Authors:M. Dajczer; Th. Vlachos Pages: 1961 - 1979 Abstract: The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0641-8 Issue No:Vol. 196, No. 6 (2017)

Authors:Andreas Debrouwere; Jasson Vindas Pages: 1983 - 2003 Abstract: We study spaces of vector-valued quasianalytic functions and solve the first Cousin problem in these spaces. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0649-0 Issue No:Vol. 196, No. 6 (2017)

Authors:Kunnath Sandeep; Cyril Tintarev Pages: 2005 - 2021 Abstract: We prove a subset of inequalities of Caffarelli–Kohn–Nirenberg type in the hyperbolic space \({{\mathbb {H}}^N}, N\ge 2\) , based on invariance with respect to a certain nonlinear scaling group, and study existence of corresponding minimizers. Earlier results concerning the Moser–Trudinger inequality are now interpreted in terms of CKN inequalities on the Poincaré disk. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0650-7 Issue No:Vol. 196, No. 6 (2017)

Authors:Gilles Evéquoz Pages: 2023 - 2042 Abstract: This paper studies for large frequency number \(k>0\) the existence and multiplicity of solutions of the semilinear problem $$\begin{aligned} -\varDelta u -k^2 u=Q(x) u ^{p-2}u\quad \text { in }\mathbb {R}^N, \quad N\ge 2. \end{aligned}$$ The exponent p is subcritical, and the coefficient Q is continuous, nonnegative and satisfies the condition $$\begin{aligned} \limsup _{ x \rightarrow \infty }Q(x)<\sup _{x\in \mathbb {R}^N}Q(x). \end{aligned}$$ In the limit \(k\rightarrow \infty \) , sequences of solutions associated with ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function Q. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0651-6 Issue No:Vol. 196, No. 6 (2017)

Authors:Vincenzo Ambrosio Pages: 2043 - 2062 Abstract: By using the penalization method and the Ljusternik–Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation $$\begin{aligned} \varepsilon ^{2s}(-\Delta )^{s} u + V(x)u = f(u)\quad \text{ in } {\mathbb {R}}^{N} \end{aligned}$$ where \(\varepsilon >0\) is a parameter, \(s\in (0, 1)\) , \(N>2s\) , \((-\Delta )^{s}\) is the fractional Laplacian, V is a positive continuous potential with local minimum, and f is a superlinear function with subcritical growth. We also obtain a multiplicity result when \(f(u)= u ^{q-2}u+\lambda u ^{r-2}u\) with \(2<q<2^{*}_{s}\le r\) and \(\lambda >0\) , by combining a truncation argument and a Moser-type iteration. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0652-5 Issue No:Vol. 196, No. 6 (2017)

Authors:Xiaodong Chen; Xiaoxiang Jiao Pages: 2063 - 2076 Abstract: In this paper, we want to construct conformal minimal surfaces and conformal minimal two-spheres in \({\mathbb {H}}P^{n}\) by the twistor map \(\pi : {\mathbb {C}}P^{2n+1} \rightarrow {\mathbb {H}}P^{n}\) . The construction is due to Eells and Wood’s conclusion about the composition of a horizontal harmonic map in 1983. Firstly, we give a characterization of horizontal holomorphic surfaces in \({\mathbb {C}}P^{5}\) . Under this characterization, we construct eight families of conformal minimal surfaces in \({\mathbb {H}}P^{2}\) . Then, we study horizontal Veronese sequences in \({\mathbb {C}}P^{4}\) and \({\mathbb {C}}P^{5}\) , and we transform the construction into solving a quadratic equation. Based on this, we get some examples of conformal minimal two-spheres in \({\mathbb {H}}P^{2}\) with constant curvature \(\frac{4}{5}\) and \(\frac{4}{13}\) . PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0653-4 Issue No:Vol. 196, No. 6 (2017)

Authors:Michele Maschio Pages: 2077 - 2089 Abstract: This paper examines special metrics on compact complex manifolds, and it is notably focused on the notion of super strong Kähler with torsion metric. This condition is related to the strong Kähler with torsion one in the same manner as the strongly Gauduchon condition is related to the Gauduchon one. Moreover, we provide sufficient and necessary conditions so that every strong Kähler with torsion metric on a compact complex manifold is in fact super strong Kähler with torsion. We prove that these conditions are verified on compact complex manifolds satisfying \(\partial \overline{\partial }\) -lemma but not on 6-dimensional compact complex nilmanifolds. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0654-3 Issue No:Vol. 196, No. 6 (2017)

Authors:Kolyan Ray; Johannes Schmidt-Hieber Pages: 2091 - 2103 Abstract: We investigate the regularity of the positive roots of a nonnegative function of one-variable. A modified Hölder space \(\mathcal {F}^\beta \) is introduced such that if \(f\in \mathcal {F}^\beta \) then \(f^\alpha \in C^{\alpha \beta }\) . This provides sufficient conditions to overcome the usual limitation in the square root case ( \(\alpha = 1/2\) ) for Hölder functions that \(f^{1/2}\) need be no more than \(C^1\) in general. We also derive bounds on the wavelet coefficients of \(f^\alpha \) , which provide a finer understanding of its local regularity. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0655-2 Issue No:Vol. 196, No. 6 (2017)

Authors:Prince Romeo Mensah Pages: 2105 - 2133 Abstract: We give an existence and asymptotic result for the so-called finite energy weak martingale solution of the compressible isentropic Navier–Stokes system driven by some random force in the whole spatial region. In particular, given a general nonlinear multiplicative noise, we establish the convergence to the incompressible system as the Mach number, representing the ratio between the average flow velocity and the speed of sound, approaches zero. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0656-1 Issue No:Vol. 196, No. 6 (2017)

Authors:Sharief Deshmukh Pages: 2135 - 2145 Abstract: It is well known that the Euclidean space \( (R^{n},\left\langle ,\right\rangle )\) , the n-sphere \(S^{n}(c)\) of constant curvature c are examples of spaces admitting many conformal vector fields, and therefore conformal vector fields are used in obtaining characterizations of these spaces. In this paper, we use nontrivial conformal vector fields on a compact and connected Riemannian manifold to characterize the sphere \(S^{n}(c)\) . Also, we use a nontrivial conformal vector field on a complete and connected Riemannian manifold and find characterizations for a Euclidean space \((R^{n},\left\langle ,\right\rangle )\) and the sphere \(S^{n}(c)\) . PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0657-0 Issue No:Vol. 196, No. 6 (2017)

Authors:Flavia Giannetti; Antonia Passarelli di Napoli; Atsushi Tachikawa Pages: 2147 - 2165 Abstract: We prove the partial Hölder continuity of the local minimizers of non-autonomous integral functionals of the type $$\begin{aligned} \int _\varOmega \varPhi \left( \big ( A^{\alpha \beta }_{ij}(x,u) D_iu^\alpha D_ju^\beta \big )^{1/2}\right) \mathrm{d}x, \end{aligned}$$ where \(\varPhi \) is an Orlicz function satisfying both the \(\varDelta _2\) and \(\nabla _2\) conditions and the function \(A(x,s) = \big (A^{\alpha \beta }_{ij}(x,s)\big )\) is uniformly elliptic, bounded and continuous. Assuming in addition that the function \(A(x,s) = \big (A^{\alpha \beta }_{ij}(x,s)\big )\) is Hölder continuous, we prove the partial Hölder continuity also of the gradient of the local minimizers. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0658-z Issue No:Vol. 196, No. 6 (2017)

Authors:Aram L. Karakhanyan; Henrik Shahgholian Pages: 2167 - 2183 Abstract: For a periodic vector field F, let \(X^\varepsilon \) solve the dynamical system $$\begin{aligned} \frac{{\hbox {d}}{\hbox {X}}^{\varepsilon }}{{\hbox {d}}t} = {{F}}\left( \frac{{X}^{\varepsilon }}{\varepsilon }\right) . \end{aligned}$$ In (Set Valued Anal 2(1–2):175–182, 1994) Ennio De Giorgi enquiers whether from the existence of the limit \( X^0(t):=\lim \nolimits _{\varepsilon \rightarrow 0} X^\varepsilon (t)\) one can conclude that \( \frac{{\hbox {d}} X^0}{{\hbox {d}}t}= {\hbox {constant}}\) . Our main result settles this conjecture under fairly general assumptions on F, which in some cases may also depend on t-variable. Once the above problem is solved, one can apply the result to the corresponding transport equation, in a standard way. This is also touched upon in the text to follow. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0659-y Issue No:Vol. 196, No. 6 (2017)

Authors:Leonardo Biliotti; Michela Zedda Pages: 2185 - 2211 Abstract: We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of noncompact real reductive Lie groups on topological spaces that admit functions similar to the Kempf–Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real noncompact reductive Lie group G on a real compact submanifold M of a Kähler manifold Z and to the action of G on measures of M. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0660-5 Issue No:Vol. 196, No. 6 (2017)

Authors:Jie Fei; Ling He Pages: 2213 - 2237 Abstract: In this paper we determine all homogeneous minimal immersions of 2-spheres in quaternionic projective spaces \({\mathbb {H}}P^n\) . PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0661-4 Issue No:Vol. 196, No. 6 (2017)

Authors:C. Bargetz; E. A. Nigsch; N. Ortner Pages: 2239 - 2251 Abstract: We apply L. Schwartz’ theory of vector-valued distributions in order to simplify, unify and generalize statements about convolvability of distributions, their regularization properties and topological properties of sets of distributions. The proofs rely on propositions on the multiplication of vector-valued distributions and on the characterization of the spaces \(\mathcal {O}_{M}(E,F)\) and \(\mathcal {O}_{C}'(E,F)\) of multipliers and convolutors for distribution spaces E and F. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0662-3 Issue No:Vol. 196, No. 6 (2017)

Authors:Calin Iulian Martin Pages: 2253 - 2260 Abstract: We show that gravity wave trains governed by the equatorial f-plane approximation propagate at the free surface of a rotational water flow of constant vorticity vector \((\Omega _1, \Omega _2, \Omega _3)\) over a flat bed only if the flow is two-dimensional. Owing to the presence of Coriolis effects, our result is also true even if the vorticity vector vanishes. This represents a striking difference when compared with the cases without geophysical effects discussed in Constantin (Europhys Lett 86:29001, 2009, Eur J Mech 30:12–16; 2011) and Martin (J Math Fluid Mech 2016. doi:10.1007/s00021-016-0306-1), where the conclusion about the two-dimensionality of the flow was possible under the assumption of constant nonvanishing vorticity vector. Another upshot is that the only nonzero component of the vorticity that may not vanish is \(\Omega _2\) , that is, the one pointing in the horizontal direction orthogonal to the direction of wave propagation. PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0663-2 Issue No:Vol. 196, No. 6 (2017)

Authors:Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen Pages: 2261 - 2301 Abstract: Given a complete noncompact Riemannian manifold \(N^n\) , we investigate whether the set of bounded Sobolev maps \((W^{1, p} \cap L^\infty ) (Q^m; N^n)\) on the cube \(Q^{m}\) is strongly dense in the Sobolev space \(W^{1, p} (Q^m; N^n)\) for \(1 \le p \le m\) . The density always holds when p is not an integer. When p is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured, for example, by a uniform Lipschitz geometry of \(N^{n}\) . As a by-product, we give necessary and sufficient conditions for the strong density of the set of smooth maps \(C^\infty (\overline{Q^m}; N^n)\) in \(W^{1, p} (Q^m; N^n)\) . PubDate: 2017-12-01 DOI: 10.1007/s10231-017-0664-1 Issue No:Vol. 196, No. 6 (2017)

Authors:Pierluigi Benevieri; Alessandro Calamai; Massimo Furi; Maria Patrizia Pera Abstract: Let E, F be real Banach spaces and S the unit sphere of E. We study a nonlinear eigenvalue problem of the type \(Lx + \varepsilon N(x) = \lambda Cx\) , where \(\varepsilon ,\lambda \) are real parameters, \(L:E \rightarrow F\) is a Fredholm linear operator of index zero, \(C:E \rightarrow F\) is a compact linear operator, and \(N:S \rightarrow F\) is a compact map. Given a solution \((x,\varepsilon ,\lambda ) \in S \times \mathbb {R}\times \mathbb {R}\) of this problem, we say that the first element x of the triple is a unit eigenvector corresponding to the eigenpair \((\varepsilon ,\lambda )\) . Assuming that \(\lambda _0 \in \mathbb {R}\) is such that the kernel of \(L -\lambda _0C\) is odd dimensional and a natural transversality condition between the operators \(L -\lambda _0C\) and C is satisfied, we prove that, in the set of all the eigenpairs, the connected component containing \((0,\lambda _0)\) is either unbounded or meets an eigenpair \((0,\lambda _1)\) , with \(\lambda _1 \not = \lambda _0\) . Our approach is topological and based on the classical Leray–Schauder degree. PubDate: 2017-11-11 DOI: 10.1007/s10231-017-0717-5

Authors:M. Bildhauer; M. Fuchs; J. Müller; X. Zhong Abstract: We prove local boundedness of generalized solutions to a large class of variational problems of linear growth including boundary value problems of minimal surface type and models from image analysis related to the procedure of TV regularization occurring in connection with the denoising of images, which might even be coupled with an inpainting process. Our main argument relies on a Moser-type iteration procedure. PubDate: 2017-11-10 DOI: 10.1007/s10231-017-0716-6

Authors:M. Dajczer; Th. Vlachos Abstract: All the results of the paper remain true as stated but there is a serious gap in the proof of Theorem 13. Equations (26) and (27) need to be corrected. PubDate: 2017-10-13 DOI: 10.1007/s10231-017-0700-1