 + {p 1 ^{+}}的λ(Ω,p,β)= u min∈W 1,p(Ω)J p(u){λ(ω p β)= min_ {u W ^ {1, p}(ω)}J_ {p} (u)}。我们证明了在给定体积的所有有界光滑开集中，当β∈(-1,0){betain(-1,0)}时，球最大化Λ≠(Ω， β) {Lambda(Omega，beta)}，当β∈[0，∞){betain[0，infty)}时，球最小化Λ≠(Ω， β) {Lambda(Omega，beta)}。
{"title":"On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1","authors":"Francesco Della Pietra, C. Nitsch, Francescantonio Oliva, C. Trombetti","doi":"10.1515/acv-2021-0085","DOIUrl":"https://doi.org/10.1515/acv-2021-0085","url":null,"abstract":"Abstract In this paper, we study the Γ-limit, as p → 1 {pto 1} , of the functional J p ( u ) = ∫ Ω | ∇ u | p + β ∫ ∂ Ω | u | p ∫ Ω | u | p , J_{p}(u)=frac{int_{Omega}lvertnabla urvert^{p}+betaint_{partialOmega% }lvert urvert^{p}}{int_{Omega}lvert urvert^{p}}, where Ω is a smooth bounded open set in ℝ N {mathbb{R}^{N}} , p > 1 {p>1} and β is a real number. Among our results, for β > - 1 {beta>-1} , we derive an isoperimetric inequality for Λ ( Ω , β ) = inf u ∈ BV ( Ω ) , u ≢ 0 | D u | ( Ω ) + min ( β , 1 ) ∫ ∂ Ω | u | ∫ Ω | u | Lambda(Omega,beta)=inf_{uinoperatorname{BV}(Omega),,unotequiv 0}% frac{lvert Durvert(Omega)+min(beta,1)int_{partialOmega}lvert urvert% }{int_{Omega}lvert urvert} which is the limit as p → 1 + {pto 1^{+}} of λ ( Ω , p , β ) = min u ∈ W 1 , p ( Ω ) J p ( u ) {lambda(Omega,p,beta)=min_{uin W^{1,p}(Omega)}J_{p}(u)} . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ( Ω , β ) {Lambda(Omega,beta)} when β ∈ ( - 1 , 0 ) {betain(-1,0)} and minimizes Λ ( Ω , β ) {Lambda(Omega,beta)} when β ∈ [ 0 , ∞ ) {betain[0,infty)} .","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"16 1","pages":"1123 - 1135"},"PeriodicalIF":1.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48582838","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 Let Ω ⊂ ℝ n {Omegasubsetmathbb{R}^{n}} be a C 1 {C^{1}} smooth compact domain. Furthermore, let F : Ω × ℝ n N → ℝ {F:Omegatimesmathbb{R}^{nN}tomathbb{R}} , F ( x , p ) {F(x,p)} , be C 0 {C^{0}} , differentiable with respect to p, and with F p := D p F {F_{p}:=D_{p}F} continuous on Ω × ℝ n N {Omegatimesmathbb{R}^{nN}} and F strictly convex in p. Consider an n N × n N {nNtimes nN} matrix A = ( A α β i j ) ∈ C 0 ( Ω ) {A=(A^{{ij}}_{alphabeta})in C^{0}(Omega)} satisfying (0.1) A α β i j ( x ) ξ α i ξ β j = A β α j i ( x ) ξ α i ξ β j ≥ λ | ξ | 2 , λ > 0 . A^{ij}_{alphabeta}(x)xi^{i}_{alpha}xi^{j}_{beta}=A^{ji}_{betaalpha}(x)% xi^{i}_{alpha}xi^{j}_{beta}geqlambdalvertxirvert^{2},quadlambda>0. Suppose that (0.2) lim | p | → ∞ 1 | p | ( D p F ( x , p ) - A ( x ) p ) = 0 , displaystylelim_{lvert prverttoinfty}frac{1}{lvert prvert}(D_{p}F(x,p% )-A(x)p)=0, (0.3) - C 0 + c 0 | p | 2 ≤ F ( x , p ) ≤ C 0 ( 1 + | p | 2 ) , displaystyle{-}C_{0}+c_{0}lvert prvert^{2}leq F(x,p)leq C_{0}(1+lvert p% rvert^{2}), (0.4) | F p ( x , p ) - F p ( x , q ) | ≤ C 0 | p - q | , displaystylelvert F_{p}(x,p)-F_{p}(x,q)rvertleq C_{0}lvert p-qrvert, (0.5) 〈 F p ( x , p ) - F p ( x , q ) , p - q 〉 ≥ c 0 | p - q | 2 displaystylelangle F_{p}(x,p)-F_{p}(x,q),p-qranglegeq c_{0}lvert p-q% rvert^{2} uniformly in x and with positive constants c 0 {c_{0}} and C 0 {C_{0}} . Consider the functional (0.6) J ( u ) := ∫ Ω F ( x , D u ( x ) ) 𝑑 x + ∫ Ω G ( x , u ) 𝑑 x , J(u):=int_{Omega}F(x,Du(x)),dx+int_{Omega}G(x,u),dx, where G ( x , ⋅ ) ∈ C 1 ( ℝ N ) {G(x,cdot,)in C^{1}(mathbb{R}^{N})} for each x ∈ Ω {xinOmega} , G ( ⋅ , u ) {G(,cdot,,u)} is measurable for each u ∈ ℝ N {uinmathbb{R}^{N}} , and (0.7) | G u ( x , u ) | ≤ C 0 ( 1 + | u | s ) lvert G_{u}(x,u)rvertleq C_{0}(1+lvert urvert^{s}) with s < n + 2 n - 2 {s 2 {n>2} , then any weak solution u ∈ W 1 , 2 ( Ω , ℝ N ) {uin W^{1,2}(Omega,mathbb{R}^{N})} of the Euler equations of J, i.e. ∑ α ∂ ∂ x α F p α i ( x , D u ) = G u i ( x , u ) , i = 1 , … , N , sum_{alpha}frac{partial}{partial x^{alpha}}F_{p^{i}_{alpha}}(x,Du)=G_{u% ^{i}}(x,u),quad i=1,ldots,N, is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.
设Ω∧∈n {Omegasubsetmathbb{R}＾{n}} 是C {c ^{1}} 光滑紧致域。更进一步，设F: Ω x, n, n，→ {f:Omegatimesmathbb{R}＾{nN}tomathbb{R}} ， F∑(x, p) {F(x,p)} ，是C 0 {c ^{0}} ，对p可导，对F可导，p = dp∑F {f＿{p}:= d＿{p}f} 在Ω上连续的 {Omegatimesmathbb{R}＾{nN}} F在p中是严格凸的，考虑n n × n n {nNtimes nN} 矩阵A = (A α ＿ β i ＿ j)∈c0 ＿ (Ω) {a =(a ^{{ij}}＿{alphabeta}）in c ^{0}（Omega）} 满足(0.1)A α减去β i减去j减去(x)减去ξ α i减去ξ β j = A β减去α j减去i减去(x)减去ξ α i减去ξ β j≥λ减去| ξ | 2， λ > 0。a ^{ij}＿{alphabeta}(x)xi＾{I}＿{alpha}xi＾{j}＿{beta}= a ^{ji}＿{betaalpha}(x)% xi^{i}_{alpha}xi^{j}_{beta}geqlambdalvertxirvert^{2},quadlambda>0. Suppose that (0.2) lim | p | → ∞ 1 | p | ( D p F ( x , p ) - A ( x ) p ) = 0 , displaystylelim_{lvert prverttoinfty}frac{1}{lvert prvert}(D_{p}F(x,p% )-A(x)p)=0, (0.3) - C 0 + c 0 | p | 2 ≤ F ( x , p ) ≤ C 0 ( 1 + | p | 2 ) , displaystyle{-}C_{0}+c_{0}lvert prvert^{2}leq F(x,p)leq C_{0}(1+lvert p% rvert^{2}), (0.4) | F p ( x , p ) - F p ( x , q ) | ≤ C 0 | p - q | , displaystylelvert F_{p}(x,p)-F_{p}(x,q)rvertleq C_{0}lvert p-qrvert, (0.5) 〈 F p ( x , p ) - F p ( x , q ) , p - q 〉 ≥ c 0 | p - q | 2 displaystylelangle F_{p}(x,p)-F_{p}(x,q),p-qranglegeq c_{0}lvert p-q% rvert^{2} uniformly in x and with positive constants c 0 {c_{0}} and C 0 {C_{0}} . Consider the functional (0.6) J ( u ) := ∫ Ω F ( x , D u ( x ) ) 𝑑 x + ∫ Ω G ( x , u ) 𝑑 x , J(u):=int_{Omega}F(x,Du(x)),dx+int_{Omega}G(x,u),dx, where G ( x , ⋅ ) ∈ C 1 ( ℝ N ) {G(x,cdot,)in C^{1}(mathbb{R}^{N})} for each x ∈ Ω {xinOmega} , G ( ⋅ , u ) {G(,cdot,,u)} is measurable for each u ∈ ℝ N {uinmathbb{R}^{N}} , and (0.7) | G u ( x , u ) | ≤ C 0 ( 1 + | u | s ) lvert G_{u}(x,u)rvertleq C_{0}(1+lvert urvert^{s}) with s < n + 2 n - 2 {s 2 {n>2} , then any weak solution u ∈ W 1 , 2 ( Ω , ℝ N ) {uin W^{1,2}(Omega,mathbb{R}^{N})} of the Euler equations of J, i.e. ∑ α ∂ ∂ x α F p α i ( x , D u ) = G u i ( x , u ) , i = 1 , … , N , sum_{alpha}frac{partial}{partial x^{alpha}}F_{p^{i}_{alpha}}(x,Du)=G_{u% ^{i}}(x,u),quad i=1,ldots,N, is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.
{"title":"On the Hölder regularity of all extrema in Hilbert’s 19th Problem","authors":"F. Tomi, A. Tromba","doi":"10.1515/acv-2021-0089","DOIUrl":"https://doi.org/10.1515/acv-2021-0089","url":null,"abstract":"Abstract Let Ω ⊂ ℝ n {Omegasubsetmathbb{R}^{n}} be a C 1 {C^{1}} smooth compact domain. Furthermore, let F : Ω × ℝ n N → ℝ {F:Omegatimesmathbb{R}^{nN}tomathbb{R}} , F ( x , p ) {F(x,p)} , be C 0 {C^{0}} , differentiable with respect to p, and with F p := D p F {F_{p}:=D_{p}F} continuous on Ω × ℝ n N {Omegatimesmathbb{R}^{nN}} and F strictly convex in p. Consider an n N × n N {nNtimes nN} matrix A = ( A α β i j ) ∈ C 0 ( Ω ) {A=(A^{{ij}}_{alphabeta})in C^{0}(Omega)} satisfying (0.1) A α β i j ( x ) ξ α i ξ β j = A β α j i ( x ) ξ α i ξ β j ≥ λ | ξ | 2 , λ > 0 . A^{ij}_{alphabeta}(x)xi^{i}_{alpha}xi^{j}_{beta}=A^{ji}_{betaalpha}(x)% xi^{i}_{alpha}xi^{j}_{beta}geqlambdalvertxirvert^{2},quadlambda>0. Suppose that (0.2) lim | p | → ∞ 1 | p | ( D p F ( x , p ) - A ( x ) p ) = 0 , displaystylelim_{lvert prverttoinfty}frac{1}{lvert prvert}(D_{p}F(x,p% )-A(x)p)=0, (0.3) - C 0 + c 0 | p | 2 ≤ F ( x , p ) ≤ C 0 ( 1 + | p | 2 ) , displaystyle{-}C_{0}+c_{0}lvert prvert^{2}leq F(x,p)leq C_{0}(1+lvert p% rvert^{2}), (0.4) | F p ( x , p ) - F p ( x , q ) | ≤ C 0 | p - q | , displaystylelvert F_{p}(x,p)-F_{p}(x,q)rvertleq C_{0}lvert p-qrvert, (0.5) 〈 F p ( x , p ) - F p ( x , q ) , p - q 〉 ≥ c 0 | p - q | 2 displaystylelangle F_{p}(x,p)-F_{p}(x,q),p-qranglegeq c_{0}lvert p-q% rvert^{2} uniformly in x and with positive constants c 0 {c_{0}} and C 0 {C_{0}} . Consider the functional (0.6) J ( u ) := ∫ Ω F ( x , D u ( x ) ) 𝑑 x + ∫ Ω G ( x , u ) 𝑑 x , J(u):=int_{Omega}F(x,Du(x)),dx+int_{Omega}G(x,u),dx, where G ( x , ⋅ ) ∈ C 1 ( ℝ N ) {G(x,cdot,)in C^{1}(mathbb{R}^{N})} for each x ∈ Ω {xinOmega} , G ( ⋅ , u ) {G(,cdot,,u)} is measurable for each u ∈ ℝ N {uinmathbb{R}^{N}} , and (0.7) | G u ( x , u ) | ≤ C 0 ( 1 + | u | s ) lvert G_{u}(x,u)rvertleq C_{0}(1+lvert urvert^{s}) with s < n + 2 n - 2 {s<frac{n+2}{n-2}} . Under these conditions, we shall show that if n > 2 {n>2} , then any weak solution u ∈ W 1 , 2 ( Ω , ℝ N ) {uin W^{1,2}(Omega,mathbb{R}^{N})} of the Euler equations of J, i.e. ∑ α ∂ ∂ x α F p α i ( x , D u ) = G u i ( x , u ) , i = 1 , … , N , sum_{alpha}frac{partial}{partial x^{alpha}}F_{p^{i}_{alpha}}(x,Du)=G_{u% ^{i}}(x,u),quad i=1,ldots,N, is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48241125","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 introduce the L p {L_{p}} q-torsional measure for p ∈ ℝ {pinmathbb{R}} and q > 1 {q>1} by the L p {L_{p}} variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L p {L_{p}} Minkowski problem with respect to the q-torsional rigidity for discrete measures and general measures when 0 < p < 1 {0 1 {q>1} .
{"title":"The Lp Minkowski problem for q-torsional rigidity","authors":"Bin Chen, Xia Zhao, Weidong Wang, P. Zhao","doi":"10.1515/acv-2022-0041","DOIUrl":"https://doi.org/10.1515/acv-2022-0041","url":null,"abstract":"Abstract In this paper, we introduce the L p {L_{p}} q-torsional measure for p ∈ ℝ {pinmathbb{R}} and q > 1 {q>1} by the L p {L_{p}} variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L p {L_{p}} Minkowski problem with respect to the q-torsional rigidity for discrete measures and general measures when 0 < p < 1 {0 1 {q>1} .","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49018671","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 the paper we prove the convergence of viscosity solutions u λ {u_{lambda}} as λ → 0 + {lambdarightarrow 0_{+}} for the parametrized degenerate viscous Hamilton–Jacobi equation H ( x , d x u , λ u ) = α ( x ) Δ u , α ( x ) ≥ 0 , x ∈ 𝕋 n H(x,d_{x}u,lambda u)=alpha(x)Delta u,quadalpha(x)geq 0,quad xinmathbb% {T}^{n} under suitable convex and monotonic conditions on H : T * M × ℝ → ℝ {H:T^{*}Mtimesmathbb{R}rightarrowmathbb{R}} . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ( x , d x u , 0 ) = α ( x ) Δ u . H(x,d_{x}u,0)=alpha(x)Delta u.
摘要本文证明了黏性解u λ的收敛性 {我们……{lambda}} 当λ→0 + {lambdarightarrow 0＿{+}} 对于参数化简并粘性Hamilton-Jacobi方程H∑(x,d x∑u， λ∑u) = α∑(x)∑Δ∑u， α∑(x)≥0,x∈ndh (x,d＿{x}你，lambda u)=alpha(x)Delta 你，quadalpha(x)geq 0，quad xinmathbb% {T}^{n} under suitable convex and monotonic conditions on H : T * M × ℝ → ℝ {H:T^{*}Mtimesmathbb{R}rightarrowmathbb{R}} . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ( x , d x u , 0 ) = α ( x ) Δ u . H(x,d_{x}u,0)=alpha(x)Delta u.
{"title":"Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity","authors":"Jian-lin Zhang","doi":"10.1515/acv-2022-0108","DOIUrl":"https://doi.org/10.1515/acv-2022-0108","url":null,"abstract":"Abstract In the paper we prove the convergence of viscosity solutions u λ {u_{lambda}} as λ → 0 + {lambdarightarrow 0_{+}} for the parametrized degenerate viscous Hamilton–Jacobi equation H ( x , d x u , λ u ) = α ( x ) Δ u , α ( x ) ≥ 0 , x ∈ 𝕋 n H(x,d_{x}u,lambda u)=alpha(x)Delta u,quadalpha(x)geq 0,quad xinmathbb% {T}^{n} under suitable convex and monotonic conditions on H : T * M × ℝ → ℝ {H:T^{*}Mtimesmathbb{R}rightarrowmathbb{R}} . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ( x , d x u , 0 ) = α ( x ) Δ u . H(x,d_{x}u,0)=alpha(x)Delta u.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45024515","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 homogeneous causal action principle on a compact domain of momentum space is introduced. The connection to causal fermion systems is worked out. Existence and compactness results are reviewed. The Euler–Lagrange equations are derived and analyzed under suitable regularity assumptions.
{"title":"The homogeneous causal action principle on a compact domain in momentum space","authors":"F. Finster, Michelle Frankl, Christoph Langer","doi":"10.1515/acv-2022-0038","DOIUrl":"https://doi.org/10.1515/acv-2022-0038","url":null,"abstract":"Abstract The homogeneous causal action principle on a compact domain of momentum space is introduced. The connection to causal fermion systems is worked out. Existence and compactness results are reviewed. The Euler–Lagrange equations are derived and analyzed under suitable regularity assumptions.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48942063","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 Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO ( M ) → M {mathrm{SO}(M)rightarrow M} of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso o ( M ) ≅ SO ( M ) {mathrm{Iso}_{o}(M)congmathrm{SO}(M)} : A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.
{"title":"A split special Lagrangian calibration associated with frame vorticity","authors":"M. Salvai","doi":"10.1515/acv-2022-0036","DOIUrl":"https://doi.org/10.1515/acv-2022-0036","url":null,"abstract":"Abstract Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO ( M ) → M {mathrm{SO}(M)rightarrow M} of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso o ( M ) ≅ SO ( M ) {mathrm{Iso}_{o}(M)congmathrm{SO}(M)} : A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49341825","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 define a homogeneous De Giorgi class of order p = 2 {p=2} that contains the solutions of evolution equations of the types ξ ( x , t ) u t + A u = 0 {xi(x,t)u_{t}+Au=0} and ( ξ ( x , t ) u ) t + A u = 0 {(xi(x,t)u)_{t}+Au=0} , where ξ > 0 {xi>0} almost everywhere and A is a suitable elliptic operator. For functions belonging to this class, we prove a Harnack inequality. As a byproduct, one can obtain Hölder continuity for solutions of a subclass of the first equation.
{"title":"Harnack inequality for parabolic equations with coefficients depending on time","authors":"F. Paronetto","doi":"10.1515/acv-2021-0055","DOIUrl":"https://doi.org/10.1515/acv-2021-0055","url":null,"abstract":"Abstract We define a homogeneous De Giorgi class of order p = 2 {p=2} that contains the solutions of evolution equations of the types ξ ( x , t ) u t + A u = 0 {xi(x,t)u_{t}+Au=0} and ( ξ ( x , t ) u ) t + A u = 0 {(xi(x,t)u)_{t}+Au=0} , where ξ > 0 {xi>0} almost everywhere and A is a suitable elliptic operator. For functions belonging to this class, we prove a Harnack inequality. As a byproduct, one can obtain Hölder continuity for solutions of a subclass of the first equation.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49409494","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 Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ( ( | D u | - ν ) + p - 1 D u | D u | ) = f in Ω T = Ω × ( 0 , T ) , u_{t}-operatorname{div}Bigl{(}(lvert Durvert-nu)_{+}^{p-1}frac{Du}{% lvert Durvert}Bigr{)}=fquadtext{in }Omega_{T}=Omegatimes(0,T), where Ω is a bounded domain in ℝ n {mathbb{R}^{n}} for n ≥ 2 {ngeq 2} , p ≥ 2 {pgeq 2} , ν is a positive constant and ( ⋅ ) + {(,cdot,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t {u_{t}} . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.
摘要受气体过滤问题应用的启发，我们研究了强退化抛物型偏微分方程的弱解的正则性 （（|Du|-Γ）+p-1 Du|Du|）=f 单位为ΩT=Ω×（0，T），u_{t}-运算符名称｛div｝Bigl｛（｝（lvert Durvert-nu）_｛+｝^｛p-1｝frac｛Du｝｛%lvert Du rvert｝Bigr｛）｝=fquadtext｛in｝Omega_｛T｝=Omegatimes（0，T），其中Ω是ℝ n｛mathbb｛R｝^｛n｝｝对于n≥2｛n geq 2｝，p≥2｝p geq 2中｝，Γ是一个正常数，（∙）+｛（，cdot，）_｛+｝}代表正部分。假设数据f属于一个合适的Lebesgue–Sobolev抛物空间，我们建立了弱解的空间梯度的非线性函数的Sobolev空间正则性，这反过来意味着弱时间导数ut{u_{t}}的存在。这里的主要新颖之处在于，上述方程的结构函数仅在半径为Γ的以原点为中心的球外满足标准增长和椭圆度条件。我们想指出的是，一方面，这里获得的第一个结果可以被认为是[L.Brasco，G.Carlier和F.Santambrogio，Congested traffic dynamics，weak flow and very degenerated ellite equipments[mr2584740的修正版]，J.Math中建立的椭圆结果的抛物型对应物。Pures Appl。（9） 93 2010，652–671]，另一方面作为不太退化抛物型方程的一些已知结果的强退化上下文的扩展。
{"title":"Regularity results for a class of widely degenerate parabolic equations","authors":"P. Ambrosio, Antonia Passarelli di Napoli","doi":"10.1515/acv-2022-0062","DOIUrl":"https://doi.org/10.1515/acv-2022-0062","url":null,"abstract":"Abstract Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ( ( | D u | - ν ) + p - 1 D u | D u | ) = f in Ω T = Ω × ( 0 , T ) , u_{t}-operatorname{div}Bigl{(}(lvert Durvert-nu)_{+}^{p-1}frac{Du}{% lvert Durvert}Bigr{)}=fquadtext{in }Omega_{T}=Omegatimes(0,T), where Ω is a bounded domain in ℝ n {mathbb{R}^{n}} for n ≥ 2 {ngeq 2} , p ≥ 2 {pgeq 2} , ν is a positive constant and ( ⋅ ) + {(,cdot,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t {u_{t}} . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45729676","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 consider the volume constrained fractional mean curvature flow of a nearly spherical set and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.
{"title":"Stability of the ball under volume preserving fractional mean curvature flow","authors":"A. Cesaroni, M. Novaga","doi":"10.1515/acv-2022-0027","DOIUrl":"https://doi.org/10.1515/acv-2022-0027","url":null,"abstract":"Abstract We consider the volume constrained fractional mean curvature flow of a nearly spherical set and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49384611","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 : 2022-04-01DOI: 10.1515/acv-2022-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/acv-2022-frontmatter2","DOIUrl":"https://doi.org/10.1515/acv-2022-frontmatter2","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67106306","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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