Authors:Vincenzo Ambrosio Abstract: We deal with the existence of \(2\pi \) -periodic solutions to the following non-local critical problem $$\begin{aligned} \left\{ \begin{array}{ll} [(-\Delta _{x}+m^{2})^{s}-m^{2s}]u=W(x) u ^{2^{*}_{s}-2}u+ f(x, u) &{}\quad \text{ in } \; (-\pi ,\pi )^{N} \\ u(x+2\pi e_{i})=u(x) &{}\quad \text{ for } \text{ all } \; x \in \mathbb {R}^{N}, \quad i=1, \dots , N, \end{array} \right. \end{aligned}$$ where \(s\in (0,1)\) , \(N \ge 4s\) , \(m\ge 0\) , \(2^{*}_{s}=\frac{2N}{N-2s}\) is the fractional critical Sobolev exponent, W(x) is a positive continuous function, and f(x, u) is a superlinear \(2\pi \) -periodic (in x) continuous function with subcritical growth. When \(m>0\) , the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder \((-\,\pi ,\pi )^{N}\times (0, \infty )\) , with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case \(m=0\) by using a careful procedure of limit. As far as we know, all these results are new. PubDate: 2018-02-22 DOI: 10.1007/s00526-018-1317-y Issue No:Vol. 57, No. 2 (2018)

Authors:Jiawei Liu; Chuanjing Zhang Abstract: In this paper, by providing the uniform gradient estimates for approximating equations, we prove the existence, uniqueness and regularity of conical parabolic complex Monge–Ampère equation with weak initial data. As an application, we obtain a regularity estimate, that is, any \(L^{\infty }\) -solution of the conical complex Monge–Ampère equation admits the \(C^{2,\alpha ,\beta }\) -regularity. PubDate: 2018-02-19 DOI: 10.1007/s00526-018-1318-x Issue No:Vol. 57, No. 2 (2018)

Authors:Rupert L. Frank; Tianling Jin; Jingang Xiong Abstract: We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains. PubDate: 2018-02-17 DOI: 10.1007/s00526-018-1304-3 Issue No:Vol. 57, No. 2 (2018)

Authors:Daguang Chen; Haizhong Li; Zhizhang Wang Abstract: Given a compact Riemannian manifold M, we consider a warped product manifold \({\bar{M}} = I \times _h M\) , where I is an open interval in \({\mathbb {R}}\) . For a positive function \(\psi \) defined on \({\bar{M}}\) , we generalize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287–1325, 2015) and Ren and Wang (On the curvature estimates for Hessian equations, 2016. arXiv:1602.06535), to obtain the curvature estimates for Hessian equations \(\sigma _k(\kappa )=\psi (V,\nu (V))\) . We also obtain some existence results for the starshaped compact hypersurface \(\Sigma \) satisfying the above equation with various assumptions. PubDate: 2018-02-16 DOI: 10.1007/s00526-018-1314-1 Issue No:Vol. 57, No. 2 (2018)

Authors:Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron Abstract: We study the existence of separable infinity harmonic functions in any cone of \(\mathbb R^N\) vanishing on its boundary under the form \(u(r,\sigma )=r^{-\beta }\psi (\sigma )\) . We prove that such solutions exist, the spherical part \(\psi \) satisfies a nonlinear eigenvalue problem on a subdomain of the sphere \(S^{N-1}\) and that the exponents \(\beta =\beta _+>0\) and \(\beta =\beta _-<0\) are uniquely determined if the domain is smooth. We extend some of our results to non-smooth domains. PubDate: 2018-02-15 DOI: 10.1007/s00526-018-1309-y Issue No:Vol. 57, No. 2 (2018)

Authors:Hardy Chan Abstract: In the long-standing inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after being multiplied with the multiplier, becomes the Euler–Lagrange equation for the Lagrangian. An answer to this problem for the case of a scalar ordinary differential equation of order \(2n, n\ge 2,\) is proposed. PubDate: 2018-02-14 DOI: 10.1007/s00526-018-1302-5 Issue No:Vol. 57, No. 2 (2018)

Authors:Raz Kupferman; Cy Maor Abstract: We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold \(({\mathcal {M}},\mathfrak {g})\) , endowed with a flat, symmetric connection \(\nabla \) . The metric \(\mathfrak {g}\) determines local equilibrium distances between neighboring points; the connection \(\nabla \) induces a lattice structure shared by all the discrete models. The limit model satisfies a fundamental rigidity property: there are no stress-free configurations, unless \(\mathfrak {g}\) is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions. PubDate: 2018-02-14 DOI: 10.1007/s00526-018-1306-1 Issue No:Vol. 57, No. 2 (2018)

Authors:Bruno Colbois; Luigi Provenzano Abstract: We consider eigenvalue problems for elliptic operators of arbitrary order 2m subject to Neumann boundary conditions on bounded domains of the Euclidean N-dimensional space. We study the dependence of the eigenvalues upon variations of mass density. In particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the \(L^{\frac{N}{2m}}\) -norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds. PubDate: 2018-02-13 DOI: 10.1007/s00526-018-1307-0 Issue No:Vol. 57, No. 2 (2018)

Authors:Daomin Cao; Shuanglong Li; Zhongyuan Liu Abstract: In this paper, we are concerned with the following nonlinear supercritical elliptic problem with variable exponent, $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u= u ^{2^*+ x ^\alpha -2}u,~&{}\text {in}~ B_1(0),\\ u=0,\quad &{}\text {on} ~\partial B_1(0), \end{array}\right. } \end{aligned}$$ where \(2^*=\frac{2N}{N-2}\) , \(0<\alpha <\min \{\frac{N}{2},N-2\}\) , and \(B_1(0)\) is the unit ball in \(\mathbb {R}^N\) , \(N\ge 3\) . For any \(k\in \mathbb {N}\) , we find, by variational methods, a pair of nodal solutions for this problem, which has exactly k nodal points. PubDate: 2018-02-13 DOI: 10.1007/s00526-018-1305-2 Issue No:Vol. 57, No. 2 (2018)

Authors:Giulio Tralli; Francesco Uguzzoni Abstract: We develop a potential theory approach for some degenerate parabolic operators in non-divergence form and with non-smooth coefficients, which are modeled on smooth Hörmander vector fields. We prove necessary and sufficient Wiener-type tests for the regularity of boundary points. As a consequence we obtain, in particular, a cone-type criterion. We also investigate the related boundary value problem and the Hölder regularity at the boundary. PubDate: 2018-02-13 DOI: 10.1007/s00526-018-1301-6 Issue No:Vol. 57, No. 2 (2018)

Authors:Luigi Ambrosio; Shouhei Honda; Jacobus W. Portegies Abstract: In this note we prove in the nonlinear setting of \({{\mathrm{CD}}}(K,\infty )\) spaces the stability of the Krasnoselskii spectrum of the Laplace operator \(-\,\Delta \) under measured Gromov–Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of \({{\mathrm{CD}}}^*(K,N)\) metric measure spaces with uniformly bounded diameter. Additionally, we show that every element \(\lambda \) in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial u satisfying the eigenvalue equation \(-\, \Delta u = \lambda u\) . PubDate: 2018-02-12 DOI: 10.1007/s00526-018-1315-0 Issue No:Vol. 57, No. 2 (2018)

Authors:Qing-Ming Cheng; Guoxin Wei Abstract: In this paper, we introduce a special class of hypersurfaces which are called \(\lambda \) -hypersurfaces related to a weighted volume preserving mean curvature flow in the Euclidean space. We prove that \(\lambda \) -hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete \(\lambda \) -hypersurfaces with polynomial area growth and \(H-\lambda \ge 0\) . The classification result generalizes the results of Huisken (J Differ Geom 31:285–299, 1990) and Colding and Minicozzi (Ann Math 175:755–833, 2012). PubDate: 2018-02-12 DOI: 10.1007/s00526-018-1303-4 Issue No:Vol. 57, No. 2 (2018)

Authors:Liangming Shen Abstract: In this note, we establish a parabolic version of Tian’s \(C^{2,\alpha }\) -estimate for conical complex Monge–Ampere equations (Tian in Chin Ann Math Ser B 38(2):687–694, 2017), which includes conical Kähler–Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kähler–Ricci flow in Shen (J Reine Angew Math, [28]). PubDate: 2018-02-12 DOI: 10.1007/s00526-018-1308-z Issue No:Vol. 57, No. 2 (2018)

Authors:Takumi Yokota Abstract: We prove the law of large numbers formulated with barycenter of probability measures for random variables with values in small balls in CAT(1)-spaces. This extends the previous results in CAT(0)-spaces and CAT(1)-spaces of small diameters by Sturm and Ohta–Pálfia. PubDate: 2018-02-12 DOI: 10.1007/s00526-018-1310-5 Issue No:Vol. 57, No. 2 (2018)

Authors:Xuemei Zhang; Yihong Du Abstract: In this paper we give sharp conditions on K(x) and f(u) for the existence of strictly convex solutions to the boundary blow-up Monge–Ampère problem $$\begin{aligned} M[u](x)=K(x)f(u) \quad \hbox {for } x \in \Omega ,\; u(x)\rightarrow +\,\infty \quad \hbox {as } \mathrm{dist}(x,\partial \Omega )\rightarrow 0. \end{aligned}$$ Here \(M[u]=\det \, (u_{x_{i}x_{j}})\) is the Monge–Ampère operator, and \(\Omega \) is a smooth, bounded, strictly convex domain in \( \mathbb {R}^N \, (N\ge 2)\) . Further results are obtained for the special case that \(\Omega \) is a ball. Our approach is largely based on the construction of suitable sub- and super-solutions. PubDate: 2018-02-10 DOI: 10.1007/s00526-018-1312-3 Issue No:Vol. 57, No. 2 (2018)

Authors:Long Li; Kai Zheng Abstract: In this paper, we first prove a generalized Matsushima’s theorem, i.e. the automorphism group is reductive on a Fano manifold admitting Kähler–Einstein metrics with cone singularities along a smooth divisor. Note that the divisor in our paper is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler–Einstein cone metrics by combining the reductivity of the automorphism group with the continuity method, the approximation trick and the bifurcation technique. Moreover, our method provides an existence theorem of Kähler–Einstein cone metrics with respect to conic Ding functional. PubDate: 2018-02-10 DOI: 10.1007/s00526-018-1313-2 Issue No:Vol. 57, No. 2 (2018)

Authors:Luis A. Caffarelli; Hui Yu Abstract: We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time asymptotics of this flow. PubDate: 2018-02-09 DOI: 10.1007/s00526-018-1311-4 Issue No:Vol. 57, No. 2 (2018)

Authors:Maria Colombo; Antonio De Rosa; Andrea Marchese Abstract: Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree, and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure \(\mu ^-\) onto a target measure \(\mu ^+\) , along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power \(\alpha \in (0,1)\) of the intensity of the flow. In this paper we address an open problem in the book Optimal transportation networks by Bernot, Caselles and Morel and we improve the stability for optimal traffic paths in the Euclidean space \(\mathbb {R}^d\) , with respect to variations of the given measures \((\mu ^-,\mu ^+)\) , which was known up to now only for \(\alpha >1-\frac{1}{d}\) . We prove it for exponents \(\alpha >1-\frac{1}{d-1}\) [in particular, for every \(\alpha \in (0,1)\) when \(d=2\) ], for a fairly large class of measures \(\mu ^+\) and \(\mu ^-\) . PubDate: 2018-01-19 DOI: 10.1007/s00526-017-1299-1 Issue No:Vol. 57, No. 1 (2018)

Authors:Martin Bauer; Martins Bruveris; Boris Kolev Abstract: Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) and on its Sobolev completions \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) . We prove local well-posedness of the geodesic equations both on the Banach manifold \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) and on the Fréchet-manifold \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) provided the order of the metric is greater or equal to one. In addition we show that the \(H^s\) -metric induces a strong Riemannian metric on the Banach manifold \({\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) of the same order s, provided \(s>\frac{3}{2}\) . These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group. PubDate: 2018-01-13 DOI: 10.1007/s00526-018-1300-7 Issue No:Vol. 57, No. 1 (2018)

Authors:M. Šilhavý Abstract: This paper deals with the weakened convexity properties, mult. ext. quasiconvexity, mult. ext. one convexity, and mult. ext. polyconvexity, for integral functionals of the form $$\begin{aligned} I(\omega _1,\ldots ,\omega _s) = \int _\Omega f(\omega _1,\ldots ,\omega _s) dx \end{aligned}$$ where \(\omega _1,\ldots ,\omega _s\) are closed differential forms on a bounded open set \(\Omega \subset {\mathbb {R}}^n\) . The main results of the paper are explicit descriptions of mult. ext. quasiaffine and mult ext. polyconvex functions. It turns out that these two classes consist, respectively, of linear and convex combinations of the set of all wedge products of exterior powers of the forms \(\omega _1,\ldots ,\omega _s\) . Thus, for example, a function \(f=f(\omega _1,\ldots ,\omega _s)\) is mult. ext. polyconvex if and only if $$\begin{aligned} f(\omega _1,\ldots ,\omega _s) = \Phi (\ldots ,\omega _1^{q_1}\wedge \cdots \wedge \omega _s^{q_s},\ldots ) \end{aligned}$$ where \(q_1,\ldots ,q_s\) ranges a finite set of integers and \(\Phi \) is a convex function. An existence theorem for the minimum energy state is proved for mult. ext. polyconvex integrals. The polyconvexity in the calculus of variations and nonlinear elasticity are particular cases of mult. ext. polyconvexity. Our main motivation comes from electro-magneto-elastic interactions in continuous bodies. There the mult. ext. polyconvexity takes the form determined by an involved direct calculation in an earlier paper of the author (Šilhavý in Math Mech Solids, 2017. http://journals.sagepub.com/doi/metrics/10.1177/1081286517696536). PubDate: 2018-01-10 DOI: 10.1007/s00526-017-1298-2 Issue No:Vol. 57, No. 1 (2018)