Published by De Gruyter

Volume 16 Issue 3

Issue of Advances in Calculus of Variations

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Publicly Available July 1, 2023

Frontmatter

Page range: i-iv

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June 1, 2021

A strict inequality for the minimization of the Willmore functional under isoperimetric constraint

Andrea Mondino, Christian Scharrer

Page range: 529-540

Abstract

Inspired by previous work of Kusner and Bauer–Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller, Mondino and Rivière, our strict inequality leads to existence of minimizers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below 8 ⁢ π {8\pi} . Besides the geometric interest, such a minimization problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.

January 6, 2022

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Huyuan Chen, Laurent Véron

Page range: 541-558

Abstract

We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢ in ⁢ Ω , u = 0 ⁢ in ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k -th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.

July 22, 2022

A note on indecomposable sets of finite perimeter

Panu Lahti

Page range: 559-570

Abstract

Bonicatto, Pasqualetto and Rajala (2020) proved that a decomposition theorem for sets of finite perimeter into indecomposable sets, known to hold in Euclidean spaces, holds also in complete metric spaces equipped with a doubling measure, supporting a Poincaré inequality, and satisfying an isotropicity condition. We show that the last assumption can be removed.

December 2, 2021

Gamma-convergence of Gaussian fractional perimeter

Alessandro Carbotti, Simone Cito, Domenico Angelo La Manna, Diego Pallara

Page range: 571-595

Abstract

We prove the Γ-convergence of the renormalised Gaussian fractional s -perimeter to the Gaussian perimeter as s → 1 - {s\to 1^{-}} . Our definition of fractional perimeter comes from that of the fractional powers of Ornstein–Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the Γ-limit does not depend on the dimension.

January 4, 2022

A speed preserving Hilbert gradient flow for generalized integral Menger curvature

Jan Knappmann, Henrik Schumacher, Daniel Steenebrügge, Heiko von der Mosel

Page range: 597-635

Abstract

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Open Access January 6, 2022

A new proof of compactness in G(S)BD

Stefano Almi, Emanuele Tasso

Page range: 637-650

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Abstract

We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.

August 30, 2022

Interpolation inequalities for partial regularity

Christoph Hamburger

Page range: 651-663

Abstract

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 second, we provide a functional-analytic proof, that also sheds light upon the A -harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.

April 20, 2022

Area formula for regular submanifolds of low codimension in Heisenberg groups

Francesca Corni, Valentino Magnani

Page range: 665-688

Abstract

We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is constructed by an arbitrary homogeneous distance. Among the arguments of the proof, we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsically differentiable functions.

April 20, 2022

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Ana Hurtado, César Rosales

Page range: 689-704

Abstract

We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 {\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}} with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 {\mathcal{S}_{0}} with boundary C 0 = { 0 } × 𝕊 1 {C_{0}=\{0\}\times\mathbb{S}^{1}} minimize the sub-Riemannian area among compact C 1 {C^{1}} surfaces with the same boundary. We also see that the only C 2 {C^{2}} solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ {\mathcal{S}_{\lambda}} with λ > 0 {\lambda>0} uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} for C 1 {C^{1}} sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

Open Access December 3, 2021

Gradient estimates for an orthotropic nonlinear diffusion equation

Pierre Bousquet, Lorenzo Brasco, Chiara Leone, Anna Verde

Page range: 705-730

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Abstract

We consider a quasilinear degenerate parabolic equation driven by the orthotropic p -Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.

August 30, 2022

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Murat Akman, Steve Hofmann, José María Martell, Tatiana Toro

Page range: 731-766

Abstract

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 version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u ) and L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-% \operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion [M. Akman, S. Hofmann, J. M. Martell and T. Toro, Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition, preprint 2021, https://arxiv.org/abs/1901.08261v3] is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaining a square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above.

Open Access January 6, 2022

Integral representation of local functionals depending on vector fields

Fares Essebei, Andrea Pinamonti, Simone Verzellesi

Page range: 767-789

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Abstract

Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X -Sobolev spaces, which admit an integral representation in terms of X , i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.

About this journal

Objective
Advances in Calculus of Variations publishes high quality original research focusing on that part of calculus of variation and related applications which combines tools and methods from partial differential equations with geometrical techniques.

Topics

existence and regularity for minimizers and critical points
variational methods for partial differential equations
geometrical aspects of calculus of variation: minimal surfaces, harmonic mappings, geometrically motivated flows, curvature equations, quasi-conformal mappings
applications of variational methods to non-linear elasticity, free boundary problems, general relativity
geometric measure theory

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