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Journal articles on the topic 'Elliptic manifold'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Bridges, Thomas J. "Canonical multi-symplectic structure on the total exterior algebra bundle." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2069 (February 8, 2006): 1531–51. http://dx.doi.org/10.1098/rspa.2005.1629.

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The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

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2

Paternain, Gabriel P. "On the topology of manifolds with completely integrable geodesic flows." Ergodic Theory and Dynamical Systems 12, no. 1 (March 1992): 109–21. http://dx.doi.org/10.1017/s0143385700006623.

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AbstractWe show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy of M is finite dimensional. We also show that rational ellipticity is shared by simply connected compact manifolds whose cotangent bundle admits a multiplicity free compact action that leaves invariant the Hamiltonian associated with some Riemannian metric. In particular it follows that if M is a Riemannian manifold whose geodesic flow is completely integrable by the Thimm method, then M is rationally elliptic. Other questions concerning the global behaviour of geodesics on homogeneous spaces are discussed.

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3

Rogov, Vasily. "Complex Geometry of Iwasawa Manifolds." International Mathematics Research Notices 2020, no. 23 (November 7, 2018): 9420–39. http://dx.doi.org/10.1093/imrn/rny230.

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Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.

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4

Terzic, Svjetlana. "On geometric formality of rationally elliptic manifolds in dimensions 6 and 7." Publications de l'Institut Math?matique (Belgrade) 103, no. 117 (2018): 211–22. http://dx.doi.org/10.2298/pim1817211t.

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We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2 = 3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2 ? 2 and b3 = 0 can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well.

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5

EZHOV, V. V., A. V. ISAEV, and G. SCHMALZ. "INVARIANTS OF ELLIPTIC AND HYPERBOLIC CR-STRUCTURES OF CODIMENSION 2." International Journal of Mathematics 10, no. 01 (February 1999): 1–52. http://dx.doi.org/10.1142/s0129167x99000021.

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We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves "almost" like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of "semi-flat" manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

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6

Fan, Haining. "Existence Theorems for a Class of Edge-Degenerate Elliptic Equations on Singular Manifolds." Proceedings of the Edinburgh Mathematical Society 58, no. 2 (February 17, 2015): 355–77. http://dx.doi.org/10.1017/s0013091514000145.

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AbstractIn this paper we establish the Nehari manifold on edge Sobolev spaces and study some of their properties. Furthermore, we use these results and the mountain pass theorem to get non-negative solutions of a class of edge-degenerate elliptic equations on singular manifolds under different conditions.

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7

Fegan, H. D., and B. Steer. "First Order Operators on Manifolds With a Group Action." Canadian Journal of Mathematics 48, no. 4 (August 1, 1996): 758–76. http://dx.doi.org/10.4153/cjm-1996-039-6.

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AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

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8

Park, Jiewon. "Matrix Inequality for the Laplace Equation." International Mathematics Research Notices 2019, no. 11 (September 27, 2017): 3485–97. http://dx.doi.org/10.1093/imrn/rnx226.

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Abstract Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci flow, Kähler–Ricci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold with nonnegative Ricci curvature. In this article, we prove a related matrix inequality on manifolds with suitable curvature and volume growth assumptions.

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9

De Filippis, Cristiana, and Giuseppe Mingione. "Manifold Constrained Non-uniformly Elliptic Problems." Journal of Geometric Analysis 30, no. 2 (September 20, 2019): 1661–723. http://dx.doi.org/10.1007/s12220-019-00275-3.

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10

Ashyralyev, A., Y. Sozen, and F. Hezenci. "A remark on elliptic differential equations on manifold." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 99, no. 3 (September 30, 2020): 75–85. http://dx.doi.org/10.31489/2020m3/75-85.

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For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.

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11

Averina, Tatiana A., Elena V. Karachanskaya, and Konstantin A. Rybakov. "Statistical analysis of diffusion systems with invariants." Russian Journal of Numerical Analysis and Mathematical Modelling 33, no. 1 (February 23, 2018): 1–13. http://dx.doi.org/10.1515/rnam-2018-0001.

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Abstract The aim of the paper is the construction and numerical solution of stochastic differential equations whose trajectories are located on a given smooth manifold with probability 1. Second order cylindrical surfaces, i.e., elliptic, hyperbolic, and parabolic cylinders serve as examples of such manifolds for the tree-dimensional space (the phase space is two-dimensional). Classes of stochastic differential equations are constructed for these surfaces and linear equations with multiplicative noise are marked in these classes. The results of modelling were used to estimate the deviations of numerical solutions from the manifold. A comparative analysis of considered examples was carried out for accuracy of eight numerical solution methods for stochastic differential equations.

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12

Tahri, Kamel. "Nonhomogeneous polyharmonic elliptic problems involving GJMS operator on Riemannian manifold." Asian-European Journal of Mathematics 13, no. 06 (May 20, 2019): 2050115. http://dx.doi.org/10.1142/s1793557120501156.

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Let [Formula: see text] be a closed Riemannian manifold, under some assumptions on [Formula: see text] and [Formula: see text] by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: [Formula: see text]. When [Formula: see text] is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.

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13

DOMENECH, G., M. L. LEVINAS, and N. SÁNCHEZ. "QUANTUM FIELD THEORY AND THE ANTIPODAL IDENTIFICATION OF SPACE TIME." International Journal of Modern Physics A 03, no. 11 (November 1988): 2567–88. http://dx.doi.org/10.1142/s0217751x88001065.

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We investigate the “elliptic interpretation” of space-time (identification of antipodal points or events) in anti-deSitter and in Rindler manifolds and its consequences for QFT. We compare and give a complete description of antipodal identification in space-times with and without event horizons. Antipodal identification relates the field theories on deSitter and on anti-deSitter spaces. In the “elliptic” Rindler manifold, imaginary time is periodic with period β/2 but the Green functions (for both identifications with and without “Conical singularity”) have period β. (Here β=2π/α, α is the acceleration.) Additional new properties for the Green functions are obtained and the new terms added to the stress tensor computed.

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14

Lescure, Jean-Marie. "Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds." Journal of K-Theory 4, no. 2 (December 1, 2008): 263–97. http://dx.doi.org/10.1017/is008007020jkt062.

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AbstractIn [7], a notion of noncommutative tangent space is associated with a conical pseudomanifold and Poincaré duality in K-theory is proved between this space and the pseudomanifold. The present paper continues this line of work. We show that an appropriate presentation of the notion of symbol on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret Poincaré duality in the singular setting as a noncommutative symbol map.

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15

McLean, Mark. "Affine varieties, singularities and the growth rate of wrapped Floer cohomology." Journal of Topology and Analysis 10, no. 03 (August 30, 2018): 493–530. http://dx.doi.org/10.1142/s1793525318500176.

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In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if [Formula: see text] is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.

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16

Brüning, Jochen, and Toshikazu Sunada. "On the spectrum of periodic elliptic operators." Nagoya Mathematical Journal 126 (June 1992): 159–71. http://dx.doi.org/10.1017/s0027763000004049.

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It was observed in [Su5] that the spectrum of a periodic Schrödinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators.

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17

Goffi, Alessandro, and Francesco Pediconi. "A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds." Journal of Geometric Analysis 31, no. 8 (February 16, 2021): 8641–65. http://dx.doi.org/10.1007/s12220-021-00607-2.

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AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

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18

Fu, Jixiang, Weimin Sheng, and Lixia Yuan. "Prescribed k-Curvature Problems on Complete Noncompact Riemannian Manifolds." International Mathematics Research Notices 2020, no. 23 (November 8, 2018): 9559–92. http://dx.doi.org/10.1093/imrn/rny262.

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Abstract To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.

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19

Pym, Brent. "Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets." Compositio Mathematica 153, no. 4 (March 13, 2017): 717–44. http://dx.doi.org/10.1112/s0010437x16008174.

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A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities$\widetilde{E}_{6},\widetilde{E}_{7}$and$\widetilde{E}_{8}$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii’s Poisson structures of type$q_{5,1}$are the only log symplectic structures on projective four-space whose singular points are all elliptic.

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20

Wiemeler, Michael. "S1-Equivariant bordism, invariant metrics of positive scalar curvature, and rigidity of elliptic genera." Journal of Topology and Analysis 12, no. 04 (January 21, 2019): 1103–56. http://dx.doi.org/10.1142/s1793525319500766.

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We construct geometric generators of the effective [Formula: see text]-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which [Formula: see text]-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the same manifold, the only obstruction to the existence of such a metric is an [Formula: see text]-genus of orbit spaces. This [Formula: see text]-genus generalizes a previous definition of Lott for orbit spaces of semi-free [Formula: see text]-actions. As a further application of our results, we give a new proof of the vanishing of the [Formula: see text]-genus of a Spin manifold with nontrivial [Formula: see text]-action originally proven by Atiyah and Hirzebruch. Moreover, based on our computations we can give a bordism-theoretic proof for the rigidity of elliptic genera originally proven by Taubes and Bott–Taubes.

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21

Dubey, Ravi Shanker, and V. B. L.Chaurasia. "Definite integrals of generalized certain class of incomplete elliptic integrals." Tamkang Journal of Mathematics 44, no. 2 (June 29, 2013): 197–208. http://dx.doi.org/10.5556/j.tkjm.44.2013.1410.

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Elliptic- integral have their importance and potential in certain problems in radiation physics and nuclear technology, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics. A number of earlier works on the subject contains remarkably large number of general families of elliptic- integrals and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, our aim here is to give a systematic account of the theory of a certain family of generalized incomplete elliptic integrals in a unique and generalized manner. The results established in this paper are of manifold generality and basic in nature. By making use of the familiar Riemann-Liouville fractional differ integral operators, we establish many explicit hypergeometric representations and apply these representation in deriving several definite integrals pertaining to their, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of generalized incomplete elliptic integrals involved therein.

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22

Basile, Nicola, and Michele Mininni. "On the existence of eigenvalues for some nonlinear elliptic and hyperbolic problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 101, no. 3-4 (1985): 297–305. http://dx.doi.org/10.1017/s0308210500020849.

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SynopsisIn this paper some eigenvalue problems for elliptic as well as hyperbolic equations are solved. The main tool used is an abstract critical point theorem on an unbounded manifold of the form {u | (Lu, u) = constant} (where L is a nonpositive selfadjoint operator), which makes use of a linking type argument on a manifold.

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23

Boughazi, Hichem. "Second-Order Elliptic Equation with Singularities." International Journal of Differential Equations 2020 (May 20, 2020): 1–16. http://dx.doi.org/10.1155/2020/4589864.

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On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

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24

Wang, Xiangsheng. "Elliptic boundary value problem on non-compact G-manifolds." International Journal of Mathematics 28, no. 04 (April 2017): 1750025. http://dx.doi.org/10.1142/s0129167x17500252.

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In this paper, an equality between the Hochs–Mathai type index and the Atiyah–Patodi–Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property of the global elliptic boundary value problems of the Atiyah–Patodi–Singer type on a non-compact manifold is studied.

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25

Qian, Zhongmin. "A gradient estimate on a manifold with convex boundary." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 1 (1997): 171–79. http://dx.doi.org/10.1017/s0308210500023568.

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We present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact Riemannian manifold with convex boundary, or on a noncompact complete manifold. Our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation, and an abstract gradient estimate for a Markov semigroup.

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26

Górka, Przemysław. "Elliptic Equation with Singular Terms on Riemannian Manifold." Letters in Mathematical Physics 79, no. 2 (January 11, 2007): 193–201. http://dx.doi.org/10.1007/s11005-006-0138-7.

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27

Bendersky, Martin. "Cobordism Span of a manifold and elliptic genera." Mathematische Zeitschrift 202, no. 4 (December 1989): 483–92. http://dx.doi.org/10.1007/bf01221586.

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28

CHANG, ZHE, and XIN LI. "TOWARDS A CPT INVARIANT QUANTUM FIELD THEORY ON ELLIPTIC DE SITTER SPACE." International Journal of Modern Physics A 23, no. 12 (May 10, 2008): 1821–38. http://dx.doi.org/10.1142/s0217751x08039608.

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Consequences of Schrödinger's antipodal identification on quantum field theory in de Sitter space are investigated. The elliptic ℤ2 identification provides observers with complete information. We show that a suitable confinement on dimension of the elliptic de Sitter space guarantees the existence of globally defined spinors and orientable dS/ℤ2 manifold. In Beltrami coordinates, we give exact solutions of scalar and spinor fields. The CPT invariance of quantum field theory on the elliptic de Sitter space is presented explicitly.

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29

SLAPAR, MARKO. "CANCELLING COMPLEX POINTS IN CODIMENSION TWO." Bulletin of the Australian Mathematical Society 88, no. 1 (August 9, 2012): 64–69. http://dx.doi.org/10.1017/s0004972712000652.

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AbstractA generically embedded real submanifold of codimension two in a complex manifold has isolated complex points that can be classified as either elliptic or hyperbolic. In this paper we show that a pair consisting of one elliptic and one hyperbolic complex point of the same sign can be cancelled by a $\mathcal {C}^{0}$small isotopy of embeddings.

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30

Chen, Robin Ming, Samuel Walsh, and Miles H. Wheeler. "Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics." Nonlinearity 35, no. 4 (February 28, 2022): 1927–85. http://dx.doi.org/10.1088/1361-6544/ac5096.

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Abstract In this paper, we present a center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders that is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution problem. Under suitable hypotheses, the resulting center manifold is finite dimensional and captures all sufficiently small bounded solutions. Compared with classical methods, the reduced ODE on the manifold is more directly related to the original physical problem and also easier to compute. The analysis is conducted directly in Hölder spaces, which is often desirable for elliptic equations. We then use this machinery to construct small bounded solutions to a variety of systems. These include heteroclinic and homoclinic solutions of the anti-plane shear problem from nonlinear elasticity; exact slow moving invasion fronts in a two-dimensional Fisher–KPP equation; and hydrodynamic bores with vorticity in a channel. The last example is particularly interesting in that we find solutions with critical layers and distinctive ‘half cat’s eye’ streamline patterns.

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31

Lee, Yong Hah. "Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds." Nagoya Mathematical Journal 162 (June 2001): 149–67. http://dx.doi.org/10.1017/s0027763000007844.

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In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

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Yuan, Xiaoping. "Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation." International Journal of Mathematics and Mathematical Sciences 2003, no. 18 (2003): 1111–36. http://dx.doi.org/10.1155/s0161171203207092.

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Matallah, Atika, Sara Litimein, and Sofiane Messirdi. "Existence of multiple solutions for a nonhomogeneous p-Laplacian elliptic equation with critical Sobolev-Hardy exponent." Boletim da Sociedade Paranaense de Matemática 40 (January 24, 2022): 1–12. http://dx.doi.org/10.5269/bspm.46206.

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This paper is concerned with the existence of multiple nontrivial solutions for nonhomogeneous p-Laplacain elliptic problems involving the critical Hardy-Sobolev exponent. The method used here is based on the Nehari manifold.

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34

Gaudin, Michel, and Ugo Moschella. "Doubly elliptic strings on the (anti) de Sitter manifold." International Journal of Geometric Methods in Modern Physics 12, no. 03 (February 27, 2015): 1550032. http://dx.doi.org/10.1142/s0219887815500322.

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We present a new class of elliptic-like strings on two-dimensional manifolds of constant curvature. Our solutions are related to a class of identities between Jacobi theta functions and to the geometry of the light-cone in one dimension more. We show in particular that two well-known fundamental identities among theta functions have a natural interpretation as expressing the Virasoro constraints of dS or AdS strings.

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35

Adriouch, K., and A. El Hamidi. "The Nehari manifold for systems of nonlinear elliptic equations." Nonlinear Analysis: Theory, Methods & Applications 64, no. 10 (May 2006): 2149–67. http://dx.doi.org/10.1016/j.na.2005.06.003.

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36

Gil, Juan B., Thomas Krainer, and Gerardo A. Mendoza. "A conic manifold perspective of elliptic operators on graphs." Journal of Mathematical Analysis and Applications 340, no. 2 (April 2008): 1296–311. http://dx.doi.org/10.1016/j.jmaa.2007.09.049.

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KOCH, HANS, and SAŠA KOCIĆ. "Renormalization of vector fields and Diophantine invariant tori." Ergodic Theory and Dynamical Systems 28, no. 5 (October 2008): 1559–85. http://dx.doi.org/10.1017/s0143385707000892.

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AbstractWe extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on 𝕋d×ℝℓ. Each Diophantine vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting 𝒲 to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

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38

Xiu, Zonghu, and Caisheng Chen. "Existence of Multiple Solutions for a -Kirchhoff Problem with Nonlinear Boundary Conditions." Scientific World Journal 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/516093.

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The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem , , = , on , where , . By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.

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39

Chaurasia, V. B. L., and R. C. Meghwal. "Unified presentation of certain families of elliptic-type integrals related to Euler integrals and generating functions." Tamkang Journal of Mathematics 43, no. 4 (December 31, 2012): 507–19. http://dx.doi.org/10.5556/j.tkjm.43.2012.1320.

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The aim of the present paper is to give unified presentation of certain families of elliptic-type integrals related to Euler integrals and generating functions. It is a unification and generalization of certain families of elliptic-type integrals which were studied in a number of earlier works on the subject due to their importance and potential in certain problems in radiation physics and nuclear technology. The results established in this paper are of manifold generality and basic in nature. Besides deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.

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Hsu, Tsing-San. "Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in." Abstract and Applied Analysis 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/864296.

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Existence and multiplicity of positive solutions for the following semilinear elliptic equation: in , , are established, where if if , , satisfy suitable conditions, and maybe changes sign in . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

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Abnoune, A., E. Azroul, and M. T. K. Abbassi. "Study of a second-order nonlinear elliptic problem generated by a divergence type operator on a compact Riemannian manifold." Filomat 32, no. 14 (2018): 4811–20. http://dx.doi.org/10.2298/fil1814811a.

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In this paper, we will study a second-order nonlinear elliptic problem generated by an operator of divergence type (or type leray-Lion) : (P1){ A(u) = f in M u = 0 on ? (1) on (M,g) a compact Riemannian manifold et ? its border.

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Harlim, John, Shixiao W. Jiang, Hwanwoo Kim, and Daniel Sanz-Alonso. "Graph-based prior and forward models for inverse problems on manifolds with boundaries." Inverse Problems 38, no. 3 (January 31, 2022): 035006. http://dx.doi.org/10.1088/1361-6420/ac3994.

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Abstract This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matérn-type Gaussian field priors that enable flexible modeling near the boundaries, representing boundary values by superposition of harmonic functions with appropriate Dirichlet boundary conditions. We also investigate the graph-based approximation of forward models from PDE parameters to observed quantities. In the construction of graph-based prior and forward models, we leverage the ghost point diffusion map algorithm to approximate second-order elliptic operators with classical boundary conditions. Numerical results validate our graph-based approach and demonstrate the need to design prior covariance models that account for boundary conditions.

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CHANG, S. L., C. S. CHIEN, and B. W. JENG. "TRACING THE SOLUTION SURFACE WITH FOLDS OF A TWO-PARAMETER SYSTEM." International Journal of Bifurcation and Chaos 15, no. 08 (August 2005): 2689–700. http://dx.doi.org/10.1142/s0218127405013630.

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We describe a special Gauss–Newton method for tracing solution manifolds with singularities of multiparameter systems. First we choose one of the parameters as the continuation parameter, and fix the others. Then we trace one-dimensional solution curves by using continuation methods. Singularities such as folds, simple and multiple bifurcations on each solution curve can be easily detected. Next, we choose an interval for the second continuation parameter, and trace one-dimensional solution curves for certain values in this interval. This constitutes a two-dimensional solution surface. The procedure can be generalized to trace a k-dimensional solution manifold. Numerical results in 1D, 2D and 3D second-order semilinear elliptic eigenvalue problems given by Lions [1982] are reported.

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Chen, Guanwei, and Shiwang Ma. "Infinitely Many Nontrivial Solutions of Resonant Cooperative Elliptic Systems with Superlinear Terms." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/349304.

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We study a class of resonant cooperative elliptic systems and replace the Ambrosetti-Rabinowitz superlinear condition with general superlinear conditions. We obtain ground state solutions and infinitely many nontrivial solutions of this system by a generalized Nehari manifold method developed recently by Szulkin and Weth.

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Kristály, Alexandru, Nikolaos S. Papageorgiou, and Csaba Varga. "Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary." Canadian Mathematical Bulletin 53, no. 4 (December 1, 2010): 674–83. http://dx.doi.org/10.4153/cmb-2010-073-x.

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AbstractWe study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

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Ancona, Alano. "Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators." Nagoya Mathematical Journal 165 (March 2002): 123–58. http://dx.doi.org/10.1017/s0027763000008187.

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Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

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Valls, Claudia. "Stability of some solutions for elliptic equations on a cylindrical domain." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2109 (June 10, 2009): 2647–62. http://dx.doi.org/10.1098/rspa.2009.0110.

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We study analytically a class of solutions for the elliptic equation where α >0 and ε is a small parameter. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every α >0, it contains solutions that are defined for large values of time and they are very close (of order O ( ε )) to a linear torus for long times (of order O ( ε −1 )). The proof uses the fact that the equation leaves invariant a smooth centre manifold and, for the restriction of the system to the centre manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

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Jost, Jürgen, Lei Liu, and Miaomiao Zhu. "A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor." Journal für die reine und angewandte Mathematik (Crelles Journal) 2022, no. 785 (February 15, 2022): 81–116. http://dx.doi.org/10.1515/crelle-2021-0085.

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Abstract In this paper, we solve a new elliptic-parabolic system arising in geometric analysis that is motivated by the nonlinear supersymmetric sigma model of quantum field theory. The corresponding action functional involves two fields, a map from a Riemann surface into a Riemannian manifold and a spinor coupled to the map. The first field has to satisfy a second-order elliptic system, which we turn into a parabolic system so as to apply heat flow techniques. The spinor, however, satisfies a first-order Dirac-type equation. We carry that equation as a nonlinear constraint along the flow. With this novel scheme, in more technical terms, we can show the existence of Dirac-harmonic maps from a compact spin Riemann surface with smooth boundary to a general compact Riemannian manifold via a heat flow method when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor.

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Gelfreich, V., and L. Lerman. "Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system." Nonlinearity 15, no. 2 (February 19, 2002): 447–57. http://dx.doi.org/10.1088/0951-7715/15/2/312.

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Lu, Feng-Yun. "The Nehari manifold and application to a semilinear elliptic system." Nonlinear Analysis: Theory, Methods & Applications 71, no. 7-8 (October 2009): 3425–33. http://dx.doi.org/10.1016/j.na.2009.02.009.

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