journal: Advances in Calculus of Variations

Impact Factor: 1.7

Veröffentlicht seit 25. Januar 2008

Advances in Calculus of Variations

ISSN: 1864-8266

Managing Editors: Frank Duzaar, Juha Kinnunen

Wissenschaftlicher Beirat: 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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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

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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.

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Band 16 Heft 4

Oktober 2023

Öffentlich zugänglich 1. Oktober 2023

Frontmatter

Seiten: i-iv

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Open Access 20. April 2022

Harnack inequality for parabolic equations with coefficients depending on time

Fabio Paronetto

Seiten: 791-821

Mehr PDF downloaden

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.

4. Dezember 2021

Morse theory and the calculus of variations

Anthony Tromba

Seiten: 823-834

Mehr

Abstract

We prove for the first time that classical Morse theory applies to functionals of the form 𝒥 ⁢ ( u ) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x \displaystyle\mathcal{J}(u)=\frac{1}{2}\int_{\Omega}A^{ij}_{\alpha\beta}(x)% \frac{\partial u^{i}}{\partial x^{\alpha}}\frac{\partial u^{j}}{\partial x^{% \beta}}\,dx+\int_{\Omega}G(x,u)\,dx where u : Ω → ℝ N {u:\Omega\to\mathbb{R}^{N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} compact with C ∞ {C^{\infty}} boundary ∂ ⁡ Ω {\partial\Omega} , u | ∂ ⁡ Ω = φ {u|_{\partial\Omega}=\varphi} , and we argue that this is the largest class to which Morse theory applies.

6. Januar 2022

Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities

Stefan Krömer, Martin Kružík, Elvira Zappale

Seiten: 835-865

Mehr

Abstract

For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v , we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

22. Juli 2022

Fractional Poincaré and localized Hardy inequalities on metric spaces

Bartłomiej Dyda, Juha Lehrbäck, Antti V. Vähäkangas

Seiten: 867-884

Mehr

Abstract

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 pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.

27. Juni 2023

Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters

Irene Fonseca, Pan Liu, Xin Yang Lu

Seiten: 885-902

Mehr

Abstract

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.

Open Access 31. Mai 2022

The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems

Filippo Riva, Giovanni Scilla, Francesco Solombrino

Seiten: 903-934

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Abstract

The notion of inertial balanced viscosity (IBV) solution to rate-independent evolutionary processes is introduced. Such solutions are characterized by an energy balance where a suitable, rate-dependent, dissipation cost is optimized at jump times. The cost is reminiscent of the limit effect of small inertial terms. Therefore, this notion proves to be a suitable one to describe the asymptotic behavior of evolutions of mechanical systems with rate-independent dissipation in the limit of vanishing inertia and viscosity. It is indeed proved, in finite dimension, that these evolutions converge to IBV solutions. If the viscosity operator is neglected, or has a nontrivial kernel, the weaker notion of inertial virtual viscosity (IVV) solutions is introduced, and the analogous convergence result holds. Again in a finite-dimensional context, it is also shown that IBV and IVV solutions can be obtained via a natural extension of the minimizing movements algorithm, where the limit effect of inertial terms is taken into account.

Open Access 15. März 2022

Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions

Antonio Giuseppe Grimaldi, Erica Ipocoana

Seiten: 935-960

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Abstract

We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫ Ω ⟨ A ⁢ ( x , D ⁢ u ) , D ⁢ ( φ - u ) ⟩ ⁢ d x ≥ 0 for all ⁢ φ ∈ K ψ ⁢ ( Ω ) , \int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{for all}\ \varphi\in\mathcal{K}_{\psi}(\Omega), where Ω is a bounded open subset of R n \mathbb{R}^{n} , ψ ∈ W 1 , p ⁢ ( Ω ) \psi\in W^{1,p}(\Omega) is a fixed function called obstacle and K ψ ⁢ ( Ω ) = { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢ a.e. in ⁢ Ω } \mathcal{K}_{\psi}(\Omega)=\{w\in W^{1,p}(\Omega):w\geq\psi\ \text{a.e. in}\ \Omega\} is the class of admissible functions. Assuming that the gradient of the obstacle belongs to some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution.

22. Juni 2022

On some variational problems involving capacity, torsional rigidity, perimeter and measure

Michiel van den Berg, Andrea Malchiodi

Seiten: 961-974

Mehr

Abstract

We investigate the existence of a maximiser among open, bounded, convex sets in R d \mathbb{R}^{d} , d ≥ 3 d\geq 3 , for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if d = 2 d=2 ), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.

1. April 2022

Sub-elliptic boundary value problems in flag domains

Tuomas Orponen, Michele Villa

Seiten: 975-1059

Mehr

Abstract

A flag domain in ℝ 3 {\mathbb{R}^{3}} is a subset of ℝ 3 {\mathbb{R}^{3}} of the form { ( x , y , t ) : y < A ⁢ ( x ) } {\{(x,y,t):y<A(x)\}} , where A : ℝ → ℝ {A\colon\mathbb{R}\to\mathbb{R}} is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn–Laplacian △ ♭ = X 2 + Y 2 {\bigtriangleup^{\flat}=X^{2}+Y^{2}} in flag domains Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} , with L 2 {L^{2}} -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 {L^{2}} -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.

31. Mai 2022

On master test plans for the space of BV functions

Francesco Nobili, Enrico Pasqualetto, Timo Schultz

Seiten: 1061-1092

Mehr

Abstract

We prove that on an arbitrary metric measure space a countable collection of test plans is sufficient to recover all BV {\operatorname{BV}} functions and their total variation measures. In the setting of non-branching 𝖢𝖣 ⁢ ( K , N ) {\mathsf{CD}(K,N)} spaces (with finite reference measure), we can additionally require these test plans to be concentrated on geodesics.

Open Access 25. Oktober 2022

Regularity results for an optimal design problem with lower order terms

Luca Esposito, Lorenzo Lamberti

Seiten: 1093-1122

Mehr PDF downloaden

Abstract

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 ∂ ⁡ E {\partial E} . Namely we show that the Hausdorff dimension of the singular set is strictly smaller than n - 1 {n-1} .

Open Access 29. Juni 2022

On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1

Francesco Della Pietra, Carlo Nitsch, Francescantonio Oliva, Cristina Trombetti

Seiten: 1123-1135

Mehr PDF downloaden

Abstract

In this paper, we study the Γ-limit, as p → 1 {p\to 1} , of the functional J p ⁢ ( u ) = ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p , J_{p}(u)=\frac{\int_{\Omega}\lvert\nabla u\rvert^{p}+\beta\int_{\partial\Omega% }\lvert u\rvert^{p}}{\int_{\Omega}\lvert u\rvert^{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_{u\in\operatorname{BV}(\Omega),\,u\not\equiv 0}% \frac{\lvert Du\rvert(\Omega)+\min(\beta,1)\int_{\partial\Omega}\lvert u\rvert% }{\int_{\Omega}\lvert u\rvert} which is the limit as p → 1 + {p\to 1^{+}} of λ ⁢ ( Ω , p , β ) = min u ∈ W 1 , p ⁢ ( Ω ) ⁡ J p ⁢ ( u ) {\lambda(\Omega,p,\beta)=\min_{u\in 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 ) {\beta\in(-1,0)} and minimizes Λ ⁢ ( Ω , β ) {\Lambda(\Omega,\beta)} when β ∈ [ 0 , ∞ ) {\beta\in[0,\infty)} .

Ahead-of-Print / Just-Accepted

Ahead-of-Print / Just-Accepted

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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)

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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. EnricoValdinoci
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

(English)

Online ISSN: 1864-8266
Print ISSN: 1864-8258
Typ: Zeitschrift
Sprache: Englisch
Verlag: De Gruyter
Erstveröffentlichung: 25. Januar 2008
Erscheinungsweise: 4 Hefte pro Jahr
Zielgruppe: Researchers in the field of calculus of variation and related applications