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Extension of convex functions from a hyperplane to a half-space Calc. Var. (IF 2.1) Pub Date : 2024-04-17 John M. Ball, Christopher L. Horner
It is shown that a possibly infinite-valued proper lower semicontinuous convex function on \(\mathbb {R}^n\) has an extension to a convex function on the half-space \(\mathbb {R}^n\times [0,\infty )\) which is finite and smooth on the open half-space \(\mathbb {R}^n\times (0,\infty )\). The result is applied to nonlinear elasticity, where it clarifies how the condition of polyconvexity of the free-energy
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Existence and asymptotic behavior for $$L^2$$ -norm preserving nonlinear heat equations Calc. Var. (IF 2.1) Pub Date : 2024-04-17 Paolo Antonelli, Piermarco Cannarsa, Boris Shakarov
We consider a nonlinear parabolic equation with a nonlocal term which preserves the \(L^2\)-norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in \(H^1\). Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in \(H^1\) to a stationary state. For a ball, we prove strong convergence to the
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Inverse mean curvature flow with a free boundary in hyperbolic space Calc. Var. (IF 2.1) Pub Date : 2024-04-17 Xiaoxiang Chai
We study the inverse mean curvature flow with a free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.
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Sobolev inequalities in manifolds with asymptotically nonnegative curvature Calc. Var. (IF 2.1) Pub Date : 2024-04-17 Yuxin Dong, Hezi Lin, Lingen Lu
Using the ABP-method as in a recent work by Brendle (Commun Pure Appl Math 76:2192–2218, 2022), we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative Ricci/sectional curvature. These inequalities generalize those given by Brendle in the case of complete Riemannian manifolds with nonnegative
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Nonlocal approximation of nonlinear diffusion equations Calc. Var. (IF 2.1) Pub Date : 2024-04-13 José Antonio Carrillo, Antonio Esposito, Jeremy Sheung-Him Wu
We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme
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Quantitative stability of harmonic maps from $${\mathbb {R}}^2$$ to $${\mathbb {S}}^2$$ with a higher degree Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Bin Deng, Liming Sun, Jun-cheng Wei
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Sharp uncertainty principles on metric measure spaces Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Bang-Xian Han, Zhe-Feng Xu
We prove the rigidity of the Heisenberg–Pauli–Weyl uncertainty principle and the Caffarelli–Kohn–Nirenberg interpolation inequality, on metric measure spaces satisfying measure contraction property. Non-trivial examples fitting our setting include Finsler manifolds with non-negative Ricci curvature and many ideal sub-Riemannian manifolds, such as Heisenberg groups, the Grushin plane and Sasakian structures
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Normalized solutions for Schrödinger equations with potentials and general nonlinearities Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Yanyan Liu, Leiga Zhao
In this paper, we are concerned with the nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+V(x)u+\lambda u=g(u)\text { in }{\mathbb {R}}^{N}\text {, }\lambda \in {\mathbb {R}}, \end{aligned}$$ with prescribed \(L^{2}\)-norm \(\int _{{\mathbb {R}}^{N}}u^{2}dx=\rho ^{2}\) and \( \lim _{|x|\rightarrow +\infty }V(x)=:V_{\infty }\le +\infty \) under general assumptions on g which allows at least
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Partial regularity for degenerate parabolic systems with general growth via caloric approximations Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Jihoon Ok, Giovanni Scilla, Bianca Stroffolini
We establish a partial regularity result for solutions of parabolic systems with general \(\varphi \)-growth, where \(\varphi \) is an Orlicz function. In this setting we can develop a unified approach that is independent of the degeneracy of system and relies on two caloric approximation results: the \(\varphi \)-caloric approximation, which was introduced in Diening et al. (Calc Var Partial Differ
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Sharp uniform bound for the quaternionic Monge-Ampère equation on hyperhermitian manifolds Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Marcin Sroka
We provide the sharp \(C^0\) estimate for the quaternionic Monge-Ampère equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends only on \(L^p\) norm of the right hand side for any \(p>2\) (as suggested by the local case studied in Sroka (Anal. PDE 13(6):1755-1776, 2020)). Moreover, the
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A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Samuel Daudin, Benjamin Seeger
The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at
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Existence of energy-variational solutions to hyperbolic conservation laws Calc. Var. (IF 2.1) Pub Date : 2024-04-13 Thomas Eiter, Robert Lasarzik
We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This
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Non-degeneracy of solution for critical Lane–Emden systems with linear perturbation Calc. Var. (IF 2.1) Pub Date : 2024-04-10 Yuxia Guo, Yichen Hu, Shaolong Peng
In this paper, we consider the following elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |v|^{p-1}v +\epsilon (\alpha u + \beta _1 v), &{}\quad {\hbox {in}}\; \Omega , \\ -\Delta v = |u|^{q-1}u+\epsilon (\beta _2 u +\alpha v), &{}\quad {\hbox {in}}\;\Omega , \\ u=v=0,&{}\quad {\hbox {on}}\; \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a smooth
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Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions Calc. Var. (IF 2.1) Pub Date : 2024-04-10 Irfan Glogić, Sarah Kistner, Birgit Schörkhuber
We study singularity formation for the heat flow of harmonic maps from \(\mathbb {R}^d\). For each \(d \ge 4\), we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.
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Trudinger–Moser and Hardy–Trudinger–Moser inequalities for the Aharonov–Bohm magnetic field Calc. Var. (IF 2.1) Pub Date : 2024-04-10 Guozhen Lu, Qiaohua Yang
The main results of this paper concern sharp constant of the Trudinger–Moser inequality in \(\mathbb {R}^{2}\) for Aharonov–Bohm magnetic fields. This is a borderline case of the Hardy type inequalities for Aharonov–Bohm magnetic fields in \(\mathbb {R}^2\) studied by A. Laptev and T. Weidl. As an application, we obtain the exact asymptotic estimates on best constants of magnetic Hardy–Sobolev inequalities
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Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure Calc. Var. (IF 2.1) Pub Date : 2024-04-09 Marin Bužančić, Elisa Davoli, Igor Velčić
An effective model is identified for thin perfectly plastic plates whose microstructure consists of the periodic assembling of two elastoplastic phases, as the periodicity parameter converges to zero. Assuming that the thickness of the plates and the periodicity of the microstructure are comparably small, a limiting description is obtained in the quasistatic regime via simultaneous homogenization and
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Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility Calc. Var. (IF 2.1) Pub Date : 2024-04-09 Helmut Abels, Mingwen Fei, Maximilian Moser
We consider the sharp interface limit of a Navier–Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility \(m_\varepsilon =\sqrt{\varepsilon }\), where the small parameter \(\varepsilon >0\) related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence
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Classical Solutions of Rayleigh–Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains Calc. Var. (IF 2.1) Pub Date : 2024-04-06 Fei Jiang, Youyi Zhao
We study the existence of unstable classical solutions of the Rayleigh–Taylor instability problem (abbr. RT problem) of an inhomogeneous incompressible viscous fluid in a bounded domain. We find that, by using an existence theory of (steady) Stokes problem and an iterative technique, the initial data of classical solutions of the linearized RT problem can be modified to new initial data, which can
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Nonrelativistic limit of normalized solutions to a class of nonlinear Dirac equations Calc. Var. (IF 2.1) Pub Date : 2024-04-05 Pan Chen, Yanheng Ding, Qi Guo, Hua-Yang Wang
In this paper, we investigate the nonrelativistic limit of normalized solutions to a nonlinear Dirac equation as given below: $$\begin{aligned} {\left\{ \begin{array}{ll} &{}-i c\sum \limits _{k=1}^3\alpha _k\partial _k u +mc^2 \beta {u}- \Gamma * (K |{u}|^\kappa ) K|{u}|^{\kappa -2}{u}- P |{u}|^{s-2}{u}=\omega {u}, \\ &{}\displaystyle \int _{\mathbb {R}^3}\vert u \vert ^2 dx =1. \end{array}\right
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Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates Calc. Var. (IF 2.1) Pub Date : 2024-04-05 Carlo Alberto Antonini
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Sharp anisotropic singular Trudinger–Moser inequalities in the entire space Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Kaiwen Guo, Yanjun Liu
In this paper, we investigate sharp singular Trudinger–Moser inequalities involving the anisotropic Dirichlet norm \(\left( \int _{{\mathbb {R}}^{N}}F^{N}(\nabla u)\;\textrm{d}x\right) ^{\frac{1}{N}}\) in Sobolev-type space \(D^{N,q}(\mathbb {R}^{N})\), \(N\ge 2\), \(q\ge 1\). Here \(F:\mathbb {R}^{N}\rightarrow [0,+\infty )\) is a convex function of class \(C^{2}(\mathbb {R}^{N}\setminus \{0\})\)
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Monge–Ampère operators and valuations Calc. Var. (IF 2.1) Pub Date : 2024-04-04
Abstract Two classes of measure-valued valuations on convex functions related to Monge–Ampère operators are investigated and classified. It is shown that the space of all valuations with values in the space of complex Radon measures on \(\mathbb {R}^n\) that are locally determined, continuous, dually epi-translation invariant as well as translation equivariant, is finite dimensional. Integral representations
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Manifolds for which Huber’s Theorem holds Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Yuxiang Li, Zihao Wang
Extensions of Huber’s Theorem to higher dimensions with \(L^\frac{n}{2}\) bounded scalar curvature have been extensively studied over the years. In this paper, we delve into the properties of conformal metrics on a punctured ball with \(\Vert R\Vert _{L^\frac{n}{2}}<+\infty \), aiming to identify necessary geometric constraints for Huber’s theorem to be applicable. Unexpectedly, such metrics are more
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Subgradient estimates for a nonlinear subparabolic equation on complete pseudo-Hermitian manifolds Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Wenjing Wu
Let \((M,J,\theta )\) be a complete noncompact pseudo-Hermitian manifold which satisfies the CR sub-Laplacian comparison property. We first obtain local subgradient estimates for positive solutions to the following nonlinear subparabolic equation: $$\begin{aligned} u_t=\Delta _bu+au\ln u+bu, \end{aligned}$$ on \(M\times [0,+\infty )\), where a, b are two real constants. As a application, we derive
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Geometric linearisation for optimal transport with strongly p-convex cost Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Lukas Koch
We prove a geometric linearisation result for minimisers of optimal transport problems where the cost-function is strongly p-convex and of p-growth. Initial and target measures are allowed to be rough, but are assumed to be close to Lebesgue measure.
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Curvature estimates for a class of Hessian quotient type curvature equations Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Jundong Zhou
In this paper, we are concerned with the hypersurface that can be locally represented as a graph and satisfies a class of Hessian quotient type curvature equations. We establish interior curvature estimates under the condition of \(0\le l
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Solenoidal improvement of Rellich-Hardy inequalities with power weights Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Naoki Hamamoto
We compute best constants in functional inequalities which we call Rellich-Hardy inequalities for solenoidal vector fields on \(\mathbb {R}^N\); this gives a solenoidal improvement of the inequalities whose best constants are known for unconstrained fields. We give a stronger version of Rellich-Leray inequality studied in our previous work (Hamamoto in Calc Var Partial Differ Equ 60:65, 2021). In order
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Higher differentiability and integrability for some nonlinear elliptic systems with growth coefficients in BMO Calc. Var. (IF 2.1) Pub Date : 2024-04-04 Gioconda Moscariello, Giulio Pascale
We consider local solutions u of nonlinear elliptic systems of the type $$\begin{aligned} \text {div} \,A(x, Du) = \text {div} \, F \qquad \text {in} \quad \Omega \subset \mathbb {R}^n, \end{aligned}$$ where \(u: \Omega \rightarrow \mathbb {R}^N\) is in a weighted \(W^{1, p}_{loc}\) space, with \(p \ge 2\), F is in a weighted \(W^{1, 2}_{loc}\) space and x \(\rightarrow \) \(A(x, \xi )\) has growth
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Topological degree for Chern–Simons Higgs models on finite graphs Calc. Var. (IF 2.1) Pub Date : 2024-04-04
Abstract Let (V, E) be a finite connected graph. We are concerned about the Chern–Simons Higgs model 0.1 $$\begin{aligned} \Delta u=\lambda e^u(e^u-1)+f, \end{aligned}$$ where \(\Delta \) is the graph Laplacian, \(\lambda \) is a real number and f is a function on V. When \(\lambda >0\) and \(f=4\pi \sum _{i=1}^N\delta _{p_i}\) , \(N\in {\mathbb {N}}\) , \(p_1,\cdots ,p_N\in V\) , the equation (0.1)
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Nonuniqueness of solutions to the $$L_p$$ chord Minkowski problem Calc. Var. (IF 2.1) Pub Date : 2024-04-04
Abstract This paper explores the nonuniqueness of solutions to the \(L_p\) chord Minkowski problem for negative p. The \(L_p\) chord Minkowski problem was recently posed by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the \(L_p\) chord measure of a convex body, and it includes the chord Minkowski problem
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Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum Calc. Var. (IF 2.1) Pub Date : 2024-03-21 Xiang He, Zuoqin Wang
In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold M of dimension \(n\ge 3\) with a non-empty smooth boundary \(\partial M\). We show that for any finite increasing sequence of real numbers \(0
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Monotonicity of Steklov eigenvalues on graphs and applications Calc. Var. (IF 2.1) Pub Date : 2024-03-21
Abstract In this paper, we obtain monotonicity of Steklov eigenvalues on graphs which as a special case on trees extends the results of He and Hua (Steklov flows on trees and applications. arXiv:2103.07696) to higher Steklov eigenvalues and gives affirmative answers to two problems proposed in He and Hua (arXiv:2103.07696). As applications of the monotonicity of Steklov eigenvalues, we obtain some
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Asymptotic behaviour of solutions to the anisotropic doubly critical equation Calc. Var. (IF 2.1) Pub Date : 2024-03-15 Francesco Esposito, Luigi Montoro, Berardino Sciunzi, Domenico Vuono
The aim of this paper is to deal with the anisotropic doubly critical equation $$\begin{aligned} -\Delta _p^H u - \frac{\gamma }{[H^\circ (x)]^p} u^{p-1} = u^{p^*-1} \qquad \text {in } {\mathbb {R}}^N, \end{aligned}$$ where H is in some cases called Finsler norm, \(H^\circ \) is the dual norm, \(1
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Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory Calc. Var. (IF 2.1) Pub Date : 2024-03-14 Xue-yun Lin, Cheng-jie Liu, Ting Zhang
In this paper, we study the well-posedness theory of the magneto-micropolar boundary layer and justify the high Reynolds numbers limit for the magneto-micropolar system with Prandtl boundary layer expansion. If the initial tangential magnetic field is nondegenerate, we obtain the local-in-time existence, uniqueness of solutions for the incompressible magneto-micropolar boundary layer equations with
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On the hydrostatic Navier–Stokes equations with Gevrey class 2 data Calc. Var. (IF 2.1) Pub Date : 2024-03-14
Abstract In this paper, we study the two-dimensional hydrostatic Navier–Stokes equations in the strip domain \({\mathbb R}\times \mathbb {T}\) . Motivated by Gérard-Varet et al. (Anal PDE 13(5):1417–1455 2020), we obtain the local well-posedness result in Gevrey class 2 when the initial data is a small perturbation of some convex function. Then we justify strictly the limit from the anisotropic Navier–Stokes
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Local well-posedness for incompressible neo-Hookean elastic equations in almost critical Sobolev spaces Calc. Var. (IF 2.1) Pub Date : 2024-03-07 Huali Zhang
Inspired by a pioneer work of Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023), we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to \(H^{\frac{n+2}{2}+}({\mathbb {R}}^n) \times H^{\frac{n}{2}+}({\mathbb {R}}^n)\) (\(n=2,3\)), where \(\frac{n+2}{2}\) and \(\frac{n}{2}\) is respectively
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Stability of quermassintegral inequalities along inverse curvature flows Calc. Var. (IF 2.1) Pub Date : 2024-03-07 Caroline VanBlargan, Yi Wang
In this paper, we consider the stability of quermassintegral inequalities along a inverse curvature flow. We choose a special rescaling of the flow such that the k-th quermassintegral is decreasing and the \(k-1\)-th quermassintegral is preserved. Along this rescaled flow, we prove that the decreasing rate of the k-th quermassintegral is faster than the Fraenkel asymmetry of the domain along the flow
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The Dirichlet problem for Lévy-stable operators with $$L^2$$ -data Calc. Var. (IF 2.1) Pub Date : 2024-03-07 Florian Grube, Thorben Hensiek, Waldemar Schefer
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On the weak Harnack estimate for nonlocal equations Calc. Var. (IF 2.1) Pub Date : 2024-03-07
Abstract We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation $$\begin{aligned} \partial _t(|u|^{p-2}u) + (-\Delta _p)^s u = 0 \end{aligned}$$ for \(p\in (1,\infty )\) and \(s \in (0,1)\) . Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even
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A homogenization result in finite plasticity Calc. Var. (IF 2.1) Pub Date : 2024-03-07
Abstract We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the \(\Gamma \) -convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to \(\textsf{SL}(3)\)
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Asymptotic stability of boundary layer to the multi-dimensional isentropic Euler-Poisson equations arising in plasma physics Calc. Var. (IF 2.1) Pub Date : 2024-03-07 Yufeng Chen, Wenjuan Ding, Junpei Gao, Mengyuan Lin, Lizhi Ruan
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Towards existence theorems to affine p-Laplace equations via variational approach Calc. Var. (IF 2.1) Pub Date : 2024-03-07 Edir Júnior Ferreira Leite, Marcos Montenegro
The present work deals with theory of critical points to the energy functional on \(W^{1,p}_0(\Omega )\) defined by $$\begin{aligned} \Phi _\mathcal{A}(u) = \frac{1}{p} \mathcal{E}^p_{p,\Omega }(u) - \int _\Omega F(x,u)\,dx, \end{aligned}$$ where \(\mathcal{E}^p_{p,\Omega }\) stands for the affine p-energy introduced for \(p > 1\) by Lutwak et al. (J Differ Geom 62:17–38, 2002). Its development is
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Multiplicity one for min–max theory in compact manifolds with boundary and its applications Calc. Var. (IF 2.1) Pub Date : 2024-03-07 Ao Sun, Zhichao Wang, Xin Zhou
We prove the multiplicity one theorem for min–max free boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free boundary hypersurface with prescribed mean curvature, which includes the regularity theory for minimizers, compactness theory, and a generic min–max theory with
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On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential Calc. Var. (IF 2.1) Pub Date : 2024-03-07
Abstract We are concerned with the existence of blowing-up solutions to the following boundary value problem $$\begin{aligned} -\Delta u= \lambda V(x) e^u-4\pi N {\varvec{\delta }}_0 \;\hbox { in } B_1,\quad u=0 \;\hbox { on }\partial B_1, \end{aligned}$$ where \(B_1\) is the unit ball in \(\mathbb {R}^2\) centered at the origin, V(x) is a positive smooth potential, N is a positive integer ( \(N\ge
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Normalized ground states for the fractional Schrödinger–Poisson system with critical nonlinearities Calc. Var. (IF 2.1) Pub Date : 2024-03-02 Yuxi Meng, Xiaoming He
In this paper we study the existence and properties of ground states for the fractional Schrödinger–Poisson system with combined power nonlinearities $$\begin{aligned}{\left\{ \begin{array}{ll}\displaystyle (-\Delta )^su-\phi |u|^{2^*_s-3}u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*_s-2}u, &{}x \in {\mathbb {R}}^{3},\\ (-\Delta )^{s}\phi =|u|^{2^*_s-1}, &{}x \in {\mathbb {R}}^{3},\end{array}\right. } \end{aligned}$$
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Singular solutions to the $$\sigma _k$$ -Yamabe equation with prescribed asymptotics Calc. Var. (IF 2.1) Pub Date : 2024-02-27 Zirui Li, Qing Han
In this work, we explore the asymptotic behaviors of positive solutions to the \(\sigma _k\)-Yamabe equation. Extending Han and Li’s previous work on the Yamabe equation, we demonstrate that for every approximate solution \({\widetilde{w}}\) of a specified order, there exists a corresponding solution w that closely approximates \({\widetilde{w}}\). Our study further presents a concrete method for constructing
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Matrix Li–Yau–Hamilton estimates under Ricci flow and parabolic frequency Calc. Var. (IF 2.1) Pub Date : 2024-02-26 Xiaolong Li, Qi S. Zhang
We prove matrix Li–Yau–Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply these estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature
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The Lavrentiev phenomenon in calculus of variations with differential forms Calc. Var. (IF 2.1) Pub Date : 2024-02-22 Anna Kh. Balci, Mikhail Surnachev
In this article we study convex non-autonomous variational problems with differential forms and corresponding function spaces. We introduce a general framework for constructing counterexamples to the Lavrentiev gap, which we apply to several models, including the double phase, borderline case of double phase potential, and variable exponent. The results for the borderline case of double phase potential
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Transportation onto log-Lipschitz perturbations Calc. Var. (IF 2.1) Pub Date : 2024-02-20 Max Fathi, Dan Mikulincer, Yair Shenfeld
We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean spaces and uniform measures on spheres as source measures. More generally, we prove results for source measures on manifolds satisfying strong curvature assumptions
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Global regularity for nonlinear systems with symmetric gradients Calc. Var. (IF 2.1) Pub Date : 2024-02-16 Linus Behn, Lars Diening
We study global regularity of nonlinear systems of partial differential equations depending on the symmetric part of the gradient with Dirichlet boundary conditions. These systems arise from variational problems in plasticity with power growth. We cover the full range of exponents \(p \in (1,\infty )\). As a novelty the degenerate case for \(p>2\) is included. We present a unified approach for all
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The weak solutions to complex Hessian equations Calc. Var. (IF 2.1) Pub Date : 2024-02-12 Wei Sun
In this paper, we shall study existence of weak solutions to complex Hessian equations on compact Hermitian manifolds. With appropriate assumptions, it is possible to obtain weak solutions in different senses.
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Dirichlet problem for Krylov type equation in conformal geometry Calc. Var. (IF 2.1) Pub Date : 2024-02-12
Abstract In this paper, we study a class of nonlinear elliptic equations in the Krylov type, which can be viewed as a generalization of the Hessian equation for Schouten tensor. After a conformal change, we considered the Dirichlet problem for a modified Schouten tensor in the smooth closed Riemannian manifold with smooth boundary. A unique k-admissible solution can be assured under some suitable settings
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Least gradient problem with Dirichlet condition imposed on a part of the boundary Calc. Var. (IF 2.1) Pub Date : 2024-02-12 Wojciech Górny
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Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision Calc. Var. (IF 2.1) Pub Date : 2024-02-10 Antonín Češík, Giovanni Gravina, Malte Kampschulte
We study the time evolution of non-linear viscoelastic solids in the presence of inertia and (self-)contact. For this problem we prove the existence of weak solutions for arbitrary times and initial data, thereby solving an open problem in the field. Our construction directly includes the physically correct, measure-valued contact forces and thus obeys conservation of momentum and an energy balance
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The $$L^p$$ -Fisher–Rao metric and Amari–C̆encov $$\alpha $$ -Connections Calc. Var. (IF 2.1) Pub Date : 2024-02-10 Martin Bauer, Alice Le Brigant, Yuxiu Lu, Cy Maor
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Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps Calc. Var. (IF 2.1) Pub Date : 2024-02-10 Hugo Lavenant, Léonard Monsaingeon, Luca Tamanini, Dmitry Vorotnikov
If \(u: \Omega \subset \mathbb {R}^d \rightarrow \textrm{X}\) is a harmonic map valued in a metric space \(\textrm{X}\) and \(\textsf{E}: \textrm{X}\rightarrow \mathbb {R}\) is a convex function, in the sense that it generates an \(\textrm{EVI}_0\)-gradient flow, we prove that the pullback \(\textsf{E}\circ u: \Omega \rightarrow \mathbb {R}\) is subharmonic. This property was known in the smooth Riemannian
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Einstein manifolds and curvature operator of the second kind Calc. Var. (IF 2.1) Pub Date : 2024-02-09 Zhi-Lin Dai, Hai-Ping Fu
We prove that a compact Einstein manifold of dimension \(n\ge 4\) with nonnegative curvature operator of the second kind is a constant curvature space by Bochner technique. Moreover, we obtain that compact Einstein manifolds of dimension \(n\ge 11\) with \(\left[ \frac{n+2}{4} \right] \)-nonnegative curvature operator of the second kind, \(4\ (\text{ resp. },8,9,10)\)-dimensional compact Einstein manifolds
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Bernstein type theorems of translating solitons of the mean curvature flow in higher codimension Calc. Var. (IF 2.1) Pub Date : 2024-02-07 Hongbing Qiu
By carrying out point-wise estimates for the mean curvature, we prove Bernstein type theorems of complete translating solitons of the mean curvature flow in higher codimension under various geometric conditions.
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An existence result for the Kazdan–Warner equation with a sign-changing prescribed function Calc. Var. (IF 2.1) Pub Date : 2024-02-07 Linlin Sun, Jingyong Zhu