journal: Advances in Calculus of Variations

Impact Factor: 1.7

Published since January 25, 2008

Advances in Calculus of Variations

ISSN: 1864-8266

Managing Editors: Frank Duzaar, Juha Kinnunen

Editorial Board: Zoltán Balogh, Verena Bögelein, Pierre Cardiliaguet, Francesca Da Lio, Gianni Dal Maso, Irene Fonseca, Ugo Gianazza, Hitoshi Ishii, Jan Kristensen, Juan Manfredi, Jose Maria Martell, Giuseppe Mingione, Kaj Nystrom, Tristan Riviére, Reiner Schaetzle, Zhongwei Shen, Luis Silvestre, Yannick Sire, Ulisse Stefanelli, Susanna Terracini, Yoshihiro Tonegawa, Nizar Touzi, Enrico Valdinoci, Guofang Wang

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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

Article formats
Research articles

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Significant impact factor, highly ranked in Journal Citation Report®
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History

The journal was founded in 2008.

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Volume 16 Issue 3

July 2023

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.

Ahead of Print / Just Accepted

Ahead of Print / Just Accepted

Volume 16 (2023)

Volume 15 (2022)

Volume 14 (2021)

Volume 13 (2020)

Volume 12 (2019)

Volume 11 (2018)

Volume 10 (2017)

Volume 9 (2016)

Volume 8 (2015)

Volume 7 (2014)

Volume 6 (2013)

Volume 5 (2012)

Volume 4 (2011)

Volume 3 (2010)

Volume 2 (2009)

Volume 1 (2008)

Journal Impact Factor	1.7	2022, Journal Citation Reports (Clarivate, 2023)
5-year Journal Impact Factor	1.7	2022, Journal Citation Reports (Clarivate, 2023)
Journal Citation Indicator	1.70	2022, Journal Citation Reports (Clarivate, 2023)
CiteScore	4.8	2022, Scopus (Elsevier B.V., 2023)
SCImago Journal Rank	1.492	2022, SJR (Scimago Lab, 2023; Data Source: Scopus)
Source Normalized Impact per Paper	1.177	2022, CWTS Journal Indicators (CWTS B.V., 2023; Data Source: Scopus)
Mathematical Citation Quotient	1.84	2022, CWTS Journal Indicators (CWTS B.V., 2023; Data Source: Scopus)

More information about journal rankings

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Editorial

Managing Editors

Frank Duzaar
Department of Data Science
University of Erlangen-Nuremberg
Cauerstrasse 11
91058 Erlangen, Germany
E-mail: frank.duzaar@fau.de

Juha Kinnunen
Department of Mathematics
Aalto University School of Science
P.O. Box 11100
00076 Aalto, Finland
E-mail: juha.k.kinnunen@aalto.fi

Editorial Board

Zoltán Balogh
Mathematisches Institut (MAI)
Universität Bern
Sidlerstrasse 5
3012 Bern, Switzerland
E-mail: zoltan.balogh@math.unibe.ch

Prof. Dr. Verena Bögelein
Department of Mathematics
University of Salzburg
Hellbrunnerstraße 34
5020 Salzburg, Austria

Pierre Cardaliaguet
Professeur
Ceremade (UMR CNRS 7534)
Université Paris-Dauphine
Place du Maréchal De Lattre De Tassigny
75775 Paris Cedex 16 - France
E-mail: cardaliaguet@ceremade.dauphine.fr

Francesca Da Lio
Department of Mathematics
ETHZ
8092 Zurich, Switzerland
E-mail: francesca.dalio@math.ethz.ch

Gianni Dal Maso
SISSA
Via Beirut 2-4
34014 Trieste, Italy
E-mail: dalmaso@sissa.it

Irene Fonseca
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA 15213, U.S.A.
E-mail: fonseca@andrew.cmu.edu

Ugo Gianazza
Deparment of Mathematics “F. Casorati”
University of Pavia
Via Ferrata 1
27100 Pavia, Italy
E-mail: gianazza@imati.cnr.it

Hitoshi Ishii
Department of Mathematics
Faculty of Education and Integrated Arts and Sciences
Waseda University
Tokyo 169-8050, Japan
E-mail: hitoshi.ishii@waseda.jp

Jan Kristensen
Mathematical Institute
24-29 St. Giles'
University of Oxford
Oxford OX1 3LB, England
E-mail: kristens@maths.ox.ac.uk

Juan Manfredi
Department of Mathematics
University of Pittsburgh
Pittsburgh, PA 15260, U.S.A.
E-mail: manfredi@pitt.edu

José María Martell
Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM
Consejo Superior de Investigaciones Cientificas
C/ Nicolas Cabrera 13-15
28049 Madrid, Spain
E-mail: chema.martell@icmat.es

Prof. Dr. Giuseppe Mingione
Dipartimento di Matematica
Università di Parma
Parco Area delle Scienze 53/a
Campus
43100 Parma, Italy
E-mail: giuseppe.mingione@unipr.it

Prof. Dr. Kaj Nyström
Department of Mathematics
Uppsala University
Box 480
751 06 Uppsala, Sweden
E-mail: kaj.nystrom@math.uu.se

Prof. Dr. Reiner Schaetzle
Universität Tübingen
Mathem. Institut
Auf der Morgenstelle 10
Tübingen, Germany
E-mail: rsch@everest.mathematik.uni-tuebingen.de

Prof. Dr. Zhongwei Shen
University of Kentucky
Department of Mathematics
737 Patterson Office Tower
Lexington, KY 40506-0027, USA
E-mail: zshen2@uky.edu

Prof. Dr. Luis Silvestre
Department of Mathematics
University of Chicago
5734 S. University Avenue
Chicago, IL 60637, U.S.A.
E-mail: luis@math.uchicago.edu

Prof. Dr. Yannick Sire
Johns Hopkins University
Dept. of Mathematics
Baltimore, USA
E-mail: ysire1@jhu.edu

Ulisse Stefanelli
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
E-mail: ulisse.stefanelli@univie.ac.at

Prof. Dr. Susanna Terracini
Dipartimento di Matematica "Giuseppe Peano"
UniversitaÌ di Torino, Italia
E-mail: susanna.terracini@unito.it

Yoshihiro Tonegawa
Department of Mathematics
Tokyo Institute of Technology
2-12-1 Ookayama, Meguro-ku,
Tokyo 152-8551, Japan
E-mail: Tonegawa@math.titech.ac.jp

Prof. Dr. Nizar Touzi
Professor, Applied Mathematics
Centre de Mathématiques Appliquées
Ecole Polytechnique
UMR CNRS 7641
91128 Palaiseau Cedex, France
E-mail: nizar.touzi@polytechnique.edu

Prof. Dr. Enrico Valdinoci
University of Western Australia
Perth, Australia
E-mail: enrico.valdinoci@uwa.edu.au

Prof. Dr. Guofang Wang
Mathematisches Institut
Albert-Ludwigs-Universität
Ernst-Zermelo-Str. 1
79104 Freiburg im Breisgau, Germany
E-mail: guofang.wang@math.uni-freiburg.de

(Deutsch)

Online ISSN: 1864-8266
Print ISSN: 1864-8258
Type: Journal
Language: English
Publisher: De Gruyter
First published: January 25, 2008
Publication Frequency: 4 Issues per Year
Audience: Researchers in the field of calculus of variation and related applications