Authors:Seonghak Kim; Baisheng Yan Abstract: Abstract As a sequel to the paper Kim and Yan (Ann Inst H Poincaré Anal Non Linéaire. doi:10.1016/j.anihpc.2017.03.001, 2017), we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward–backward diffusion equations with diffusion fluxes violating Fourier’s inequality. PubDate: 2017-04-26 DOI: 10.1007/s00526-017-1155-3 Issue No:Vol. 56, No. 3 (2017)

Authors:Naoyuki Koike Abstract: Abstract First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type, the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature in the \(C^{\infty }\) -topology in infinite time. PubDate: 2017-04-25 DOI: 10.1007/s00526-017-1156-2 Issue No:Vol. 56, No. 3 (2017)

Authors:Raghavendra Venkatraman Abstract: Abstract We prove the existence of non-constant time periodic vortex solutions to the Gross–Pitaevskii equations for small but fixed \(\varepsilon > 0.\) The vortices of these solutions follow periodic orbits to the point vortex system of ordinary differential equations for all time. The construction uses two approaches—constrained minimization techniques adapted from Gelantalis and Sternberg (J Math Phys 53:083701, 2012) and topological minimax techniques adapted from Lin and Lin (Sel Math New Ser 3:99–113, 1997), applied to a formulation of the problem within a rotational ansatz. PubDate: 2017-04-25 DOI: 10.1007/s00526-017-1168-y Issue No:Vol. 56, No. 3 (2017)

Authors:Huagui Duan; Hui Liu Abstract: Abstract In this paper, we firstly generalize some theories developed by Ekeland and Hofer (Commun Math Phys 113:419–467, 1987) for closed characteristics on compact convex hypersurfaces in \(\mathbf{R}^{2n}\) to star-shaped hypersurfaces. As applications we use Ekeland–Hofer theory and index iteration theory to prove that if a compact star-shaped hypersuface in \(\mathbf{R}^4\) satisfying some suitable pinching condition carries exactly two geometrically distinct closed characteristics, then both of them must be elliptic. We also conclude that the theory developed by Long and Zhu (Ann Math 155:317–368, 2002) still holds for dynamically convex star-shaped hypersurfaces, and combining it with the results in Wang et al. (Duke Math J 139:411–462, 2007), Liu et al. (J Funct Anal 266:5598–5638, 2014) and Wang (Adv Math 297:93–148, 2016), we obtain that there exist at least n closed characteristics on every dynamically convex star-shaped hypersurface in \(\mathbf{R}^{2n}\) for \(n=3, 4\) . PubDate: 2017-04-25 DOI: 10.1007/s00526-017-1173-1 Issue No:Vol. 56, No. 3 (2017)

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš Abstract: Abstract We study a nonlinear boundary value problem driven by the p-Laplacian plus an indefinite potential with Robin boundary condition. The reaction term is a Carathéodory function which is asymptotically resonant at \(\pm \infty \) with respect to a nonprincipal Ljusternik–Schnirelmann eigenvalue. Using variational methods, together with Morse theory and truncation-perturbation techniques, we show that the problem has at least three nontrivial smooth solutions, two of which have a fixed sign. PubDate: 2017-04-25 DOI: 10.1007/s00526-017-1164-2 Issue No:Vol. 56, No. 3 (2017)

Authors:Ningkui Sun; Bendong Lou; Maolin Zhou Abstract: Abstract We consider a reaction–diffusion–advection equation of the form: \(u_t=u_{xx}-\beta (t)u_x+f(t,u)\) for \(x\in (g(t),h(t))\) , where \(\beta (t)\) is a T-periodic function representing the intensity of the advection, f(t, u) is a Fisher–KPP type of nonlinearity, T-periodic in t, g(t) and h(t) are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both \(\beta \) and f are independent of t) was recently studied by Gu et al. (J Funct Anal 269:1714–1768, 2015). In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing–spreading dichotomy result holds when \(\beta \) is small; a vanishing–transition–virtual spreading trichotomy result holds when \(\beta \) is a medium-sized function; all solutions vanish when \(\beta \) is large. Here the partition of \(\beta (t)\) depends not only on the “size” \(\bar{\beta }:= \frac{1}{T}\int _0^T \beta (t) dt\) of \(\beta (t)\) but also on its “shape” \(\tilde{\beta }(t) := \beta (t) - \bar{\beta }\) . PubDate: 2017-04-24 DOI: 10.1007/s00526-017-1165-1 Issue No:Vol. 56, No. 3 (2017)

Authors:Lukas Döring; Christof Melcher Abstract: Abstract We examine lower order perturbations of the harmonic map problem from \(\mathbb {R}^2\) to \(\mathbb {S}^2\) including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform background state. Energy functionals of this type arise in the context of magnetic systems without inversion symmetry. In the almost conformal regime, where these perturbations are weighted with a small parameter, we examine the existence of relative minimizers in a non-trivial homotopy class, so-called chiral skyrmions, strong compactness of almost minimizers, and their asymptotic limit. Finally we examine dynamic stability and compactness of almost minimizers in the context of the Landau–Lifshitz–Gilbert equation including spin-transfer torques arising from the interaction with an external current. PubDate: 2017-04-24 DOI: 10.1007/s00526-017-1172-2 Issue No:Vol. 56, No. 3 (2017)

Authors:Haizhong Li; Yong Wei Abstract: Abstract In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface \(\Sigma \) is strictly mean convex and star-shaped, then the flow hypersurface \(\Sigma _t\) converges to a large coordinate sphere as \(t\rightarrow \infty \) exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler–Schwarzschild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken–Ilmanen’s (J Differ Geom 80:433–451, 2008) result. PubDate: 2017-04-24 DOI: 10.1007/s00526-017-1160-6 Issue No:Vol. 56, No. 3 (2017)

Authors:Mario Bukal; Igor Velčić Abstract: Abstract We provide a framework for simultaneous homogenization and dimension reduction in the setting of linearized elasticity as well as non-linear elasticity for the derivation of homogenized von Kármán plate and bending rod models. The framework encompasses even perforated domains and domains with oscillatory boundary, provided that the corresponding extension operator can be constructed. Locality property of \(\varGamma \) -closure is established, i.e. every energy density obtained by the homogenization process can be in almost every point obtained as the limit of periodic energy densities. PubDate: 2017-04-24 DOI: 10.1007/s00526-017-1167-z Issue No:Vol. 56, No. 3 (2017)

Authors:Jianyi Chen; Zhitao Zhang Abstract: Abstract This paper is concerned with the Dirichlet problem of the asymptotically linear wave equation $$\begin{aligned} u_{tt}-\Delta u = g(t,x,u) \end{aligned}$$ in a n-dimensional ball with radius R, where \(n>1\) and g(t, x, u) is radially symmetric in x and T-periodic in time. An interesting feature is that the solvable of the problem depends on the space dimension n and the arithmetical properties of R and T. Based on the spectral properties of the radially symmetric wave operator, we use the saddle point reduction and variational methods to construct at least three radially symmetric solutions with time period T, when T is a rational multiple of R and g(t, x, u) satisfies some monotonicity and asymptotically linear conditions. PubDate: 2017-04-22 DOI: 10.1007/s00526-017-1154-4 Issue No:Vol. 56, No. 3 (2017)

Authors:Richard Gratwick Abstract: Abstract We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general of the form of a standard Lipschitz “variation”. Part of this investigation, but of interest in its own right, is an example of a nowhere locally Lipschitz minimizer which serves as a counter-example to any putative Tonelli partial regularity statement. Under these low assumptions we find it nonetheless remains possible to derive necessary conditions for minimizers, in terms of approximate continuity and equality of the one-sided derivatives. PubDate: 2017-04-13 DOI: 10.1007/s00526-017-1135-7 Issue No:Vol. 56, No. 3 (2017)

Authors:Duvan Henao; Apala Majumdar; Adriano Pisante Abstract: Abstract We study global minimizers of the Landau–de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the \(t\rightarrow \infty \) limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829–838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of “strongly biaxial” regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial \(\mathbf{Q}\) -tensors cannot be stable critical points of the LdG energy in this limit. PubDate: 2017-04-04 DOI: 10.1007/s00526-017-1142-8 Issue No:Vol. 56, No. 2 (2017)

Authors:Rupert L. Frank; Marius Lemm; Barry Simon Abstract: Abstract We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two-body potential which admits a bound state. The particles are confined to a domain with Dirichlet boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross–Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain. PubDate: 2017-04-03 DOI: 10.1007/s00526-017-1140-x Issue No:Vol. 56, No. 2 (2017)

Authors:B. Krummel; F. Maggi Abstract: Abstract It was proved by Almgren that among boundaries whose mean curvature is bounded from above, perimeter is uniquely minimized by balls. We obtain sharp stability estimates for Almgren’s isoperimetric principle and, as an application, we deduce a sharp description of boundaries with almost constant mean curvature under a total perimeter bound which prevents bubbling. PubDate: 2017-03-24 DOI: 10.1007/s00526-017-1139-3 Issue No:Vol. 56, No. 2 (2017)

Authors:Chungen Liu; Qiang Ren Abstract: Abstract In this paper, the prescribed \(\sigma \) -curvature problem $$\begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned}$$ is considered. When \({\tilde{K}}(x)\) is some axis symmetric function on \({\mathbb {S}}^N\) , by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for \(0<\sigma \le 1\) and \(N> 2\sigma +2\) . PubDate: 2017-03-23 DOI: 10.1007/s00526-017-1141-9 Issue No:Vol. 56, No. 2 (2017)

Authors:Hoai-Minh Nguyen; Quoc-Hung Nguyen Abstract: Abstract This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary. PubDate: 2017-03-22 DOI: 10.1007/s00526-017-1143-7 Issue No:Vol. 56, No. 2 (2017)

Authors:Jihoon Ok Abstract: Abstract We prove local Hölder continuity results for \(\omega \) -minimizers of a class of functionals with non-standard growth, characterized by the fact of having a double type of degeneracy, and thereby extending to \(\omega \) -minimizers the results obtained in Colombo and Mingione (Arch Ration Mech Anal 215(2):443–496, 2015) for ordinary minimizers. As a side benefit of the proof, we also consider a class of functionals with Orlicz-growth and prove regularity for \(\omega \) -minimizers. PubDate: 2017-03-21 DOI: 10.1007/s00526-017-1137-5 Issue No:Vol. 56, No. 2 (2017)

Authors:Hynek Kovařík; Konstantin Pankrashkin Abstract: Abstract Let \(\Omega \subset \mathbb {R}^\nu \) , \(\nu \ge 2\) , be a \(C^{1,1}\) domain whose boundary \(\partial \Omega \) is either compact or behaves suitably at infinity. For \(p\in (1,\infty )\) and \(\alpha >0\) , define $$\begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega \nabla u ^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } u ^p\mathrm {d}\sigma }{\displaystyle \int _\Omega u ^p\mathrm {d} x}, \end{aligned}$$ where \(\mathrm {d}\sigma \) is the surface measure on \(\partial \Omega \) . We show the asymptotics $$\begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned}$$ where \(H_\mathrm {max}\) is the maximum mean curvature of \(\partial \Omega \) . The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities. PubDate: 2017-03-21 DOI: 10.1007/s00526-017-1138-4 Issue No:Vol. 56, No. 2 (2017)

Authors:Daguang Chen; Qing-Ming Cheng Abstract: Abstract In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere \(S^{n+1}(1)\) . We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to \(S^1(r)\times S^{n-1}(\sqrt{1-r^2})\) when \(H\ne 0\) or \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to a Clifford torus \( S^{n-k}\left( \sqrt{\dfrac{n-k}{n}}\right) \times S^k\left( \sqrt{\dfrac{k}{n}}\right) \) , for \(k=1, 2, \ldots , n-1\) when \(H=0\) . PubDate: 2017-03-21 DOI: 10.1007/s00526-017-1132-x Issue No:Vol. 56, No. 2 (2017)