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 Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>→</m:mo> <m:mi>M</m:mi> </m:mrow> </m:math> {\mathrm{SO}(M)\rightarrow M} of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>Iso</m:mi> <m:mi>o</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≅</m:mo> <m:mrow> <m:mi>SO</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\mathrm{Iso}_{o}(M)\cong\mathrm{SO}(M)} : A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

Marcos Salvai

A split special Lagrangian calibration associated with frame vorticity

与框架涡度相关的分裂特殊拉格朗日校准

Abstract By using the monotonicity of the log Sobolev functionals, we prove a no breathers theorem for noncompact harmonic Ricci flows under conditions on infimum of log Sobolev functionals and curvatures. As an application, we obtain a no breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature.

Jiarui Chen

Qun Chen

No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature

非紧致调和Ricci流的无呼吸子定理与渐近非负Ricci曲率

Abstract Given <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ε</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {{\varepsilon}_{0}&gt;0} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>I</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>ℕ</m:mi> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {I\in\mathbb{N}\cup\{0\}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>K</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {K_{0},H_{0}\geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Inj</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:msub> <m:mi>ε</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> {\operatorname{Inj}(X)\geq{\varepsilon}_{0}} and with the supremum of absolute sectional curvature at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {K_{0}} , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>↬</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {M\looparrowright X} be a complete immersed surface of constant mean curvature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {H\in[0,H_{0}]} with index at most I . For such <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>↬</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {M\looparrowright X} , we prove a structure theorem which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M , where the norm of the second fundamental form takes on large local maximum values.

William H. Meeks III

Joaquín Pérez

Hierarchy structures in finite index CMC surfaces

有限指数CMC曲面中的层次结构

Abstract Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>-</m:mo> <m:mrow> <m:mi>div</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo maxsize="160%" minsize="160%">(</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>-</m:mo> <m:mi>ν</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mfrac> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mfrac> </m:mrow> <m:mo maxsize="160%" minsize="160%">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="normal">Ω</m:mi> <m:mi>T</m:mi> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>×</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> u_{t}-\operatorname{div}\Bigl{(}(\lvert Du\rvert-\nu)_{+}^{p-1}\frac{Du}{% \lvert Du\rvert}\Bigr{)}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p\geq 2} , ν is a positive constant and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> </m:msub> </m:math> {(\,\cdot\,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> </m:math> {u_{t}} . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.

Pasquale Ambrosio

Antonia Passarelli di Napoli

Regularity results for a class of widely degenerate parabolic equations

一类广泛退化抛物方程的正则性结果

Abstract The model introduced in [45] in the framework of the theory on stress-driven rearrangement instabilities (SDRI) [3, 43] for the morphology of crystalline materials under stress is considered. As in [45] and in agreement with the models in [50, 55], a mismatch strain, rather than a Dirichlet condition as in [19], is included into the analysis to represent the lattice mismatch between the crystal and possible adjacent (supporting) materials. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of the restriction to a finite number m of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous results for epitaxially strained thin films and material cavities [6, 35, 34, 45]. Due to the lack of compactness and lower semicontinuity for the sequences of m -minimizers, i.e., minimizers among configurations with at most m connected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence of m -minimizers’ energies to the energy infimum as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {m\to\infty} . Finally, regularity properties for the morphology satisfied by every minimizer are established.

Shokhrukh Y. Kholmatov

Paolo Piovano

Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain

二维SDRI模型中极小值的存在性：失配应变下的润湿和去润湿状态

Abstract We extend the Calderón–Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>A</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> {\mathcal{L}^{s}_{A}u=f} , where the operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>A</m:mi> <m:mi>s</m:mi> </m:msubsup> </m:math> {\mathcal{L}^{s}_{A}} is formally given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msubsup> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>A</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> <m:mrow> <m:mfrac> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mfrac> <m:mo>⁢</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⊗</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>y</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> \mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-% y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>s</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0&lt;s&lt;1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}} taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {A(\,\cdot\,,y)} is uniformly Holder continuous and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo>inf</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {\inf_{x\in\mathbb{R}^{n}}A(x,x)&gt;0} , then for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:mrow> </m:math> {f\in L^{p}_{\rm loc}} , for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p\geq 2} , the solution vector <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>H</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:mrow> </m:math> {u\in H^{2s-\delta,p}_{\rm loc}} for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\delta\in(0,s)} .

Tadele Mengesha

Armin Schikorra

Adisak Seesanea

Sasikarn Yeepo

Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

Calderón-Zygmund理论对于具有Hölder正则系数的高度耦合线性非局部方程系统的研究

Abstract Given a bounded open connected set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {\Omega\subset\mathbb{R}^{2}} with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo>,</m:mo> <m:mi>γ</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {\alpha,\beta,\gamma\in\mathbb{R}^{2}} , vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Det</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{Det}(\nabla\kern 0.569055ptu)} and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Det</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{Det}(\nabla\kern 0.569055ptu)} . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions.

Riccardo Scala

Giuseppe Scianna

On the L 1-relaxed area of graphs of BV piecewise constant maps taking three values

关于取三个值的BV分段常值映射图形的L 1松弛面积

Abstract Consider a positive functional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi>t</m:mi> <m:mi>T</m:mi> </m:msubsup> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:msup> <m:mi>y</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {F(y):=\int_{t}^{T}L(s,y(s),y^{\prime}(s))\,ds} , defined on the space of Sobolev functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <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:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>;</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {W^{1,p}([t,T];{\mathbb{R}}^{n})} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {p\geq 1} . This functional is minimized among functions y that may satisfy one or both endpoint conditions. The Lagrangian L is allowed to assume the value <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {+\infty} . In numerous applications minimizers may not be explicit or even may not exist. In such circumstances, it is crucial to know that the infimum of F can be approximated using a sequence of Lipschitz functions that meet the given boundary conditions. However, there are instances where this approximation is not feasible, even with polynomial Lagrangians that meet Tonelli’s existence conditions: this situation is referred to as the Lavrentiev phenomenon. Some results are present in the literature if one requires the Lipschitz approximations to preserve just one endpoint constraint or when the Lagrangian is finite valued. The present paper deals with problems with two endpoint constraints and Lagrangians that are allowed to take extended values. As a byproduct, our Lipschitz approximations may preserve given state constraints. The extended-valued case is challenging since the phenomenon may occur even when the Lagrangian is constant on its effective domain, whose topology becomes relevant. Once assumed that the Lagrangian is radial convex on the rays of the last variable, our findings offer new insights, even when the Lagrangian L is real-valued, autonomous, and there are no state constraints.

Carlo Mariconda

Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints

避免约束下的一维非自治泛函的Lavrentiev间隙

Abstract In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

Vanessa Hüsken

On prescribing the number of singular points in a Cosserat-elastic solid

关于在Cosserat弹性固体中规定奇点数量的研究

Abstract Given is a bounded domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> {\partial\Omega} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function <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:mo stretchy="false">(</m:mo> <m:msup> <m:mi>u</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>m</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbf{u}=(u^{1},\dots,u^{m})} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {m\geq 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem.

短名
Journal Impact	1.34
国际分区	MATHEMATICS, APPLIED(Q2)
期刊索引	SCI Q1中科院 3 区

ISSN	1864-8258, 1864-8266
h-index	25
国内分区	数学(3区)数学数学(3区)数学应用数学(3区)

涉及主题	数学数学分析纯数学物理组合数学几何学量子力学计算机科学工程类哲学离散数学
出版信息	出版商: Walter de Gruyter GmbH，出版周期: ，期刊类型: journal，开源期刊: 否
基本数据	创刊年份: 2008，原创研究文献占比: 100.00%，自引率:0.00%， Gold OA占比: 17.90%
平均审稿周期	网友分享经验：>12周，或约稿
平均录用比例	网友分享经验：容易

Advances in Calculus of Variations

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