ADVANCES IN CALCULUS OF VARIATIONS 发表了高质量的原创研究，重点关注变分法和相关应用的部分，将偏微分方程的工具和方法与几何技术相结合。

The journal "Advances in Calculus of Variations" publishes high-quality original research focusing on the calculus of variations and its related applications, integrating tools and methods from partial differential equations with geometric techniques.

《Advances in Calculus of Variations》期刊发表高质量的原创研究，主要关注变分法及其相关应用，结合偏微分方程的工具与几何技术。

Advances in Calculus of Variations

Abstract We continue our study, started in [A. Češík, G. Gravina and M. Kampschulte, Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision, Calc. Var. Partial Differential Equations 63 2024, 2, Paper No. 55], of (self-)collisions of viscoelastic solids in an inertial regime. We show existence of weak solutions with a corresponding contact force measure in the case of solids with only Lipschitz-regular boundaries. This necessitates a careful study of different concepts of tangent and normal cones and the role these play both in the proofs and in the formulation of the problem itself. Consistent with our previous approach, we study contact without resorting to penalization, i.e., by only relying on a strict non-interpenetration condition. Additionally, we improve the strategies of our previous proof, eliminating the need for regularization terms across all levels of approximation.

Antonín Češík

Giovanni Gravina

Malte Kampschulte

Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries

粘弹性固体在Lipschitz边界上的惯性（自）碰撞

Abstract In this work we prove that the non-negative functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {u\in L^{s}_{\rm loc}(\Omega)} , for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {s&gt;0} , belonging to the De Giorgi classes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">⨍</m:mo> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>σ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo stretchy="false">|</m:mo> <m:mo>∇</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo>-</m:mo> <m:mi>k</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>-</m:mo> </m:msub> <m:mpadded width="+1.7pt"> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>p</m:mi> </m:msup> </m:mpadded> <m:mi>d</m:mi> <m:mi>x</m:mi> <m:mo>⩽</m:mo> <m:mpadded width="+1.7pt"> <m:mfrac> <m:mi>c</m:mi> <m:msup> <m:mi>σ</m:mi> <m:mi>q</m:mi> </m:msup> </m:mfrac> </m:mpadded> <m:mi mathvariant="normal">Λ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mfrac> <m:mi>k</m:mi> <m:mi>r</m:mi> </m:mfrac> <m:mo maxsize="210%" minsize="210%">)</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> <m:msup> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mfrac> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mi>B</m:mi> <m:mi>r</m:mi> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>u</m:mi> <m:mo>⩽</m:mo> <m:mi>k</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:mfrac> <m:mo maxsize="210%" minsize="210%">)</m:mo> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>δ</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mrow> </m:math> \barint_{B_{r(1-\sigma)}(x_{0})}|\nabla(u-k)_{-}|^{p}\,dx\leqslant\frac{c}{% \sigma^{q}}\,\Lambda(x_{0},r,k)\bigg{(}\frac{k}{r}\bigg{)}^{p}\bigg{(}\frac{|B% _{r}(x_{0})\cap\{u\leqslant k\}|}{|B_{r}(x_{0})|}\bigg{)}^{1-\delta}, under proper assumptions on Λ, satisfy a weak Harnack inequality with a constant depending on the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>s</m:mi> </m:msup> </m:math> {L^{s}} -norm of u . Under suitable assumptions on Λ, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying the above condition; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.

Simone Ciani

Eurica Henriques

Igor I. Skrypnik

The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

椭圆泛函的无界极小化子的弱Harnack不等式与广义Orlicz增长

Abstract We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of Γ-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semidiscrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector.

Sergio Conti

Adriana Garroni

Michael Ortiz

A discrete crystal model in three dimensions: The line-tension limit for dislocations

三维离散晶体模型：位错的线张力极限

Abstract We prove that in any rank one symmetric space of non-compact type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>H</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>ℂ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>H</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>ℍ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>H</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>𝕆</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {M\in\{\mathbb{R}H^{n},\mathbb{C}H^{m},\mathbb{H}H^{m},\mathbb{O}H^{2}\}} , geodesic spheres are uniformly quantitatively stable with respect to small <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {C^{1}} -volume preserving perturbations. We quantify the gain of perimeter in terms of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {W^{1,2}} -norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in M . As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter.

Lauro Silini

Quantitative C 1-stability of spheres in rank one symmetric spaces of non-compact type

非紧型秩一对称空间中球面的定量C 1稳定性

Abstract We investigate some of the effects of the lack of compactness in the critical Folland–Stein–Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi’s Γ-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe [M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 1984, 4, 511–517].

Giampiero Palatucci

Mirco Piccinini

Letizia Temperini

Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group

Struwe全局紧性和海森伯群中临界Sobolev嵌入的能量逼近

Abstract We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional quantity, related to the trace of the Hopf differential of the harmonic extension of the original map. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.

Filippo Gaia

The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

分数霍普夫微分与半狄利克雷能量的弱稳态公式

Abstract In this paper, we consider the BV least gradient problem with Dirichlet condition on a part <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⊂</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> </m:math> {\Gamma\subset\partial\Omega} and Neumann boundary condition on its complementary part <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>\</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:mrow> </m:math> {\partial\Omega\backslash\Gamma} . We will show that in the plane this problem is equivalent to an optimal transport problem with import/export taxes on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>\</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> </m:mrow> </m:math> {\partial\Omega\backslash\Gamma} . Thanks to this equivalence, we will be able to show existence and uniqueness of a solution to this mixed least gradient problem and we will also prove some Sobolev regularity on this solution. We note that these results generalize those in [S. Dweik, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> W^{1,p} regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary, Nonlinear Anal. 223 2022, Article ID 113012], where we studied the pure Dirichlet version of this problem.

Abbreviation
Journal Impact	1.34
Quartiles(Global)	MATHEMATICS, APPLIED(Q2)

ISSN	1864-8258, 1864-8266
h-index	25

Publication Information	Publisher: Walter de Gruyter GmbH，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2008，Proportion of original research papers: 100.00%，Self Citation Rate:0.00%， Gold OA Rate: 17.90%
Average review cycle	网友分享经验：>12周，或约稿
Average recruitment ratio	网友分享经验：容易

Advances in Calculus of Variations

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Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries

The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

A discrete crystal model in three dimensions: The line-tension limit for dislocations

Quantitative C ¹-stability of spheres in rank one symmetric spaces of non-compact type

Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group

The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

The least gradient problem with Dirichlet and Neumann boundary conditions

Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries

The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

A discrete crystal model in three dimensions: The line-tension limit for dislocations