Journal articles: 'Mean curvature operator'

1

Mawhin, Jean. "Nonlinear boundary value problems involving the extrinsic mean curvature operator." Mathematica Bohemica 139, no. 2 (2014): 299–313. http://dx.doi.org/10.21136/mb.2014.143856.

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Bessa, Gregório P., Luquésio P. Jorge, Barnabé P. Lima, and José F. Montenegro. "Fundamental tone estimates for elliptic operators in divergence form and geometric applications." Anais da Academia Brasileira de Ciências 78, no. 3 (September 2006): 391–404. http://dx.doi.org/10.1590/s0001-37652006000300001.

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We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

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Yang, Dan, and Yu Fu. "Biharmonic hypersurfaces in pseudo-Riemannian space forms." International Journal of Geometric Methods in Modern Physics 13, no. 07 (July 25, 2016): 1650094. http://dx.doi.org/10.1142/s0219887816500948.

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Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

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Harrell II, Evans M., and Michael Loss. "On the Laplace Operator Penalized by Mean Curvature." Communications in Mathematical Physics 195, no. 3 (August 1998): 643–50. http://dx.doi.org/10.1007/s002200050406.

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DE LIMA, HENRIQUE FERNANDES, ANTONIO FERNANDO DE SOUSA, and MARCO ANTONIO LÁZARO VELÁSQUEZ. "STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER-ORDER MEAN CURVATURES LINEARLY RELATED IN CONFORMALLY STATIONARY SPACETIMES." International Journal of Mathematics 24, no. 14 (December 2013): 1350109. http://dx.doi.org/10.1142/s0129167x13501097.

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In this paper, we establish the notion of (r, s)-stability concerning spacelike hypersurfaces with higher-order mean curvatures linearly related in conformally stationary spacetimes of constant sectional curvature. In this setting, we characterize (r, s)-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of an operator naturally attached to the higher-order mean curvatures. Moreover, we obtain sufficient conditions which assure the (r, s)-stability of complete spacelike hypersurfaces immersed in the de Sitter space.

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Matsutani, Shigeki. "A constant mean curvature surface and the Dirac operator." Journal of Physics A: Mathematical and General 30, no. 11 (June 7, 1997): 4019–29. http://dx.doi.org/10.1088/0305-4470/30/11/028.

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Etemad, A. "First eigenvalue of the Laplace operator and mean curvature." Ukrainian Mathematical Journal 60, no. 7 (July 2008): 1172–75. http://dx.doi.org/10.1007/s11253-008-0111-y.

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Yang, Dan, Jinchao Yu, Jingjing Zhang, and Xiaoying Zhu. "A class of hypersurfaces in $ \mathbb{E}^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda\vec{H} $." AIMS Mathematics 7, no. 1 (2022): 39–53. http://dx.doi.org/10.3934/math.2022003.

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<abstract>A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured [<xref ref-type="bibr" rid="b1">1</xref>]: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.</abstract>

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Ecker, Klaus. "On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetimes." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 55, no. 1 (August 1993): 41–59. http://dx.doi.org/10.1017/s1446788700031918.

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AbstractWe prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.

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Qi, Xuesen, and Ximin Liu. "Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow." Open Mathematics 18, no. 1 (January 1, 2020): 1518–30. http://dx.doi.org/10.1515/math-2020-0090.

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Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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Qi, Xuesen, and Ximin Liu. "Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow." Open Mathematics 18, no. 1 (December 29, 2020): 1518–30. http://dx.doi.org/10.1515/math-2020-0090.

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Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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Pashaie, Firooz. "Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1." Proyecciones (Antofagasta) 40, no. 3 (June 1, 2021): 711–19. http://dx.doi.org/10.22199/issn.0717-6279-3584.

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In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

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Slesar, Vladimir. "Riemannian foliations and the kernel of the basic Dirac operator." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 2 (June 1, 2012): 145–58. http://dx.doi.org/10.2478/v10309-012-0046-z.

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Abstract In this paper, in the special setting of a Riemannian foliation en- dowed with a bundle-like metric, we obtain conditions that force the vanishing of the kernel of the basic Dirac operator associated to the metric; this way we extend the traditional setting of Riemannian foli- ations with basic-harmonic mean curvature, where Bochner technique and vanishing results are known to work. Beside classical conditions concerning the positivity of some curvature terms we obtain new rela- tions between the mean curvature form and the kernel of the basic Dirac operator

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Cano-Casanova, Santiago, Julián López-Gómez, and Kazuhiro Takimoto. "A weighted quasilinear equation related to the mean curvature operator." Nonlinear Analysis: Theory, Methods & Applications 75, no. 15 (October 2012): 5905–23. http://dx.doi.org/10.1016/j.na.2012.06.004.

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Corsato, Chiara, Franco Obersnel, and Pierpaolo Omari. "The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space." Georgian Mathematical Journal 24, no. 1 (March 1, 2017): 113–34. http://dx.doi.org/10.1515/gmj-2016-0078.

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AbstractWe discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

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CHEN, BANG-YEN. "Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions." Glasgow Mathematical Journal 41, no. 1 (March 1999): 33–41. http://dx.doi.org/10.1017/s0017089599970271.

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First we define the notion of k-Ricci curvature of a Riemannian n-manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.

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Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Nonhomogeneous Nonlinear Dirichlet Problems with ap-Superlinear Reaction." Abstract and Applied Analysis 2012 (2012): 1–28. http://dx.doi.org/10.1155/2012/918271.

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We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reactionf(z,ζ), whose primitivef(z,ζ)isp-superlinear near±∞, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like thep-Laplacian, the(p,q)-Laplacian, and thep-generalized mean curvature operator.

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Das, Sudip Kumar, Mirza Cenanovic, and Junfeng Zhang. "A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces." Proceedings of the International Geometry Center 12, no. 1 (February 28, 2019): 70–78. http://dx.doi.org/10.15673/tmgc.v12i1.1377.

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In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.

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Chen, Zhihui, and Yaotian Shen. "Infinitely many solutions of dirichlet problem for p-mean curvature operator." Applied Mathematics-A Journal of Chinese Universities 18, no. 2 (June 2003): 161–72. http://dx.doi.org/10.1007/s11766-003-0020-7.

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Güler, Erhan, Hasan Hacısalihoğlu, and Young Kim. "The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space." Symmetry 10, no. 9 (September 12, 2018): 398. http://dx.doi.org/10.3390/sym10090398.

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We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

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Park, Jung-Ho, Ji-Hye Moon, Sanghun Park, and Seung-Hyun Yoon. "GeoStamp: Detail Transfer Based on Mean Curvature Field." Mathematics 10, no. 3 (February 4, 2022): 500. http://dx.doi.org/10.3390/math10030500.

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A shape detail transfer is the process of extracting the geometric details of a source region and transferring it onto a target region. In this paper, we present a simple and effective method, called GeoStamp, for transferring shape details using a Poisson equation. First, the mean curvature field on a source region is computed by using the Laplace–Beltrami operator and is defined as the shape details of the source region. Subsequently, the source and target regions are parameterized on a common 2D domain, and a mean curvature field on the target region is interpolated by the correspondence between two regions. Finally, we solve the Poisson equation using the interpolated mean curvature field and the Laplacian matrix of the target region. Consequently, the mean curvature field of the target region is replaced with that of the source region, which results in the transfer of shape details from the source region to the target region. We demonstrate the effectiveness of our technique by showing several examples and also show that our method is quite useful for adding shape details to a surface patch filling a hole in a triangular mesh.

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Pashaie, Firooz. "On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$." Tamkang Journal of Mathematics 51, no. 4 (November 1, 2020): 313–32. http://dx.doi.org/10.5556/j.tkjm.51.2020.3188.

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A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

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Du, Li. "Classification of f-biharmonic submanifolds in Lorentz space forms." Open Mathematics 19, no. 1 (January 1, 2021): 1299–314. http://dx.doi.org/10.1515/math-2021-0084.

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Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

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Polymerakis, Panagiotis. "Spectral Estimates for Riemannian Submersions with Fibers of Basic Mean Curvature." Journal of Geometric Analysis 31, no. 10 (March 22, 2021): 9951–80. http://dx.doi.org/10.1007/s12220-021-00634-z.

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AbstractFor Riemannian submersions with fibers of basic mean curvature, we compare the spectrum of the total space with the spectrum of a Schrödinger operator on the base manifold. Exploiting this concept, we study submersions arising from actions of Lie groups. In this context, we extend the state-of-the-art results on the bottom of the spectrum under Riemannian coverings. As an application, we compute the bottom of the spectrum and the Cheeger constant of connected, amenable Lie groups.

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Sheng, Weimin, and Haobin Yu. "Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow." Communications in Contemporary Mathematics 19, no. 01 (November 24, 2016): 1550092. http://dx.doi.org/10.1142/s0219199715500923.

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We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

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Ling, Jiaoxiu, and Zhan Zhou. "Positive solutions of the discrete Dirichlet problem involving the mean curvature operator." Open Mathematics 17, no. 1 (September 14, 2019): 1055–64. http://dx.doi.org/10.1515/math-2019-0081.

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Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

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Chen, Bang-Yen. "Mean curvature and shape operator of isometric immersions in real-space-forms." Glasgow Mathematical Journal 38, no. 1 (January 1996): 87–97. http://dx.doi.org/10.1017/s001708950003130x.

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According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

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Jebelean, Petru, and Calin-Constantin Şerban. "Fisher–Kolmogorov type perturbations of the mean curvature operator in Minkowski space." Electronic Journal of Qualitative Theory of Differential Equations, no. 81 (2020): 1–12. http://dx.doi.org/10.14232/ejqtde.2020.1.81.

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Dai, Guowei. "Global structure of one-sign solutions for problem with mean curvature operator." Nonlinearity 31, no. 11 (October 18, 2018): 5309–28. http://dx.doi.org/10.1088/1361-6544/aadf43.

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Jebelean, Petru, and Calin-Constantin Şerban. "Fisher–Kolmogorov type perturbations of the mean curvature operator in Minkowski space." Electronic Journal of Qualitative Theory of Differential Equations, no. 81 (2020): 1–12. http://dx.doi.org/10.14232/ejqtde.2020.1.81.

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Bereanu, Cristian, Petru Jebelean, and Jean Mawhin. "Corrigendum to: The Dirichlet problem with mean curvature operator in Minkowski space." Advanced Nonlinear Studies 16, no. 1 (February 1, 2016): 173–74. http://dx.doi.org/10.1515/ans-2015-5030.

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Bonanno, Gabriele, Roberto Livrea, and Jean Mawhin. "Existence results for parametric boundary value problems involving the mean curvature operator." Nonlinear Differential Equations and Applications NoDEA 22, no. 3 (October 4, 2014): 411–26. http://dx.doi.org/10.1007/s00030-014-0289-7.

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Zhao, Liang. "The first eigenvalue of Laplace operator under powers of mean curvature flow." Science China Mathematics 53, no. 7 (May 27, 2010): 1703–10. http://dx.doi.org/10.1007/s11425-010-3123-7.

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Du, Li, and Jinjun Ren. "On η-biharmonic hypersurfaces in pseudo-Riemannian space forms." Mathematica Slovaca 72, no. 5 (October 1, 2022): 1259–72. http://dx.doi.org/10.1515/ms-2022-0086.

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Abstract In this paper, η-biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms are studied. We prove that such hypersurfaces with diagonalizable shape operator have constant mean curvature, which gives an affirmative partial answer to the conjecture in [Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type $\begin{array}{} \displaystyle M^3_2 \end{array}$ in $\begin{array}{} \displaystyle \mathbb{E}^4_2 \end{array}$ with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106]. As a result, we give several partial classification results.

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Li, Yanlin, Akram Ali, Fatemah Mofarreh, Abimbola Abolarinwa, and Rifaqat Ali. "Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms." Journal of Function Spaces 2021 (October 12, 2021): 1–10. http://dx.doi.org/10.1155/2021/6195939.

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This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the ϕ -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the ϕ -Laplacian operator on closed oriented m -dimensional slant submanifolds in a Sasakian space form M ~ 2 k + 1 ε is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the ϕ -Laplacian on slant submanifold in a sphere S 2 n + 1 with ε = 1 and ϕ = 2 .

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Wang, Jianxia, and Zhan Zhou. "Existence of Solutions for the Discrete Dirichlet Problem Involving p-Mean Curvature Operator." Discrete Dynamics in Nature and Society 2020 (August 29, 2020): 1–10. http://dx.doi.org/10.1155/2020/6591523.

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This work is to discuss the Dirichlet boundary value problem of the difference equation with p -mean curvature operator. Under some determinate growth conditions on the nonlinear term, the existence of one solution or two nontrivial solutions is obtained via variational methods and some analysis techniques. Examples are also given to illustrate our theorems in the last section.

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Gurban, Daniela, Petru Jebelean, and Călin Şerban. "Nontrivial Solutions for Potential Systems Involving the Mean Curvature Operator in Minkowski Space." Advanced Nonlinear Studies 17, no. 4 (October 1, 2017): 769–80. http://dx.doi.org/10.1515/ans-2016-6025.

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AbstractIn this paper, we use the critical point theory for convex, lower semicontinuous perturbations of{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator{u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

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Khan, Meraj Ali, Ali H. Alkhaldi, and Mohd Aquib. "Estimation of eigenvalues for the $ \alpha $-Laplace operator on pseudo-slant submanifolds of generalized Sasakian space forms." AIMS Mathematics 7, no. 9 (2022): 16054–66. http://dx.doi.org/10.3934/math.2022879.

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<abstract>In this study, we seek to establish new upper bounds for the mean curvature and constant sectional curvature of the first positive eigenvalue of the $ \alpha $-Laplacian operator on Riemannian manifolds. More precisely, various methods are used to determine the first eigenvalue for the $ \alpha $-Laplacian operator on the closed oriented pseudo-slant submanifolds in a generalized Sasakian space form. From our findings for the Laplacian, we extend many Reilly-like inequalities to the $ \alpha $-Laplacian on pseudo slant submanifold in a unit sphere.</abstract>

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DZIUK, GERHARD. "CONVERGENCE OF A SEMI-DISCRETE SCHEME FOR THE CURVE SHORTENING FLOW." Mathematical Models and Methods in Applied Sciences 04, no. 04 (August 1994): 589–606. http://dx.doi.org/10.1142/s0218202594000339.

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Convergence for a spatial discretization of the curvature flow for curves in possibly higher codimension is proved in L∞((0, T), L2(ℝ/2π)) ∩ L2((0, T) H1(ℝ/2π)). Asymptotic convergence in these norms is achieved for the position vector and its time derivative which is proportional to curvature. The underlying algorithm rests on a formulation of mean curvature flow which uses the Laplace-Beltrami operator and leads to tridiagonal linear systems which can be easily solved.

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Wang, Jianxia, and Zhan Zhou. "Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator." Mathematics 8, no. 3 (March 9, 2020): 381. http://dx.doi.org/10.3390/math8030381.

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In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.

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Dosla, Zuzana, Serena Matucci, and Pavel Řehák. "Decaying positive global solutions of second order difference equations with mean curvature operator." Electronic Journal of Qualitative Theory of Differential Equations, no. 72 (2020): 1–16. http://dx.doi.org/10.14232/ejqtde.2020.1.72.

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Wang, Zhenguo, and Qiuying Li. "Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator." Open Mathematics 20, no. 1 (January 1, 2022): 1195–202. http://dx.doi.org/10.1515/math-2022-0509.

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Abstract In this article, by using critical point theory, we prove the existence of multiple T T -periodic solutions for difference equations with the mean curvature operator: − Δ ( ϕ c ( Δ u ( t − 1 ) ) ) + q ( t ) u ( t ) = λ f ( t , u ( t ) ) , t ∈ Z , -\Delta ({\phi }_{c}\left(\Delta u\left(t-1)))+q\left(t)u\left(t)=\lambda f\left(t,u\left(t)),\hspace{1em}t\in {\mathbb{Z}}, where Z {\mathbb{Z}} is the set of integers. As a T T -periodic problem, it does not require the nonlinear term is unbounded or bounded, and thus, our results are supplements to some well-known periodic problems. Finally, we give one example to illustrate our main results.

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Wang, Shaohong, and Zhan Zhou. "Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator." Mathematics 9, no. 14 (July 19, 2021): 1691. http://dx.doi.org/10.3390/math9141691.

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Abstract:

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

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Dai, Guowei. "Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain." Indiana University Mathematics Journal 67, no. 6 (2018): 2103–21. http://dx.doi.org/10.1512/iumj.2018.67.7546.

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Ma, Ruyun, Tianlan Chen, and Hongliang Gao. "On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator." Electronic Journal of Qualitative Theory of Differential Equations, no. 98 (2016): 1–10. http://dx.doi.org/10.14232/ejqtde.2016.1.98.

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Dosla, Zuzana, Serena Matucci, and Pavel Řehák. "Decaying positive global solutions of second order difference equations with mean curvature operator." Electronic Journal of Qualitative Theory of Differential Equations, no. 72 (2020): 1–16. http://dx.doi.org/10.14232/ejqtde.2020.1.72.

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Azzollini, A. "Ground state solution for a problem with mean curvature operator in Minkowski space." Journal of Functional Analysis 266, no. 4 (February 2014): 2086–95. http://dx.doi.org/10.1016/j.jfa.2013.10.002.

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Ma, Ruyun. "Positive solutions for Dirichlet problems involving the mean curvature operator in Minkowski space." Monatshefte für Mathematik 187, no. 2 (November 2, 2017): 315–25. http://dx.doi.org/10.1007/s00605-017-1133-z.

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Ma, Ruyun, Man Xu, and Zhiqian He. "Nonconstant positive radial solutions for Neumann problem involving the mean extrinsic curvature operator." Journal of Mathematical Analysis and Applications 484, no. 2 (April 2020): 123728. http://dx.doi.org/10.1016/j.jmaa.2019.123728.

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Pašić, Mervan. "New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/735360.

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We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations withϕ-Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensionalp-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria.

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Journal articles on the topic 'Mean curvature operator'

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