journal: Advances in Calculus of Variations

Impact Factor: 1.4

Zeitschrift

Advances in Calculus of Variations

Managing Editors: Juha Kinnunen und Verena Bögelein
Wissenschaftlicher Beirat: Zoltán Balogh , 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 , Reiner Schaetzle , Zhongwei Shen , Yannick Sire , Ulisse Stefanelli , Susanna Terracini , Yoshihiro Tonegawa , Nizar Touzi , Guofang Wang und Frank Duzaar

Sprache: Englisch

Erstveröffentlichung: 25. Januar 2008

Erscheinungsweise: 4 Hefte pro Jahr

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Über diese Zeitschrift

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

Ihre Vorteile

Your benefits:

Highly specialized journal
Rigorous and thorough peer-review
Significant impact factor, highly ranked in Journal Citation Report®
Ahead of print online publication

Geschichte

The journal was founded in 2008.

Other publications in the field

Open Access
Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities

1. Oktober 2025

Costante Bellettini, Konstantinos Leskas

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We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension 8 the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt–Simon approximation theorem.
Open Access
Existence theory for linear-growth variational integrals with signed measure data

23. September 2025

Eleonora Ficola, Thomas Schmidt

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We develop a semicontinuity-based existence theory in BV \smash{\mathrm{BV}} for a general class of scalar linear-growth variational integrals with additional signed-measure terms. The results extend and refine previous considerations for anisotropic total variations and area-type cases, and they pave the way for a variational approach to the corresponding Euler–Lagrange equations, which involve the signed measure as right-hand-side datum.
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On nonexpansiveness of metric projection operators on Wasserstein spaces

16. September 2025

Anshul Adve, Alpár R. Mészáros

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In this paper, we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces ( P p ⁢ ( R d ) , W p ) (\mathcal{P}_{p}(\mathbf{R}^{d}),W_{p}) . When d = 1 d=1 , as ( P 2 ⁢ ( R ) , W 2 ) (\mathcal{P}_{2}(\mathbf{R}),W_{2}) is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When d > 1 d>1 , we show the failure of this property in two regimes: when p > 1 p>1 is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces ( P p ⁢ ( R d ) , W p ) (\mathcal{P}_{p}(\mathbf{R}^{d}),W_{p}) when d ≥ 2 d\geq 2 and p ∈ ( 1 , + ∞ ) p\in(1,+\infty) are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion . As a corollary of the former, they have nonnegative upper Alexandrov curvature , in a precise sense that we define here. In our analysis, a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.
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Harnack inequality for degenerate elliptic equations with matrix weights

11. September 2025

Giuseppe Di Fazio, Maria Stella Fanciullo, Scott Rodney, Pietro Zamboni

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Harnack inequality and consequent regularity is proved for weak solutions of certain degenerate elliptic equations.
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Compactness of Palais–Smale sequences with controlled Morse index for a Liouville-type functional

29. August 2025

Francesco Malizia

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We prove that Palais–Smale sequences for Liouville-type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. As a byproduct, we obtain a new proof of existence of solutions for Liouville-type mean-field equations in a supercritical regime. Moreover, we also discuss an extension of this result to the case of singular Liouville equations.
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Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV

29. August 2025

Marco Caroccia, Nicolas Van Goethem

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We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincaré inequality, might be employed to completely linearize blow-ups along at least one sequence. We show how to implement such argument by applying it to derive affine blow-up limits for BD {\mathrm{BD}} and BV {\mathrm{BV}} functions around Cantor points. In doing so we identify a specific subset of points – called totally singular points having blow-ups with completely singular gradient measure D ⁢ p = D s ⁢ p {Dp=D^{s}p} , ℰ ⁢ p = ℰ s ⁢ p {{\mathcal{E}}p={\mathcal{E}}^{s}p} – at which such linearization fails.
Open Access
The Capacitary John–Nirenberg Inequality Revisited

29. August 2025

Riju Basak, You-Wei Benson Chen, Prasun Roychowdhury, Daniel Spector

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In this paper, we establish maximal function estimates, Lebesgue differentiation theory, Calderón–Zygmund decompositions, and John–Nirenberg inequalities for translation invariant Hausdorff contents. We further identify a key structural component of these results – a packing condition satisfied by these Hausdorff contents which compensates for the non-linearity of the capacitary integrals. We prove that for any outer capacity, this packing condition is satisfied if and only if the capacity is equivalent to its induced Hausdorff content. Finally, we use this equivalence to extend the preceding theory to general outer capacities which are assumed to satisfy this packing condition.
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A simple proof of the 1-dimensional flat chain conjecture

29. August 2025

Andrea Marchese, Andrea Merlo

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We provide a new proof of the fact that metric 1-currents with compact support in the Euclidean space correspond to Federer–Fleming flat chains; that is, the 1-dimensional case of the so-called flat chain conjecture . While previous proofs rely on the delicate task of constructing Lipschitz functions with small L ∞ {L^{\infty}} -norm but large derivative along certain directions at all points of a given Lebesgue null set (the so-called width functions ), our approach is based primarily on Poincaré’s lemma. This perspective allows us to identify a regularity question concerning the solvability of the equation d ⁢ ω = π {d\omega=\pi} for differential k -forms, a question that is closely related to the general validity of the flat chain conjecture in higher dimensions.
Open Access
A continuous model of transportation in the Heisenberg group

29. August 2025

Michele Circelli, Albert Clop

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We present a minimization problem with a horizontal divergence-type constraint in the Heisenberg group. Our study explores its dual formulation and examines its relationship with the congested optimal transport problem, for 1 < p < + ∞ {1<p<+\infty} , as well as the Monge–Kantorovich problem, in the limit case p = 1 {p=1} .
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A Sobolev gradient flow for the area-normalised Dirichlet energy of H ¹ maps

29. August 2025

Shinya Okabe, Philip Schrader, Glen Wheeler, Valentina-Mira Wheeler

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In this article we study the H 1 {H^{1}} -gradient flow for the energy E ⁢ [ X ] {E[X]} given by the quotient of the Dirichlet energy and the signed enclosed area of an H 1 {H^{1}} map X : 𝕊 → ℝ 2 {X:\mathbb{S}\rightarrow\mathbb{R}^{2}} . We prove that solutions with initially positive signed enclosed area exist eternally, and converge as t → ∞ {t\rightarrow\infty} to a (possibly multiply-covered) circle. In this way we recover an improved parametrised isoperimetric inequality for H 1 {H^{1}} maps.
Open Access
Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory

29. August 2025

Elisa Davoli, Leon Happ

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In this paper we prove a strong two-scale approximation result for sphere-valued maps in the space L 2 ⁢ ( Ω ; W 0 1 , 2 ⁢ ( Q 0 ; 𝕊 2 ) ) {L^{2}(\Omega;W^{1,2}_{0}(Q_{0};\mathbb{S}^{2}))} , where Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} is an open domain and Q 0 ⊂ Q {Q_{0}\subset Q} an open subset of the unit cube Q = ( 0 , 1 ) 3 {Q=(0,1)^{3}} . The proof relies on a generalization of the seminal argument by F. Bethuel and X. M. Zheng to the two-scale setting. We then present an application to a variational problem in high-contrast micromagnetics.
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Universality of renormalisable mappings in two dimensions: the case of polar convex integrands

1. August 2025

Christopher Irving, Benoît Van Vaerenbergh

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We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form ∫ Ω f ⁢ ( | D ⁢ u | ) ⁢ d x {\int_{\Omega}f(|\mathrm{D}u|)\,\mathrm{d}x} . The analysis relies on the condition that the vortex map x | x | {\frac{x}{\lvert x\rvert}} has finite energy and that t ↦ f ⁢ ( t ) {t\mapsto f(\sqrt{t})} is concave. We derive the leading order asymptotics and provide a detailed description of the convergence of W 1 , 1 {\mathrm{W}^{1,1}} -almost minimisers, leading to a characterization of second-order asymptotics. At the core of the method, we prove a ball merging construction (following Jerrard and Sandier’s approach) for a general class of convex integrands. We therefore generalize the approximation by p -harmonic mappings when p ↗ 2 {p\nearrow 2} and can also cover linearly growing functionals, including those of area type.
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Convergence of a heterogeneous Allen–Cahn equation to weighted mean curvature flow

17. Juni 2025

Likhit Ganedi, Alice Marveggio, Kerrek Stinson

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We consider a variational model for heterogeneous phase separation, based on a diffuse interface energy with moving wells. Our main result identifies the asymptotic behavior of the first variation of the phase field energies as the width of the diffuse interface vanishes. This convergence result allows us to deduce a Gibbs–Thomson relation for heterogeneous surface tensions. Proceeding from this information, we prove that (weak) solutions of the Allen–Cahn equation with space dependent potential converge to a BV solution of weighted mean curvature flow, under an energy convergence hypothesis. Additionally, relying on the relative energy technique, we establish a weak-strong uniqueness principle for solutions of weighted mean curvature flow.
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On the asymptotic behavior of a diffraction problem with a thin layer

29. Mai 2025

Paolo Acampora, Emanuele Cristoforoni

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We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set Ω and a thin layer Σ locally described by ε ⁢ h {\varepsilon h} , where h is a positive function defined on the boundary ∂ ⁡ Ω {\partial\Omega} , and ε is the ellipticity constant of the differential operator in the thin layer Σ. We study the problem in the limit for ε going to zero and prove a first-order asymptotic development by Γ-convergence of the associated energy functional.
Open Access
Characterization of sets of finite local and non-local perimeter via non-local heat equation

29. Mai 2025

Andrea Kubin, Domenico Angelo La Manna

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In this paper we provide a characterization of sets with finite local and non-local perimeter via a Γ-convergence result. As an application, we give a short proof of the isoperimetric inequality, both in the local and in the non-local cases.
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Degenerate parabolic p-Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions

29. April 2025

David Cruz-Uribe, Kabe Moen, Yuanzhen Shao

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In this paper we study the degenerate parabolic p -Laplacian, ∂ t ⁡ u - v - 1 ⁢ div ⁡ ( | Q ⁢ ∇ ⁡ u | p - 2 ⁢ Q ⁢ ∇ ⁡ u ) = 0 {\partial_{t}u-v^{-1}\operatorname{div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=0} , where the degeneracy is controlled by a matrix Q and a weight v . With mild integrability assumptions on Q and v , we prove the existence and uniqueness of solutions on any interval [ 0 , T ] {[0,T]} . If we further assume the existence of a degenerate Sobolev inequality with gain, the degeneracy again controlled by v and Q , then we can prove both finite time extinction and ultracontractive bounds. Moreover, we show that there is equivalence between the existence of ultracontractive bounds and the weighted Sobolev inequality.
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An elliptic approximation for phase separation in a fractured material

29. April 2025

Kerrek Stinson, Solveig Wittig

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We consider a free-boundary and free-discontinuity energy connecting phase separation and fracture in an elastic material. The energy excludes the contribution of phase boundaries in the cracked region, providing a heuristic approximation of the interfacial energy in the current material configuration. Our primary result shows that the sharp energy may be recovered via Γ-convergence from a modified Cahn–Hilliard energy coupled with an Ambrosio–Tortorelli-type approximation of the (linear) elastic and fracture energy.
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The L ¹-relaxed area of the graph of the vortex map: Optimal upper bound

29. April 2025

Giovanni Bellettini, Alaa Elshorbagy, Riccardo Scala

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We compute an upper bound for the value of the L 1 {L^{1}} -relaxed area of the graph of the vortex map u : B l ⁢ ( 0 ) ⊂ ℝ 2 → ℝ 2 {u:{\rm B}_{l}(0)\subset\mathbb{R}^{2}\to\mathbb{R}^{2}} , u ⁢ ( x ) := x | x | {u(x):=\frac{x}{|x|}} , x ≠ 0 {x\neq 0} , for all values of l > 0 {l>0} . Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius l in a certain range, in particular l not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved.

Journal Impact Factor	1.4	2024, Journal Citation Reports (Clarivate, 2025)
5-year Journal Impact Factor	1.5	2024, Journal Citation Reports (Clarivate, 2025)
Journal Citation Indicator	1.07	2024, Journal Citation Reports (Clarivate, 2025)
CiteScore	3.2	2024, Scopus (Elsevier B.V., 2025)
SCImago Journal Rank	1.534	2024, SJR (Scimago Lab, 2025; Data Source: Scopus)
Source Normalized Impact per Paper	1.203	2024, CWTS Journal Indicators (CWTS B.V., 2025; Data Source: Scopus)
Mathematical Citation Quotient	1.84

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Managing Editors

Juha Kinnunen
Department of Mathematics
Aalto University,
Posfach 11100
00076 Aalto, Finland
E-mail: juha.k.kinnunen@aalto.fi

Verena Bögelein
Department of Mathematics
University of Salzburg
Hellbrunnerstraße 34
5020 Salzburg, Austria
E-mail: verena.boegelein@plus.ac.at

Founding Editors

Prof. Frank Duzaar
Prof. Nicola Fusco

Editorial Board

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

Pierre Cardaliaguet
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

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

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

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

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

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

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

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

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

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eISSN:

1864-8266

Sprache:

Englisch

Verlag:

De Gruyter

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Erstveröffentlichung:

25. Januar 2008

Erscheinungsweise:

4 Hefte pro Jahr

eISSN: 1864-8266

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Advances in Calculus of Variations

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