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1 2 3 4 | Last

 Calculus of Variations and Partial Differential Equations   [SJR: 2.455]   [H-I: 44]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0835 - ISSN (Online) 0944-2669    Published by Springer-Verlag  [2353 journals]
• A multidimensional Birkhoff theorem for time-dependent Tonelli
Hamiltonians
• Authors: Marie-Claude Arnaud; Andrea Venturelli
Abstract: Abstract Let M be a closed and connected manifold, $$H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}$$ a Tonelli 1-periodic Hamiltonian and $${\mathscr {L}}\subset T^*M$$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $${\mathscr {L}}$$ is invariant by the time-one map of H, then $${\mathscr {L}}$$ is a graph over M. An interesting consequence in the autonomous case is that in this case, $${\mathscr {L}}$$ is invariant by all the time t maps of the Hamiltonian flow of H.
PubDate: 2017-07-21
DOI: 10.1007/s00526-017-1210-0
Issue No: Vol. 56, No. 4 (2017)

• Construction of type II blow-up solutions for a semilinear parabolic
system with higher dimension
Abstract: Abstract This paper is concerned with blow-up solutions for a semilinear parabolic system with a power type nonlinearity. Non self-similar blow-up solution is constructed by the matched asymptotic expansions. One component of this solution converges to the singular steady state, and another component converges to zero in self-similar variables.
PubDate: 2017-07-19
DOI: 10.1007/s00526-017-1213-x
Issue No: Vol. 56, No. 4 (2017)

• Selection problems of $${{\mathbb {Z}}}^2$$ Z 2 -periodic entropy
solutions and viscosity solutions
• Authors: Kohei Soga
Abstract: Abstract $${{\mathbb {Z}}}^2$$ -periodic entropy solutions of hyperbolic scalar conservation laws and $${{\mathbb {Z}}}^2$$ -periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate $${{\mathbb {Z}}}^2$$ -periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural $${{\mathbb {Z}}}^2$$ -periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.
PubDate: 2017-07-17
DOI: 10.1007/s00526-017-1208-7
Issue No: Vol. 56, No. 4 (2017)

• Parabolic Lipschitz truncation and caloric approximation
• Authors: L. Diening; S. Schwarzacher; B. Stroffolini; A. Verde
Abstract: Abstract We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show that almost p-caloric functions are close to p-caloric functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.
PubDate: 2017-07-17
DOI: 10.1007/s00526-017-1209-6
Issue No: Vol. 56, No. 4 (2017)

• The regularity of conformal target harmonic maps
• Authors: Tristan Rivière
Abstract: Abstract We establish that any weakly conformal $$W^{1,2}$$ map from a Riemann surface S into a closed oriented sub-manifold $$N^n$$ of an euclidian space $${\mathbb {R}}^m$$ realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form S into $$N^n$$ .
PubDate: 2017-07-14
DOI: 10.1007/s00526-017-1215-8
Issue No: Vol. 56, No. 4 (2017)

• Local uniqueness and periodicity for the prescribed scalar curvature
problem of fractional operator in $${\mathbb {R}}^{N}$$ R N
• Authors: Yuxia Guo; Jianjun Nie; Miaomiao Niu; Zhongwei Tang
Abstract: Abstract Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent: 0.1 \begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned} For $$N\ge 4$$ and $$\sigma \in (\frac{1}{2}, 1),$$ we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).
PubDate: 2017-07-14
DOI: 10.1007/s00526-017-1194-9
Issue No: Vol. 56, No. 4 (2017)

• On global and local minimizers of prestrained thin elastic rods
• Authors: Marco Cicalese; Matthias Ruf; Francesco Solombrino
Abstract: Abstract We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By $$\Gamma$$ -convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations.
PubDate: 2017-07-13
DOI: 10.1007/s00526-017-1197-6
Issue No: Vol. 56, No. 4 (2017)

• The concentration behavior of ground state solutions for a fractional
Schrödinger–Poisson system
• Authors: Yuanyang Yu; Fukun Zhao; Leiga Zhao
Abstract: Abstract In this paper, we study the following fractional Schrödinger–Poisson system 0.1 \begin{aligned} \left\{ \begin{array}{ll} \varepsilon ^{2s}(-\Delta )^s u +V(x)u+\phi u=K(x) u ^{p-2}u,\,\,\text {in}~\mathbb {R}^3,\\ \\ \varepsilon ^{2s}(-\Delta )^s \phi =u^2,\,\,\text {in}~\mathbb {R}^3, \end{array} \right. \end{aligned} where $$\varepsilon >0$$ is a small parameter, $$\frac{3}{4}<s<1$$ , $$4<p<2_s^*:=\frac{6}{3-2s}$$ , $$V(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)$$ has positive global minimum, and $$K(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)$$ is positive and has global maximum. We prove the existence of a positive ground state solution by using variational methods for each $$\varepsilon >0$$ sufficiently small, and we determine a concrete set related to the potentials V and K as the concentration position of these ground state solutions as $$\varepsilon \rightarrow 0$$ . Moreover, we considered some properties of these ground state solutions, such as convergence and decay estimate.
PubDate: 2017-07-13
DOI: 10.1007/s00526-017-1199-4
Issue No: Vol. 56, No. 4 (2017)

• Injectivity and self-contact in second-gradient nonlinear elasticity
• Authors: Aaron Z. Palmer; Timothy J. Healey
Abstract: Abstract We prove the existence of globally injective weak solutions in mixed boundary-value problems of second-gradient nonlinear elastostatics via energy minimization. This entails the treatment of self-contact. In accordance with the classical (first-gradient) theory, the model incorporates the unbounded growth of the potential energy density as the local volume ratio approaches zero. We work in a class of admissible vector-valued deformations that are injective on the interior of the domain. We first establish a rigorous Euler–Lagrange variational inequality at a minimizer. We then define a self-contact coincidence set for an admissible deformation in a natural way, which we demonstrate to be confined to a closed subset of the boundary of the domain. We then prove the existence of a non-negative (Radon) measure, vanishing outside of the coincidence set, which represents the normal contact-reaction force distribution. With this in hand, we obtain the weak form of the equilibrium equations at a minimizer.
PubDate: 2017-07-12
DOI: 10.1007/s00526-017-1212-y
Issue No: Vol. 56, No. 4 (2017)

• On geodesic completeness for Riemannian metrics on smooth probability
densities
• Authors: Martin Bauer; Sarang Joshi; Klas Modin
Abstract: Abstract The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are the Otto metric, yielding the $$L^2$$ -Wasserstein distance of optimal mass transport, and the Fisher–Rao metric, predominant in the theory of information geometry. On the space of smooth probability densities, none of these Riemannian metrics are geodesically complete—a property desirable for example in imaging applications. That is, the existence interval for solutions to the geodesic flow equations cannot be extended to the whole real line. Here we study a class of Hamilton–Jacobi-like partial differential equations arising as geodesic flow equations for higher-order Sobolev type metrics on the space of smooth probability densities. We give order conditions for global existence and uniqueness, thereby providing geodesic completeness. The system we study is an interesting example of a flow equation with loss of derivatives, which is well-posed in the smooth category, yet non-parabolic and fully non-linear. On a more general note, the paper establishes a link between geometric analysis on the space of probability densities and analysis of Euler–Arnold equations in topological hydrodynamics.
PubDate: 2017-07-11
DOI: 10.1007/s00526-017-1195-8
Issue No: Vol. 56, No. 4 (2017)

• A sharp bound on the Hausdorff dimension of the singular set of a uniform
measure
• Authors: A. Dali Nimer
Abstract: Abstract The study of the geometry of n-uniform measures in $$\mathbb {R}^{d}$$ has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension $$n-3$$ . In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most $$n-3$$ .
PubDate: 2017-07-11
DOI: 10.1007/s00526-017-1206-9
Issue No: Vol. 56, No. 4 (2017)

• Concentration profile of endemic equilibrium of a
• Authors: Kousuke Kuto; Hiroshi Matsuzawa; Rui Peng
Abstract: Abstract Cui and Lou (J Differ Equ 261:3305–3343, 2016) proposed a reaction–diffusion–advection SIS epidemic model in heterogeneous environments, and derived interesting results on the stability of the DFE (disease-free equilibrium) and the existence of EE (endemic equilibrium) under various conditions. In this paper, we are interested in the asymptotic profile of the EE (when it exists) in the three cases: (i) large advection; (ii) small diffusion of the susceptible population; (iii) small diffusion of the infected population. We prove that in case (i), the density of both the susceptible and infected populations concentrates only at the downstream behaving like a delta function; in case (ii), the density of the susceptible concentrates only at the downstream behaving like a delta function and the density of the infected vanishes on the entire habitat, and in case (iii), the density of the susceptible is positive while the density of the infected vanishes on the entire habitat. Our results show that in case (ii) and case (iii), the asymptotic profile is essentially different from that in the situation where no advection is present. As a consequence, we can conclude that the impact of advection on the spatial distribution of population densities is significant.
PubDate: 2017-07-11
DOI: 10.1007/s00526-017-1207-8
Issue No: Vol. 56, No. 4 (2017)

• A general pinching principle for mean curvature flow and applications
• Authors: Mat Langford
Abstract: Abstract We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken (J Differ Geom 20(1):237–266, 1984), the convexity estimates of Huisken–Sinestrari (Acta Math 183(1):45–70, 1999) and the cylindrical estimate of Huisken–Sinestrari (Invent Math 175(1):137–221, 2009; see also Andrews and Langford in Anal PDE 7(5):1091–1107, 2014; Huisken and Sinestrari in J Differ Geom 101(2):267–287, 2015). Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is $$(m+1)$$ -convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders $$\mathbb {R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}$$ , $$t<1$$ . In particular, this yields a sharp estimate for the largest principal curvature, which we use to obtain a new proof of a sharp estimate for the inscribed curvature for embedded solutions (Brendle in Invent Math 202(1):217–237, 2015; Haslhofer and Kleiner in Int Math Res Not 15:6558–6561, 2015; Langford in Proc Am Math Soc 143(12):5395–5398, 2015). Making use of a recent idea of Huisken–Sinestrari (2015), we then obtain a series of sharp estimates for ancient solutions. In particular, we obtain a convexity estimate for ancient solutions which allows us to strengthen recent characterizations of the shrinking sphere due to Huisken–Sinestrari (2015) and Haslhofer–Hershkovits (Commun Anal Geom 24(3):593–604, 2016).
PubDate: 2017-07-10
DOI: 10.1007/s00526-017-1193-x
Issue No: Vol. 56, No. 4 (2017)

• Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type
problems with general potentials
• Authors: X. H. Tang; Sitong Chen
Abstract: Abstract This paper is dedicated to studying the following Kirchhoff-type problem 0.1 \begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left( a+b\int _{\mathbb {R}^3} \nabla u ^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), &{} x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{array} \right. \end{aligned} where $$a>0,\,b\ge 0$$ are two constants, V(x) is differentiable and $$f\in \mathcal {C}(\mathbb {R}, \mathbb {R})$$ . By introducing some new tricks, we prove that the above problem admits a ground state solution of Nehari–Pohozaev type and a least energy solution under some mild assumptions on V and f. Our results generalize and improve the ones in Guo (J Differ Equ 259:2884–2902, 2015) and Li and Ye (J Differ Equ 257:566–600, 2014) and some other related literature.
PubDate: 2017-07-10
DOI: 10.1007/s00526-017-1214-9
Issue No: Vol. 56, No. 4 (2017)

• Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with
applications to the Dirac-harmonic heat flow
• Authors: Jürgen Jost; Lei Liu; Miaomiao Zhu
Abstract: Abstract Dirac-harmonic maps couple a second order harmonic map type system with a first nonlinear Dirac equation. We consider approximate Dirac-harmonic maps $$\{(\phi _n,\psi _n)\}$$ , that is, maps that satisfy the Dirac-harmonic system up to controlled error terms. We show that such approximate Dirac-harmonic maps defined on a Riemann surface, that is, in dimension 2, continue to satisfy the basic properties of blow-up analysis like the energy identity and the no neck property. The assumptions are such that they hold for solutions of the heat flow of Dirac-harmonic maps. That flow turns the harmonic map type system into a parabolic system, but simply keeps the Dirac equation as a nonlinear first order constraint along the flow. As a corollary of the main result of this paper, when such a flow blows up at infinite time at interior points, we obtain an energy identity and the no neck property.
PubDate: 2017-07-10
DOI: 10.1007/s00526-017-1202-0
Issue No: Vol. 56, No. 4 (2017)

• Korn’s inequality and John domains
• Authors: Renjin Jiang; Aapo Kauranen
Abstract: Abstract Let $$\Omega \subset \mathbb {R}^n$$ , $$n\ge 2$$ , be a bounded domain satisfying the separation property. We show that the following conditions are equivalent: $$\Omega$$ is a John domain; for a fixed $$p\in (1,\infty )$$ , the Korn inequality holds for each $$\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)$$ satisfying $$\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0$$ , $$1\le i,j\le n$$ , \begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned} for all $$p\in (1,\infty )$$ , $$(K_p)$$ holds on $$\Omega$$ ; for a fixed $$p\in (1,\infty )$$ , for each $$f\in L^p(\Omega )$$ with vanishing mean value on $$\Omega$$ , there exists a solution $$\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)$$ to the equation $$\mathrm {div}\,\mathbf {v}=f$$ with \begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned} for all $$p\in (1,\infty )$$ , $$(DE_p)$$ holds on $$\Omega$$ . For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.
PubDate: 2017-07-10
DOI: 10.1007/s00526-017-1196-7
Issue No: Vol. 56, No. 4 (2017)

• Homotopically invisible singular curves
• Authors: Andrei A. Agrachev; Francesco Boarotto; Antonio Lerario
Abstract: Abstract Given a smooth manifold M and a totally nonholonomic distribution $$\Delta \subset TM$$ of rank $$d\ge 3$$ , we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map $$F\,{:}\,\gamma \mapsto \gamma (1)$$ defined on the space $$\Omega$$ of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy $$J\,{:}\,\Omega (y)\rightarrow \mathbb R$$ , where $$\Omega (y)=F^{-1}(y)$$ is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets $$\{\gamma \in \Omega (y)\, \,J(\gamma )\le E\}$$ . We call them homotopically invisible. It turns out that for $$d\ge 3$$ generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications).
PubDate: 2017-07-10
DOI: 10.1007/s00526-017-1203-z
Issue No: Vol. 56, No. 4 (2017)

• Paneitz operator for metrics near $$S^{3}$$ S 3
• Authors: Fengbo Hang; Paul C. Yang
Abstract: Abstract We derive the first and second variation formula for the Green’s function pole’s value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the second eigenvalue and Sobolev inequality of Paneitz operator.
PubDate: 2017-07-10
DOI: 10.1007/s00526-017-1201-1
Issue No: Vol. 56, No. 4 (2017)

• A minimal partition problem with trace constraint in the Grushin plane
• Authors: Valentina Franceschi
Abstract: Abstract We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the least amount of perimeter, under an additional “one-dimensional” constraint on the intersections of their boundaries. We prove existence of regular solutions for this problem, and we characterize them in terms of isoperimetric sets, showing differences with the Euclidean case. The problem arises from the study of quantitative isoperimetric inequalities and has connections with the theory of minimal clusters.
PubDate: 2017-07-08
DOI: 10.1007/s00526-017-1198-5
Issue No: Vol. 56, No. 4 (2017)

• The existence and nonexistence of global solutions for a semilinear heat
equation on graphs
• Authors: Yong Lin; Yiting Wu
Abstract: Abstract Let $$G=(V,E)$$ be a finite or locally finite connected weighted graph, $$\Delta$$ be the usual graph Laplacian. Using heat kernel estimates, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on G \begin{aligned} \left\{ \begin{array}{lc} u_t=\Delta u + u^{1+\alpha } &{}\, \text {in }(0,+\infty )\times V,\\ u(0,x)=a(x) &{}\, \text {in }V. \end{array} \right. \end{aligned} We conclude that, for a graph satisfying curvature dimension condition $$\textit{CDE}'(n,0)$$ and $$V(x,r)\simeq r^m$$ , if $$0<m\alpha <2$$ , then the non-negative solution u is not global, and if $$m\alpha >2$$ , then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results apply to the lattice $${\mathbb {Z}}^m$$ .
PubDate: 2017-07-07
DOI: 10.1007/s00526-017-1204-y
Issue No: Vol. 56, No. 4 (2017)

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