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Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions Adv. Calc. Var. (IF 1.7) Pub Date : 2024-02-20 Roberto Alicandro, Lucia De Luca, Mariapia Palombaro, Marcello Ponsiglione
We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure
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On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary Adv. Calc. Var. (IF 1.7) Pub Date : 2024-02-20 Jan-Henrik Metsch
We consider the area-preserving Willmore evolution of surfaces ϕ that are close to a half-sphere with a small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a “half-spherical” shape. Additionally, we investigate the asymptotic behavior of the flow and prove that for large times the barycenter of the surfaces approximately
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Non-local BV functions and a denoising model with L 1 fidelity Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-30 Konstantinos Bessas, Giorgio Stefani
We study a general total variation denoising model with weighted L 1 {L^{1}} fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel K, and the approximation term is given by the L 1 {L^{1}} norm with respect to a non-singular measure with positively lower-bounded L ∞ {L^{\infty}} density. We provide a detailed analysis of the space of non-local
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Another proof of the existence of homothetic solitons of the inverse mean curvature flow Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-30 Shu-Yu Hsu
We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in ℝ n × ℝ {\mathbb{R}^{n}\times\mathbb{R}} , n ≥ 2 {n\geq 2} , of the form ( r , y ( r ) ) {(r,y(r))} or ( r ( y ) , y ) {(r(y),y)} , where r = | x | {r=|x|} , x ∈ ℝ n {x\in\mathbb{R}^{n}} , is the radially symmetric coordinate and y ∈ ℝ {y\in\mathbb{R}} . More precisely for any 1 n
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Generalized minimizing movements for the varifold Canham–Helfrich flow Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-12 Katharina Brazda, Martin Kružík, Ulisse Stefanelli
The gradient flow of the Canham–Helfrich functional is tackled via the generalized minimizing movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered C 1 , 1 {C^{1,1}} surfaces, we provide a Li–Yau-type estimate for the Canham–Helfrich energy and prove the conservation of
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Quasiconformal, Lipschitz, and BV mappings in metric spaces Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-12 Panu Lahti
Consider a mapping f : X → Y {f\colon X\to Y} between two metric measure spaces. We study generalized versions of the local Lipschitz number Lip f {\operatorname{Lip}f} , as well as of the distortion number H f {H_{f}} that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping f ∈ BV loc ( X ; Y ) {f\in\mathrm{BV}_{\mathrm{loc}}(X;Y)}
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Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-11 Marin Bužančić, Elisa Davoli, Igor Velčić
We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than
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Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-10 Edir Júnior Ferreira Leite, Marcos Montenegro
The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 𝒜 {\Delta_{p}^{\cal A}} , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p {L^{p}} energy ℰ p , Ω {{\cal E}_{p,\Omega}} from
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Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-10 Qing Guo
We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ u ∂ t ( x , t ) + ( - Δ ) p s u ( x , t ) = f ( t , u ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ( x , t )
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Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-10 Dominik Engl, Stefan Krömer, Martin Kružík
We perform via Γ-convergence a 2d-1d dimension reduction analysis of a single-slip elastoplastic body in large deformations. Rigid plastic and elastoplastic regimes are considered. In particular, we show that limit deformations can essentially freely bend even if subjected to the most restrictive constraints corresponding to the elastically rigid single-slip regime. The primary challenge arises in
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A sub-Riemannian maximum modulus theorem Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-01 Federico Buseghin, Nicolò Forcillo, Nicola Garofalo
In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one.
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The 2+1-convex hull of a~finite set Adv. Calc. Var. (IF 1.7) Pub Date : 2024-01-01 Pablo Angulo, Carlos García-Gutiérrez
Rank-one convexity is a weak form of convexity related to convex integration and the elusive notion of quasiconvexity, but more amenable both in theory and practice. However, exact algorithms for computing the rank one convex hull of a finite set are only known for some special cases of separate convexity with a finite number of directions. Both inner approximations either with laminates or T 4 {T_{4}}
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Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term Adv. Calc. Var. (IF 1.7) Pub Date : 2023-11-29 Stefano Biagi, Francesco Esposito, Luigi Montoro, Eugenio Vecchi
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by p-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.
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Sobolev embeddings and distance functions Adv. Calc. Var. (IF 1.7) Pub Date : 2023-11-27 Lorenzo Brasco, Francesca Prinari, Anna Chiara Zagati
On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p \mathcal{D}^{{1,p}}_{0} into L q L^{q} and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 𝑝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when
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Minimizers of 3D anisotropic interaction energies Adv. Calc. Var. (IF 1.7) Pub Date : 2023-11-22 José Antonio Carrillo, Ruiwen Shu
We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of the recent work [J. A. Carrillo and R. Shu, Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D, Comm. Pure Appl. Math. (2023), 10.1002/cpa.22162] in two dimensions to the present setting. For potentials
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Discrete approximation of nonlocal-gradient energies Adv. Calc. Var. (IF 1.7) Pub Date : 2023-11-19 Andrea Braides, Andrea Causin, Margherita Solci
We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.
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Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity Adv. Calc. Var. (IF 1.7) Pub Date : 2023-10-26 Shuntaro Tsubouchi
In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p-Laplacian-type diffusion operator with 1 < p < ∞ {1
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Relaxed many-body optimal transport and related asymptotics Adv. Calc. Var. (IF 1.7) Pub Date : 2023-10-26 Ugo Bindini, Guy Bouchitté
Optimization problems on probability measures in ℝ d {\mathbb{R}^{d}} are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, for example in Density Functional Theory, the interaction cost is repulsive and described by a two-point function c ( x , y ) = ℓ ( | x - y | ) {c(x,y)=\ell(\lvert x-y\rvert)} where ℓ : ℝ + → [ 0 , ∞ ] {
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A Weierstrass extremal field theory for the fractional Laplacian Adv. Calc. Var. (IF 1.7) Pub Date : 2023-10-26 Xavier Cabré, Iñigo U. Erneta, Juan-Carlos Felipe-Navarro
In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian in the
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Sobolev contractivity of gradient flow maximal functions Adv. Calc. Var. (IF 1.7) Pub Date : 2023-10-26 Simon Bortz, Moritz Egert, Olli Saari
We prove that the energy dissipation property of gradient flows extends to semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the p-parabolic extension does not increase the p-norm of the gradient when p > 2 {p>2} . We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable
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Minimizing movements for anisotropic and inhomogeneous mean curvature flows Adv. Calc. Var. (IF 1.7) Pub Date : 2023-10-03 Antonin Chambolle, Daniele De Gennaro, Massimiliano Morini
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
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Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data Adv. Calc. Var. (IF 1.7) Pub Date : 2023-10-03 Iwona Chlebicka, Flavia Giannetti, Anna Zatorska-Goldstein
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 × ℝ n → ℝ n {{\mathcal{A}}:\Omega\times{\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
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Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains Adv. Calc. Var. (IF 1.7) Pub Date : 2023-08-31 Xavier Pellet, Lucia Scardia, Caterina Ida Zeppieri
In this paper, we study the asymptotic behaviour of a family of random free-discontinuity energies E ε {E_{\varepsilon}} defined in a randomly perforated domain, as ε goes to zero. The functionals E ε {E_{\varepsilon}} model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to
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Regularity results for a class of widely degenerate parabolic equations Adv. Calc. Var. (IF 1.7) Pub Date : 2023-08-24 Pasquale Ambrosio, Antonia Passarelli di Napoli
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 Du\rvert-\nu)_{+}^{p-1}\frac{Du}{% \lvert Du\rvert}\Bigr{)}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain
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Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball Adv. Calc. Var. (IF 1.7) Pub Date : 2023-08-24 Chao Xia, Xuwen Zhang
In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most n - 3 {n-3} . With this inequality, we classify all volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular
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Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain Adv. Calc. Var. (IF 1.7) Pub Date : 2023-08-24 Shokhrukh Y. Kholmatov, Paolo Piovano
The model introduced in [45] in the framework of the theory on stress-driven rearrangement instabilities (SDRI) [3, 43] for the morphology of crystalline materials under stress is considered. As in [45] and in agreement with the models in [50, 55], a mismatch strain, rather than a Dirichlet condition as in [19], is included into the analysis to represent the lattice mismatch between the crystal and
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On the Hölder regularity of all extrema in Hilbert’s 19th Problem Adv. Calc. Var. (IF 1.7) Pub Date : 2023-08-23 Friedrich Tomi, Anthony Tromba
Let Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} be a C 1 {C^{1}} smooth compact domain. Furthermore, let F : Ω × ℝ n N → ℝ {F:\Omega\times\mathbb{R}^{nN}\to\mathbb{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 {\Omega\times\mathbb{R}^{nN}} and F strictly convex in p. Consider an n N × n N {nN\times
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Hierarchy structures in finite index CMC surfaces Adv. Calc. Var. (IF 1.7) Pub Date : 2023-07-25 William H. Meeks III, Joaquín Pérez
Given ε 0 > 0 {{\varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {I\in\mathbb{N}\cup\{0\}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}\geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ ε 0 {\operatorname{Inj}(X)\geq{\varepsilon}_{0}} and with the supremum of absolute sectional curvature at most K 0 {K_{0}} , and let M ↬ X {M\looparrowright X} be a complete immersed surface of constant
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A split special Lagrangian calibration associated with frame vorticity Adv. Calc. Var. (IF 1.7) Pub Date : 2023-07-24 Marcos Salvai
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)\cong\mathrm{SO}(M)} :
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No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature Adv. Calc. Var. (IF 1.7) Pub Date : 2023-07-24 Jiarui Chen, Qun Chen
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.
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Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity Adv. Calc. Var. (IF 1.7) Pub Date : 2023-06-26 José C. Bellido, Javier Cueto, Carlos Mora-Corral
We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but
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Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters Adv. Calc. Var. (IF 1.7) Pub Date : 2023-06-26 Irene Fonseca, Pan Liu, Xin Yang Lu
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.
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On functions of bounded β-dimensional mean oscillation Adv. Calc. Var. (IF 1.7) Pub Date : 2023-05-31 You-Wei Chen, Daniel Spector
In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}\subset\mathbb{R}^{d}\to\mathbb{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_{Q\subset Q_{% 0}}\inf_{c\
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On isosupremic vectorial minimisation problems in L ∞ with general nonlinear constraints Adv. Calc. Var. (IF 1.7) Pub Date : 2023-05-31 Ed Clark, Nikos Katzourakis
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
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A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1 Adv. Calc. Var. (IF 1.7) Pub Date : 2023-05-05 Haizhong Li, Ruijia Zhang
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}+\lvert\overline{\nabla}\rho\rvert^{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}
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A generalized fractional Pohozaev identity and applications Adv. Calc. Var. (IF 1.7) Pub Date : 2023-05-02 Sidy Moctar Djitte, Mouhamed Moustapha, Tobias Weth
We prove a fractional Pohozaev-type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to the nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity
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Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity Adv. Calc. Var. (IF 1.7) Pub Date : 2023-04-26 Jianlu Zhang
In the paper we prove the convergence of viscosity solutions u λ {u_{\lambda}} as λ → 0 + {\lambda\rightarrow 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,\quad\alpha(x)\geq 0,\quad x\in\mathbb% {T}^{n} under suitable convex and monotonic conditions on H : T
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Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets Adv. Calc. Var. (IF 1.7) Pub Date : 2023-03-30 Jakob Fuchs, Tim Laux
In this work, we analyze Merriman, Bence and Osher’s thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoğlu and Otto, we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the
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The first Grushin eigenvalue on cartesian product domains Adv. Calc. Var. (IF 1.7) Pub Date : 2023-03-30 Paolo Luzzini, Luigi Provenzano, Joachim Stubbe
In this paper, we consider the first eigenvalue λ 1 ( Ω ) {\lambda_{1}(\Omega)} of the Grushin operator Δ G := Δ x 1 + | x 1 | 2 s Δ x 2 {\Delta_{G}:=\Delta_{x_{1}}+\lvert x_{1}\rvert^{2s}\Delta_{x_{2}}} with Dirichlet boundary conditions on a bounded domain Ω of ℝ d = ℝ d 1 + d 2 {\mathbb{R}^{d}=\mathbb{R}^{d_{1}+d_{2}}} . We prove that λ 1 ( Ω ) {\lambda_{1}(\Omega)} admits a unique minimizer
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A characterization of gauge balls in ℍ n by horizontal curvature Adv. Calc. Var. (IF 1.7) Pub Date : 2023-03-30 Chiara Guidi, Vittorio Martino, Giulio Tralli
In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n {{\mathbb{H}^{n}}} via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 {\mathbb{H}^{1}} under an assumption on the location of the singular set, and in ℍ n {\mathbb{H}^{n}} for n ≥ 2 {n\geq 2} in the proper class of horizontally umbilical hypersurfaces
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On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1 Adv. Calc. Var. (IF 1.7) Pub Date : 2023-02-27 Xiangyu Liang
In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau’s problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau’s problem. It gives a very good description of local behavior
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A twist in sharp Sobolev inequalities with lower order remainder terms Adv. Calc. Var. (IF 1.7) Pub Date : 2023-01-27 Emmanuel Hebey
Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 3 {n\geq 3} . Let also A be a smooth symmetrical positive ( 0 , 2 ) {(0,2)} -tensor field in M. By the Sobolev embedding theorem, we can write that there exist K , B > 0 {K,B>0} such that for any u ∈ H 1 ( M ) {u\in H^{1}(M)} , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ∥ ∇ A u ∥ L 2 2 + B ∥ u ∥ L 1 2 \|u\|_{L^{2^{\star}}}^{2}\leq
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The homogeneous causal action principle on a compact domain in momentum space Adv. Calc. Var. (IF 1.7) Pub Date : 2023-01-27 Felix Finster, Michelle Frankl, Christoph Langer
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.
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Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity Adv. Calc. Var. (IF 1.7) Pub Date : 2023-01-27 Shaoguang Shi, Zhichun Zhai, Lei Zhang
In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ( x ) {u(x)} is so-called ( α , p ) {(\alpha
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Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case Adv. Calc. Var. (IF 1.7) Pub Date : 2022-12-07 Gianni Dal Maso, Rodica Toader
We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.
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The Lp Minkowski problem for q-torsional rigidity Adv. Calc. Var. (IF 1.7) Pub Date : 2022-11-23 Bin Chen, Xia Zhao, Weidong Wang, Peibiao Zhao
In this paper, we introduce the L p {L_{p}} q-torsional measure for p ∈ ℝ {p\in\mathbb{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 <
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Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains Adv. Calc. Var. (IF 1.7) Pub Date : 2022-11-11 Jimmy Lamboley, Arian Novruzi, Michel Pierre
We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ( Ω ) {E_{f}(\Omega)} of the Laplacian in the domain Ω or the first eigenvalue λ 1 ( Ω ) {\lambda_{1}(\Omega)} of the Dirichlet-Laplacian. Usually, one considers minimization
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Functional inequalities and applications to doubly nonlinear diffusion equations Adv. Calc. Var. (IF 1.7) Pub Date : 2022-11-11 Iwona Chlebicka, Nikita Simonov
We study weighted inequalities of Hardy and Hardy–Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.
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Homogenization of high-contrast composites under differential constraints Adv. Calc. Var. (IF 1.7) Pub Date : 2022-10-24 Elisa Davoli, Martin Kružík, Valerio Pagliari
We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical
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Regularity results for an optimal design problem with lower order terms Adv. Calc. Var. (IF 1.7) Pub Date : 2022-10-24 Luca Esposito, Lorenzo Lamberti
We study the regularity of the interface for optimal energy configurations of functionals involving bulk energies with an additional perimeter penalization of the interface. Here we allow a more general structure for the energy functional in the bulk term. For a minimal configuration ( E , u ) {(E,u)} , the Hölder continuity of u is well known. We give an estimate for the singular set of the boundary
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An inequality for the normal derivative of the Lane–Emden ground state Adv. Calc. Var. (IF 1.7) Pub Date : 2022-10-10 Rupert L. Frank, Simon Larson
We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 0\leq q-1\leq 1 , that is, minimizers of the Dirichlet integral among L q L^{q} -normalized functions. Our main result is a sharp lower bound on the L 2 L^{2} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets
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Stability of the ball under volume preserving fractional mean curvature flow Adv. Calc. Var. (IF 1.7) Pub Date : 2022-10-07 Annalisa Cesaroni, Matteo Novaga
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.
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Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds Adv. Calc. Var. (IF 1.7) Pub Date : 2022-09-29 Giuseppina Barletta, Andrea Cianchi, Vladimir Maz’ya
Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L q {L^{q}} - or L ∞ {L^{\infty}} -bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function
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No Lavrentiev gap for some double phase integrals Adv. Calc. Var. (IF 1.7) Pub Date : 2022-08-29 Filomena De Filippis, Francesco Leonetti
We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ( u ) ≔ ∫ Ω f ( x , D u ( x ) ) 𝑑 x , \mathcal{F}(u)\coloneqq\int_{\Omega}f(x,Du(x))\,dx, where the density f ( x , z ) {f(x,z)} is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n {x\in\Omega\subset\mathbb{R}^{n}} , it satisfies the ( p , q ) {(p,q)} -growth conditions | z | p ⩽ f ( x , z ) ⩽ L ( 1 + | z |
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Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition Adv. Calc. Var. (IF 1.7) Pub Date : 2022-08-29 Murat Akman, Steve Hofmann, José María Martell, Tatiana Toro
Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative
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Properties of the free boundaries for the obstacle problem of the porous medium equations Adv. Calc. Var. (IF 1.7) Pub Date : 2022-08-29 Sunghoon Kim, Ki-Ahm Lee, Jinwan Park
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
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Interpolation inequalities for partial regularity Adv. Calc. Var. (IF 1.7) Pub Date : 2022-08-29 Christoph Hamburger
We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p {L^{p}} norms in combination with an L p {L^{p}} estimate with low exponent p > 1 {p>1} . For the
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On the existence of canonical multi-phase Brakke flows Adv. Calc. Var. (IF 1.7) Pub Date : 2022-07-22 Salvatore Stuvard, Yoshihiro Tonegawa
This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature
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Fractional Poincaré and localized Hardy inequalities on metric spaces Adv. Calc. Var. (IF 1.7) Pub Date : 2022-07-21 Bartłomiej Dyda, Juha Lehrbäck, Antti V. Vähäkangas
We prove fractional Sobolev–Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning
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Concavity properties for solutions to p-Laplace equations with concave nonlinearities Adv. Calc. Var. (IF 1.7) Pub Date : 2022-07-21 William Borrelli, Sunra Mosconi, Marco Squassina
We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the p-Laplace operator and a general nonlinearity satisfying concavity-type assumptions. This provides an extension of results previously known in the literature only for the torsion and the first eigenvalue equations. In the semilinear case p = 2 {p=2} the results are