Advances in Calculus of Variations最新文献

{"title":"Properties of the free boundaries for the obstacle problem of the porous medium equations","authors":"Sunghoon Kim, Ki-ahm Lee, Jinwan Park","doi":"10.1515/acv-2021-0113","DOIUrl":"https://doi.org/10.1515/acv-2021-0113","url":null,"abstract":"Abstract In this paper, we study the existence and interior W 2 , p {W^{2,p}} -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } {partial{u>phi}} to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m {u_{t}=Delta u^{m}} ( m > 1 {m>1} ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } {partial{u>phi}} and ∂ ⁡ { u > 0 } {partial{u>0}} , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } {partial{u>phi}} separate from the free boundary ∂ ⁡ { u > 0 } {partial{u>0}} . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 {C^{1}} -regularity of the free boundary ∂ ⁡ { u > ϕ } {partial{u>phi}} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47653952","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":"On functions of bounded β-dimensional mean oscillation","authors":"You-Wei Chen, Daniel Spector","doi":"10.1515/acv-2022-0084","DOIUrl":"https://doi.org/10.1515/acv-2022-0084","url":null,"abstract":"Abstract In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}subsetmathbb{R}^{d}tomathbb{R}} which are integrable on β-dimensional subsets of the cube Q 0 {Q_{0}} : ∥ u ∥ BMO β ⁢ ( Q 0 ) := sup Q ⊂ Q 0 ⁡ inf c ∈ ℝ ⁡ 1 l ⁢ ( Q ) β ⁢ ∫ Q | u - c | ⁢ 𝑑 ℋ ∞ β , displaystyle|u|_{mathrm{BMO}^{beta}(Q_{0})}vcentcolon=sup_{Qsubset Q_{% 0}}inf_{cinmathbb{R}}frac{1}{l(Q)^{beta}}int_{Q}|u-c|,dmathcal{H}^{% beta}_{infty}, where the supremum is taken over all finite subcubes Q parallel to Q 0 {Q_{0}} , l ⁢ ( Q ) {l(Q)} is the length of the side of the cube Q, and ℋ ∞ β {mathcal{H}^{beta}_{infty}} is the Hausdorff content. In the case β = d {beta=d} we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ ( 0 , d ] {betain(0,d]} one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c , C > 0 {c,C>0} such that ℋ ∞ β ⁢ ( { x ∈ Q : | u ⁢ ( x ) - c Q | > t } ) ≤ C ⁢ l ⁢ ( Q ) β ⁢ exp ⁡ ( - c ⁢ t ∥ u ∥ BMO β ⁢ ( Q 0 ) ) displaystylemathcal{H}^{beta}_{infty}({xin Q:|u(x)-c_{Q}|>t})leq Cl(Q)% ^{beta}expbiggl{(}-frac{ct}{|u|_{mathrm{BMO}^{beta}(Q_{0})}}biggr{)} for every t > 0 {t>0} , u ∈ BMO β ⁢ ( Q 0 ) {uinmathrm{BMO}^{beta}(Q_{0})} , Q ⊂ Q 0 {Qsubset Q_{0}} , and suitable c Q ∈ ℝ {c_{Q}inmathbb{R}} . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44010899","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":"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}