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 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  [2352 journals]
• A new and extended variational characterization of the Fučík Spectrum
with application to nonresonance and resonance problems
• Authors: Pavel Drábek; Stephen B. Robinson
Abstract: We consider the boundary value problem \begin{aligned} -\Delta u= & {} \alpha u^{+}-\beta u^{-} \quad \hbox { in } \Omega ,\\ u= & {} 0 \quad \hbox { on }\partial \Omega , \end{aligned} where $$\Omega$$ is a smooth bounded domain in $${\mathbb {R}}^N,u^{\pm }:=\max \{\pm \,u,0\}$$ , and $$(\alpha ,\beta )\in {\mathbb {R}}^2$$ . When this problem has a nontrivial solution, then $$(\alpha ,\beta )$$ is an element of the Fučík Spectrum, $$\Sigma$$ . Our main result is to extend the variational characterization, due to Castro and Chang in 2010, of several curves in $$\Sigma$$ . As an application we prove existence theorems for nonresonance and resonance problems relative to these extended spectral curves.
PubDate: 2017-11-10
DOI: 10.1007/s00526-017-1276-8
Issue No: Vol. 57, No. 1 (2017)

• On the 4D nonlinear Schrödinger equation with combined terms under
the energy threshold
• Authors: Changxing Miao; Tengfei Zhao; Jiqiang Zheng
Abstract: In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767–808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 265–298, 2015) to four dimension without radial assumption and the proof of scattering is based on the interaction Morawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension $$d =4$$ for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
PubDate: 2017-11-09
DOI: 10.1007/s00526-017-1264-z
Issue No: Vol. 56, No. 6 (2017)

• First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs
• Authors: Bobo Hua; Yan Huang; Zuoqin Wang
Abstract: Following Escobar (J Funct Anal 150(2):544–556, 1997) and Jammes (Ann l’Inst Fourier 65(3):1381–1385, 2015), we introduce two types of isoperimetric constants and give lower bound estimates for the first nontrivial eigenvalues of Dirichlet-to-Neumann operators on finite graphs with boundary respectively.
PubDate: 2017-11-09
DOI: 10.1007/s00526-017-1260-3
Issue No: Vol. 56, No. 6 (2017)

• Index and topology of minimal hypersurfaces in $$\mathbb {R}^n$$ R n
• Authors: Chao Li
Abstract: In this paper, we consider immersed two-sided minimal hypersurfaces in $$\mathbb {R}^n$$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $$n=4$$ , we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective Morse index bound by purely topological invariants, and is a generalization of Li and Wang (Math Res Lett 9(1):95–104, 2002). Using our index estimates and ideas from the recent work of Chodosh–Ketover–Maximo (Minimal surfaces with bounded index, 2015. arXiv:1509.06724), we prove compactness and finiteness results of minimal hypersurfaces in $$\mathbb {R}^4$$ with finite index.
PubDate: 2017-11-09
DOI: 10.1007/s00526-017-1272-z
Issue No: Vol. 56, No. 6 (2017)

• Rigidity and curvature estimates for graphical self-shrinkers
• Authors: Qiang Guang; Jonathan J. Zhu
Abstract: Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It is presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: for $$2\le n \le 6$$ , any smooth, complete self-shrinker $$\Sigma ^n\subset \mathbf {R}^{n+1}$$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key component of this paper is the procurement of linear curvature estimates for almost stable shrinkers, and it is this step that is responsible for the restriction on n. Our methods also yield uniform curvature bounds for translating solitons of the mean curvature flow.
PubDate: 2017-11-08
DOI: 10.1007/s00526-017-1277-7
Issue No: Vol. 56, No. 6 (2017)

• Weighted Lorentz estimates for parabolic equations with non-standard
growth on rough domains
• Authors: The Anh Bui; Xuan Thinh Duong
Abstract: The main aim of this paper is to prove the Calderón–Zygmund estimates for a general nonlinear parabolic equation of p(x, t)-Laplacian type in the weighted Lorentz spaces. Note that we only require some mild conditions on the nonlinearity of coefficients and the underlying domain. The result for these nonlinear parabolic equations is new even in the particular case when the growth p(x, t) is a constant.
PubDate: 2017-11-08
DOI: 10.1007/s00526-017-1273-y
Issue No: Vol. 56, No. 6 (2017)

• Bubbling analysis for approximate Lorentzian harmonic maps from Riemann
surfaces
• Authors: Xiaoli Han; Jürgen Jost; Lei Liu; Liang Zhao
Abstract: For a sequence of approximate harmonic maps $$(u_n,v_n)$$ (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form $$g_N -\beta dt^2$$ for some Riemannian metric $$g_N$$ and some positive function $$\beta$$ on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.
PubDate: 2017-11-07
DOI: 10.1007/s00526-017-1271-0
Issue No: Vol. 56, No. 6 (2017)

• Ricci curvature and orientability
• Authors: Shouhei Honda
Abstract: In this paper we define an orientation of a measured Gromov–Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between $$L^2$$ -convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani–Wenger, the pointed flat convergence by Lang–Wenger, and the Gromov–Hausdorff convergence, which is a generalization of a recent work by Matveev–Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.
PubDate: 2017-11-07
DOI: 10.1007/s00526-017-1258-x
Issue No: Vol. 56, No. 6 (2017)

• Interior Calderón–Zygmund estimates for solutions to general parabolic
equations of p -Laplacian type
• Authors: Truyen Nguyen
Abstract: We study general parabolic equations of the form $$u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,( \mathbf {F} ^{p-2} \mathbf {F})+ f$$ whose principal part depends on the solution itself. The vector field $$\mathbf {A}$$ is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderón–Zygmund estimates for locally bounded weak solutions to the equations when $$p>2n/(n+2)$$ . This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285–320, 2007).
PubDate: 2017-11-07
DOI: 10.1007/s00526-017-1265-y
Issue No: Vol. 56, No. 6 (2017)

• Regularity for elliptic systems of differential forms and applications
• Authors: Swarnendu Sil
Abstract: We prove existence and up to the boundary regularity estimates in $$L^{p}$$ and Hölder spaces for weak solutions of the linear system \begin{aligned} \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text { in } \varOmega , \end{aligned} with either $$\nu \wedge \omega$$ and $$\nu \wedge \delta \left( B\omega \right)$$ or $$\nu \lrcorner B\omega$$ and $$\nu \lrcorner \left( A d\omega \right)$$ prescribed on $$\partial \varOmega .$$ The proofs are in the spirit of ‘Campanato method’ and thus avoid potential theory and do not require a verification of Agmon–Douglis–Nirenberg or Lopatinskiĭ–Shapiro type conditions. Applications to a number of related problems, such as general versions of the time-harmonic Maxwell system, stationary Stokes problem and the ‘div-curl’ systems, are included.
PubDate: 2017-11-07
DOI: 10.1007/s00526-017-1269-7
Issue No: Vol. 56, No. 6 (2017)

• Time-convexity of the entropy in the multiphasic formulation of the
incompressible Euler equation
• Authors: Hugo Lavenant
Abstract: We study the multiphasic formulation of the incompressible Euler equation introduced by Brenier: infinitely many phases evolve according to the compressible Euler equation and are coupled through a global incompressibility constraint. In a convex domain, we are able to prove that the entropy, when averaged over all phases, is a convex function of time, a result that was conjectured by Brenier. The novelty in our approach consists in introducing a time-discretization that allows us to import a flow interchange inequality previously used by Matthes, McCann and Savaré to study first order in time PDE, namely the JKO scheme associated with non-linear parabolic equations.
PubDate: 2017-11-04
DOI: 10.1007/s00526-017-1262-1
Issue No: Vol. 56, No. 6 (2017)

• Standing waves for a coupled nonlinear Hartree equations with nonlocal
interaction
• Authors: Jun Wang; Junping Shi
Abstract: Standing wave solutions of coupled nonlinear Hartree equations with nonlocal interaction are considered. Such systems arises from mathematical models in Bose–Einstein condensates theory and nonlinear optics. The existence and non-existence of positive ground state solutions are proved under optimal conditions on parameters, and various qualitative properties of ground state solutions are shown. The uniqueness of the positive solution or the positive ground state solution are also obtained in some cases.
PubDate: 2017-11-04
DOI: 10.1007/s00526-017-1268-8
Issue No: Vol. 56, No. 6 (2017)

• Short time existence of the heat flow for Dirac-harmonic maps on closed
manifolds
• Authors: Johannes Wittmann
Abstract: The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.
PubDate: 2017-11-04
DOI: 10.1007/s00526-017-1270-1
Issue No: Vol. 56, No. 6 (2017)

• Stable self-similar blowup in the supercritical heat flow of harmonic maps
• Authors: Paweł Biernat; Roland Donninger; Birgit Schörkhuber
Abstract: We consider the heat flow of corotational harmonic maps from $$\mathbb {R}^3$$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.
PubDate: 2017-11-04
DOI: 10.1007/s00526-017-1256-z
Issue No: Vol. 56, No. 6 (2017)

• Index theory for heteroclinic orbits of Hamiltonian systems
• Authors: Xijun Hu; Alessandro Portaluri
Abstract: Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors’ knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrized by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrized by bounded intervals.
PubDate: 2017-11-04
DOI: 10.1007/s00526-017-1259-9
Issue No: Vol. 56, No. 6 (2017)

• Michell trusses in two dimensions as a $$\Gamma$$ Γ -limit of optimal
design problems in linear elasticity
• Authors: Heiner Olbermann
Abstract: We reconsider the minimization of the compliance of a two dimensional elastic body with traction boundary conditions for a given weight. It is well known how to rewrite this optimal design problem as a nonlinear variational problem. We take the limit of vanishing weight by sending a suitable Lagrange multiplier to infinity in the variational formulation. We show that the limit, in the sense of $$\Gamma$$ -convergence, is a certain Michell truss problem. This proves a conjecture by Kohn and Allaire.
PubDate: 2017-11-03
DOI: 10.1007/s00526-017-1266-x
Issue No: Vol. 56, No. 6 (2017)

• Multiplicity and bifurcation of positive solutions for nonhomogeneous
semilinear fractional Laplacian problems
• Authors: Bingliang Li; Yongqiang Fu
Abstract: In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem \begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned} where $$\lambda >0$$ and $$\lim _{ x \rightarrow \infty }f(x,u)=\overline{f}(u)$$ uniformly on any compact subset of $$[0,\infty )$$ . We prove that under suitable conditions on f and h, there exists $$0<\lambda ^*<+\infty$$ such that the problem has at least two positive solutions if $$\lambda \in (0,\lambda ^*)$$ , a unique positive solution if $$\lambda =\lambda ^*$$ , and no solution if $$\lambda >\lambda ^*$$ . We also obtain the bifurcation of positive solutions for the problem at $$(\lambda ^*,u^*)$$ and further analyse the set of positive solutions.
PubDate: 2017-11-02
DOI: 10.1007/s00526-017-1257-y
Issue No: Vol. 56, No. 6 (2017)

• The Cheeger constant of a Jordan domain without necks
• Authors: Gian Paolo Leonardi; Robin Neumayer; Giorgio Saracco
Abstract: We show that the maximal Cheeger set of a Jordan domain $$\Omega$$ without necks is the union of all balls of radius $$r = h(\Omega )^{-1}$$ contained in $$\Omega$$ . Here, $$h(\Omega )$$ denotes the Cheeger constant of $$\Omega$$ , that is, the infimum of the ratio of perimeter over area among subsets of $$\Omega$$ , and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set $$\Omega ^r$$ is equal to $$\pi r^2$$ . The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.
PubDate: 2017-11-01
DOI: 10.1007/s00526-017-1263-0
Issue No: Vol. 56, No. 6 (2017)

• Green’s function for elliptic systems: existence and
Delmotte–Deuschel bounds
• Authors: Joseph G. Conlon; Arianna Giunti; Felix Otto
Abstract: This paper is divided into two parts: In the main deterministic part, we prove that for an open domain $$D \subset \mathbb {R}^d$$ with $$d \ge 2$$ , for every (measurable) uniformly elliptic tensor field a and for almost every point $$y \in D$$ , there exists a unique Green’s function centred in y associated to the vectorial operator $$-\nabla \cdot a\nabla$$ in D. This result implies the existence of the fundamental solution for elliptic systems when $$d>2$$ , i.e. the Green function for $$-\nabla \cdot a\nabla$$ in $$\mathbb {R}^d$$ . In the second part, we introduce a shift-invariant ensemble $$\langle \cdot \rangle$$ over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for $$\langle G(\cdot ; x,y) \rangle$$ , $$\langle \nabla _x G(\cdot ; x,y) \rangle$$ and $$\langle \nabla _x\nabla _y G(\cdot ; x,y) \rangle$$ . These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.
PubDate: 2017-10-25
DOI: 10.1007/s00526-017-1255-0
Issue No: Vol. 56, No. 6 (2017)

• Minkowski valuations on convex functions
• Authors: Andrea Colesanti; Monika Ludwig; Fabian Mussnig
Abstract: A classification of $${\text {SL}}(n)$$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new $${\text {SL}}(n)$$ covariant Minkowski valuation on convex functions is defined and characterized.
PubDate: 2017-10-23
DOI: 10.1007/s00526-017-1243-4
Issue No: Vol. 56, No. 6 (2017)

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