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Статті в журналах з теми "Minkowski mean curvature operator"

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Автор: Grafiati

Опубліковано: 11 березня 2023

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1

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

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Анотація:

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

He, Zhiqian, and Liangying Miao. "Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space." AIMS Mathematics 6, no. 6 (2021): 6171–79. http://dx.doi.org/10.3934/math.2021362.

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10

Gurban, Daniela, Petru Jebelean, and Cǎlin Şerban. "Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space." Discrete & Continuous Dynamical Systems - A 40, no. 1 (2020): 133–51. http://dx.doi.org/10.3934/dcds.2020006.

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11

Bereanu, Cristian, Petru Jebelean, and Cǎlin-Constantin Şerban. "The Dirichlet problem for discontinuous perturbations of the mean curvature operator in Minkowski space." Electronic Journal of Qualitative Theory of Differential Equations, no. 35 (2015): 1–7. http://dx.doi.org/10.14232/ejqtde.2015.1.35.

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12

Shen, Wen-guo. "Unilateral global interval bifurcation for problem with mean curvature operator in Minkowski space and its applications." Applied Mathematics-A Journal of Chinese Universities 37, no. 2 (June 2022): 159–76. http://dx.doi.org/10.1007/s11766-022-3580-0.

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13

Chen, Tianlan, and Lei Duan. "Ambrosetti–Prodi type results for a Neumann problem with a mean curvature operator in Minkowski spaces." Rocky Mountain Journal of Mathematics 50, no. 5 (October 2020): 1627–35. http://dx.doi.org/10.1216/rmj.2020.50.1627.

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14

Bereanu, Cristian, Petru Jebelean, and Pedro J. Torres. "Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space." Journal of Functional Analysis 265, no. 4 (August 2013): 644–59. http://dx.doi.org/10.1016/j.jfa.2013.04.006.

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15

Liang, Zaitao, and Yanjuan Yang. "Radial Convex Solutions of a Singular Dirichlet Problem with the Mean Curvature Operator in Minkowski Space." Acta Mathematica Scientia 39, no. 2 (March 2019): 395–402. http://dx.doi.org/10.1007/s10473-019-0205-7.

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16

ALÍAS, LUIS J., and A. GERVASIO COLARES. "Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 3 (November 2007): 703–29. http://dx.doi.org/10.1017/s0305004107000576.

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Анотація:

AbstractIn this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

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17

Ma, Ruyun, and Man Xu. "Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space." Discrete & Continuous Dynamical Systems - B 24, no. 6 (2019): 2701–18. http://dx.doi.org/10.3934/dcdsb.2018271.

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18

Ma, Ruyun, and Man Xu. "Positive rotationally symmetric solutions for a Dirichlet problem involving the higher mean curvature operator in Minkowski space." Journal of Mathematical Analysis and Applications 460, no. 1 (April 2018): 33–46. http://dx.doi.org/10.1016/j.jmaa.2017.11.049.

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19

Bereanu, Cristian, Petru Jebelean, and Pedro J. Torres. "Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space." Journal of Functional Analysis 264, no. 1 (January 2013): 270–87. http://dx.doi.org/10.1016/j.jfa.2012.10.010.

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20

Shen, Wenguo. "Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin." Journal of Function Spaces 2020 (April 13, 2020): 1–11. http://dx.doi.org/10.1155/2020/9801931.

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Анотація:

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x<R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

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21

Xu, Man, and Ruyun Ma. "Existence of infinitely many radial nodal solutions for a Dirichlet problem involving mean curvature operator in Minkowski space." Electronic Journal of Qualitative Theory of Differential Equations, no. 27 (2020): 1–14. http://dx.doi.org/10.14232/ejqtde.2020.1.27.

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22

Bereanu, Cristian, Petru Jebelean, and Jean Mawhin. "Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces." Mathematische Nachrichten 283, no. 3 (February 26, 2010): 379–91. http://dx.doi.org/10.1002/mana.200910083.

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23

Gurban, Daniela, and Petru Jebelean. "Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities." Journal of Differential Equations 266, no. 9 (April 2019): 5377–96. http://dx.doi.org/10.1016/j.jde.2018.10.030.

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24

Bereanu, C., P. Jebelean, and J. Mawhin. "Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces." Proceedings of the American Mathematical Society 137, no. 01 (July 1, 2008): 161–69. http://dx.doi.org/10.1090/s0002-9939-08-09612-3.

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25

Gomes, D., and E. Capelas De Oliveira. "The second-order Klein-Gordon field equation." International Journal of Mathematics and Mathematical Sciences 2004, no. 69 (2004): 3775–81. http://dx.doi.org/10.1155/s0161171204406565.

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Анотація:

We introduce and discuss the generalized Klein-Gordon second-order partial differential equation in the Robertson-Walker space-time, using the Casimir second-order invariant operator written in hyperspherical coordinates. The de Sitter and anti-de Sitter space-times are recovered by means of a convenient choice of the parameter associated to the space-time curvature. As an application, we discuss a few properties of the solutions. We also discuss the case where we have positive frequency exponentials and the creation and annihilation operators of particles with known quantum numbers. Finally, we recover the Minkowskian case, that is, the case of null curvature.

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26

Azak, Ayşe Zeynep, Murat Tosun, and Melek Masal. "Null parallel p-equidistant b-scrolls." Boletim da Sociedade Paranaense de Matemática 32, no. 2 (September 11, 2014): 23. http://dx.doi.org/10.5269/bspm.v32i2.20119.

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In this paper, null parallel p-equidistant B-scrolls are defined in 3-dimensional Minkowski space R_1^3 . We prove necessary and sufficient conditions for these B-scrolls to be equivalent of their Cartan frames. The relations between matrices of the shape operators and the algebraic invariants (Gauss, mean curvatures, principal curvatures) of these B-scrolls are shown. Besides we give the relations between second Gauss curvatures, mean curvatures and the distribution parameters of non-developable null parallel p-equidistant B-scrolls. Finally, an example is given related to the null parallel p-equidistant B-scrolls in R_1^3.

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27

Wang, Zenggui. "Hyperbolic mean curvature flow in Minkowski space." Nonlinear Analysis: Theory, Methods & Applications 94 (January 2014): 259–71. http://dx.doi.org/10.1016/j.na.2013.05.017.

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28

Zeng, Fanqi, Qun He, and Bin Chen. "The mean curvature flow in Minkowski spaces." Science China Mathematics 61, no. 10 (August 17, 2018): 1833–50. http://dx.doi.org/10.1007/s11425-017-9376-6.

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29

Kuruoğlu, Nuri, and Melek Masal. "Timelike parallel p_i-equidistant ruled surfaces by a timelike base curve in the Minkowski 3-space R^3_1." Acta et Commentationes Universitatis Tartuensis de Mathematica 11 (December 31, 2007): 3–11. http://dx.doi.org/10.12697/acutm.2007.11.01.

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Анотація:

Timelike parallel pi-equidistant ruled surfaces are introduced and relations about polar planes, natural curvatures and natural torsions are given. In addition, relations between distribution parameters, shape operators, Gaussian and mean curvatures of these ruled surfaces are obtained. After all, an example related to the parallel timelike p2-equidistant ruled surfaces is given.

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30

Ganchev, Georgi, and Velichka Milousheva. "Surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space." Filomat 33, no. 4 (2019): 1135–45. http://dx.doi.org/10.2298/fil1904135g.

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We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

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31

Brendle, Simon. "Hypersurfaces in Minkowski space with vanishing mean curvature." Communications on Pure and Applied Mathematics 55, no. 10 (July 17, 2002): 1249–79. http://dx.doi.org/10.1002/cpa.10044.

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32

Sarkar, Prakash. "Quantifying the Cosmic Web using the Shapefinder diagonistic." Proceedings of the International Astronomical Union 11, S308 (June 2014): 250–53. http://dx.doi.org/10.1017/s1743921316009960.

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Анотація:

AbstractOne of the most successful method in quantifying the structures in the Cosmic Web is the Minkowski Functionals. In 3D, there are four minkowski Functionals: Area, Volume, Integrated Mean Curvature and the Integrated Gaussian Curvature. For defining the Minkowski Functionals one should define a surface. We have developed a method based on Marching cube 33 algorithm to generate a surface from a discrete data sets. Next we calculate the Minkowski Functionals and Shapefinder from the triangulated polyhedral surface. Applying this methodology to different data sets , we obtain interesting results related to geometry, morphology and topology of the large scale structure

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33

Wu, B. Y. "Some results on Finsler submanifolds." International Journal of Mathematics 27, no. 03 (March 2016): 1650021. http://dx.doi.org/10.1142/s0129167x1650021x.

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In this paper we study the submanifold theory in terms of Chern connection. We introduce the notions of the second fundamental form and mean curvature for Finsler submanifolds, and establish the fundamental equations by means of moving frame for the hypersurface case. We provide the estimation of image radius for compact submanifold, and prove that there exists no compact minimal submanifold in any complete noncompact and simply connected Finsler manifold with nonpositive flag curvature. We also characterize the Minkowski hyperplanes, Minkowski hyperspheres and Minkowski cylinders as the only hypersurfaces in Minkowski space with parallel second fundamental form.

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34

Yıldız, Önder Gökmen, Selman Hızal, and Mahmut Akyiğit. "Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 3 (December 1, 2018): 99–108. http://dx.doi.org/10.2478/auom-2018-0035.

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AbstractIn this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ${\Bbb R}_1^3$with a positive density function. We get a result for minimal case. Also, we give examples of a helicoidal surface with weighted mean curvature and Gaussian curvature.

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35

KOSSOWSKI, MAREK. "RESTRICTIONS ON ZERO MEAN CURVATURE SURFACES IN MINKOWSKI SPACE." Quarterly Journal of Mathematics 42, no. 1 (1991): 315–24. http://dx.doi.org/10.1093/qmath/42.1.315.

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36

yu Wang, Yo, Ya eng Wang, and Jing Liu. "Lyapunov-type inequalities for differential equation involving one-dimensional Minkowski-curvature operator." Journal of Mathematical Inequalities, no. 2 (2021): 591–603. http://dx.doi.org/10.7153/jmi-2021-15-43.

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37

Azzollini, Antonio. "Ground state solutions for the Hénon prescribed mean curvature equation." Advances in Nonlinear Analysis 8, no. 1 (June 14, 2018): 1227–34. http://dx.doi.org/10.1515/anona-2017-0233.

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Анотація:

Abstract In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in {{\mathbb{R}^{N}}} , both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space.

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38

Zhang, Xuemei, and Meiqiang Feng. "Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space." Communications in Contemporary Mathematics 21, no. 03 (May 2019): 1850003. http://dx.doi.org/10.1142/s0219199718500037.

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Анотація:

In this paper, bifurcation diagrams and exact multiplicity of positive solution are obtained for the one-dimensional prescribed mean curvature equation in Minkowski space in the form of [Formula: see text] where [Formula: see text] is a bifurcation parameter, [Formula: see text], the radius of the one-dimensional ball [Formula: see text], is an evolution parameter. Moreover, we make a comparison between the bifurcation diagram of one-dimensional prescribed mean curvature equation in Euclid space and Minkowski space. Our methods are based on a detailed analysis of time maps.

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39

Palmas, Oscar, Francisco J. Palomo, and Alfonso Romero. "On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 1 (October 20, 2017): 199–210. http://dx.doi.org/10.1017/s0308210517000063.

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Анотація:

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

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40

Svane, Anne Marie. "ESTIMATION OF MINKOWSKI TENSORS FROM DIGITAL GREY-SCALE IMAGES." Image Analysis & Stereology 33, no. 2 (August 26, 2014): 51. http://dx.doi.org/10.5566/ias.1124.

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Анотація:

It was shown in Svane (2014b) that local algorithms based on grey-scale images sometimes lead to asymptotically unbiased estimators for surface area and integrated mean curvature. This paper extends the results to estimators for Minkowski tensors. In particular, asymptotically unbiased local algorithms for estimation of all volume and surface tensors and certain mean curvature tensors are given. This requires an extension of the asymptotic formulas of Svane (2014b) to estimators with position dependent weights.

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41

PYO, JUNCHEOL, and KEOMKYO SEO. "SPACELIKE CAPILLARY SURFACES IN THE LORENTZ–MINKOWSKI SPACE." Bulletin of the Australian Mathematical Society 84, no. 3 (August 9, 2011): 362–71. http://dx.doi.org/10.1017/s0004972711002528.

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AbstractFor a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz–Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe [‘Sufficient conditions for constant mean curvature surfaces to be round’, Math. Ann.323(1) (2002), 143–156]. Using the concept of the rotation index at the interior and boundary umbilical points and applying the Poincaré–Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than four vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilical surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover, we prove that the only immersed spacelike disk type capillary surface inside a de Sitter surface in $\Bbb L^3$ is part of (spacelike) plane or a hyperbolic plane.

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42

INOGUCHI, Jun-ichi. "Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space." Tokyo Journal of Mathematics 21, no. 1 (June 1998): 141–52. http://dx.doi.org/10.3836/tjm/1270041992.

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43

Dursun, Ugur. "ROTATION HYPERSURFACES IN LORENTZ-MINKOWSKI SPACE WITH CONSTANT MEAN CURVATURE." Taiwanese Journal of Mathematics 14, no. 2 (April 2010): 685–705. http://dx.doi.org/10.11650/twjm/1500405814.

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44

Klyachin, V. A. "Zero mean curvature surfaces of mixed type in Minkowski space." Izvestiya: Mathematics 67, no. 2 (April 30, 2003): 209–24. http://dx.doi.org/10.1070/im2003v067n02abeh000425.

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Azzollini, A. "On a prescribed mean curvature equation in Lorentz–Minkowski space." Journal de Mathématiques Pures et Appliquées 106, no. 6 (December 2016): 1122–40. http://dx.doi.org/10.1016/j.matpur.2016.04.003.

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Ganchev, Georgi, and Velichka Milousheva. "Timelike surfaces with zero mean curvature in Minkowski 4-space." Israel Journal of Mathematics 196, no. 1 (August 2013): 413–33. http://dx.doi.org/10.1007/s11856-012-0169-y.

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Xia, Chao. "Inverse anisotropic mean curvature flow and a Minkowski type inequality." Advances in Mathematics 315 (July 2017): 102–29. http://dx.doi.org/10.1016/j.aim.2017.05.020.

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Umeda, Yuhei. "Constant-Mean-Curvature Surfaces with Singularities in Minkowski 3-Space." Experimental Mathematics 18, no. 3 (January 2009): 311–23. http://dx.doi.org/10.1080/10586458.2009.10129050.

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Aarons, Mark A. S. "Mean curvature flow with a forcing term in minkowski space." Calculus of Variations and Partial Differential Equations 25, no. 2 (October 28, 2005): 205–46. http://dx.doi.org/10.1007/s00526-005-0351-8.

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López, Rafael. "The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space." Mathematics 7, no. 12 (December 9, 2019): 1211. http://dx.doi.org/10.3390/math7121211.

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Анотація:

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

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