Journal articles: 'Minimal surface equation'

1

Fouladgar, K., and Leon Simon. "The symmetric minimal surface equation." Indiana University Mathematics Journal 69, no. 1 (2020): 331–66. http://dx.doi.org/10.1512/iumj.2020.69.8412.

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2

SHAVOKHINA, N. S. "FEDOROV UNIVERSAL EQUATIONS IN THE STRING AND MEMBRANE THEORIES." International Journal of Modern Physics B 04, no. 01 (January 1990): 93–111. http://dx.doi.org/10.1142/s0217979290000048.

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It is shown that the equation of minimal hypersurface in the Euclidean (or pseudo-Euclidean) space can be written as the universal Fedorov matrix equation with first-order partial derivatives. Time-like minimal surfaces in the pseudo-Euclidean Minkowski space describe the free motion of relativistic strings and membranes, whereas space-like minimal surfaces describe the potential in the nonlinear Born electrostatic. All of them are imaginary images of minimal surface of the Euclidean space. Spherically symmetric surfaces are found to be all the three types, the hypercatenoid of any dimensionality and its imaginary images. The Fedorov equations provide rich information on the minimal surfaces.

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3

Simon, Leon. "Entire solutions of the minimal surface equation." Journal of Differential Geometry 30, no. 3 (1989): 643–88. http://dx.doi.org/10.4310/jdg/1214443827.

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4

ALÍAS, LUIS J., and BENNETT PALMER. "A duality result between the minimal surface equation and the maximal surface equation." Anais da Academia Brasileira de Ciências 73, no. 2 (June 2001): 161–64. http://dx.doi.org/10.1590/s0001-37652001000200002.

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In this note we show how classical Bernstein's theorem on minimal surfaces in the Euclidean space can be seen as a consequence of Calabi-Bernstein's theorem on maximal surfaces in the Lorentz-Minkowski space (and viceversa). This follows from a simple but nice duality between solutions to their corresponding differential equations.

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Bellettini, Giovanni, Matteo Novaga, and Giandomenico Orlandi. "Eventual regularity for the parabolic minimal surface equation." Discrete and Continuous Dynamical Systems 35, no. 12 (May 2015): 5711–23. http://dx.doi.org/10.3934/dcds.2015.35.5711.

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6

Grundland, Alfred, and Alexander Hariton. "Algebraic Aspects of the Supersymmetric Minimal Surface Equation." Symmetry 9, no. 12 (December 18, 2017): 318. http://dx.doi.org/10.3390/sym9120318.

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7

Hwang, Jenn-Fang. "Phragmen-Lindelof Theorem for the Minimal Surface Equation." Proceedings of the American Mathematical Society 104, no. 3 (November 1988): 825. http://dx.doi.org/10.2307/2046800.

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8

Dierkes, Ulrich, and Nico Groh. "Symmetric solutions of the singular minimal surface equation." Annals of Global Analysis and Geometry 60, no. 2 (June 21, 2021): 431–53. http://dx.doi.org/10.1007/s10455-021-09785-2.

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AbstractWe classify all rotational symmetric solutions of the singular minimal surface equation in both cases $$\alpha <0$$ α < 0 and $$\alpha >0$$ α > 0 . In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.

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9

Ziemer, William. "The nonhomogeneous minimal surface equation involving a measure." Pacific Journal of Mathematics 167, no. 1 (January 1, 1995): 183–200. http://dx.doi.org/10.2140/pjm.1995.167.183.

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10

Hwang, Jenn-Fang. "A uniqueness theorem for the minimal surface equation." Pacific Journal of Mathematics 176, no. 2 (December 1, 1996): 357–64. http://dx.doi.org/10.2140/pjm.1996.176.357.

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11

Hwang, Jenn-Fang. "Catenoid-like solutions for the minimal surface equation." Pacific Journal of Mathematics 183, no. 1 (March 1, 1998): 91–102. http://dx.doi.org/10.2140/pjm.1998.183.91.

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12

Hwang, Jenn-Fang. "Phragmén-Lindelöf theorem for the minimal surface equation." Proceedings of the American Mathematical Society 104, no. 3 (March 1, 1988): 825. http://dx.doi.org/10.1090/s0002-9939-1988-0964864-x.

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13

Dierkes, Ulrich. "On solutions of the singular minimal surface equation." Annali di Matematica Pura ed Applicata (1923 -) 198, no. 2 (August 21, 2018): 505–16. http://dx.doi.org/10.1007/s10231-018-0779-z.

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14

Nurminen, Janne. "An inverse problem for the minimal surface equation." Nonlinear Analysis 227 (February 2023): 113163. http://dx.doi.org/10.1016/j.na.2022.113163.

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15

Ahmad, Daud, M. Khalid Mahmood, Qin Xin, Ferdous M. O. Tawfiq, Sadia Bashir, and Arsha Khalid. "A Computational Model for q -Bernstein Quasi-Minimal Bézier Surface." Journal of Mathematics 2022 (September 20, 2022): 1–21. http://dx.doi.org/10.1155/2022/8994112.

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A computational model is presented to find the q -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on q -Bernstein–Bézier surfaces leads the way to the new generalizations of q -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The q -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the q -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q -Bernstein polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q -Bernstein–Bézier minimal surface.

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16

Lipkovski, Jana, and Aleksandar Lipkovski. "Form-finding software and minimal surface equation: A comparative approach." Filomat 29, no. 10 (2015): 2447–55. http://dx.doi.org/10.2298/fil1510447l.

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The shape of membrane and cable-net structures is usually modeled by geometry of minimal surfaces. Using central finite differences method, a nonlinear iterative process for finding the minimal surface with given fixed boundary conditions is developed and implemented in Mathematica?. Having in mind form-finding of membrane structures, the results are compared with the results obtained by commercial package EASY?, made by Technet GmbH, Germany.

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17

Reznikov, A. G. "Linearization and explicit solutions of the minimal surface equation." Publicacions Matemàtiques 36 (January 1, 1992): 39–46. http://dx.doi.org/10.5565/publmat_36192_03.

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18

Brendle, Simon. "On the Lagrangian minimal surface equation and related problems." Surveys in Differential Geometry 18, no. 1 (2013): 1–18. http://dx.doi.org/10.4310/sdg.2013.v18.n1.a1.

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19

Neu, John C. "Kinks and the minimal surface equation in Minkowski space." Physica D: Nonlinear Phenomena 43, no. 2-3 (July 1990): 421–34. http://dx.doi.org/10.1016/0167-2789(90)90145-f.

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20

Pokrovskii, A. V. "Removable Singularities of Solutions of the Minimal Surface Equation." Functional Analysis and Its Applications 39, no. 4 (October 2005): 296–300. http://dx.doi.org/10.1007/s10688-005-0050-4.

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21

Griffin, Sarah Field. "Minimal Surfaces: A Derivation of the Minimal Surface Equation for an Arbitrary ${\mathbb{C}}^2$ Coordinate Chart." Missouri Journal of Mathematical Sciences 13, no. 3 (October 2001): 145–53. http://dx.doi.org/10.35834/2001/1303145.

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22

Güler, Erhan. "Family of Enneper Minimal Surfaces." Mathematics 6, no. 12 (November 26, 2018): 281. http://dx.doi.org/10.3390/math6120281.

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We consider a family of higher degree Enneper minimal surface E m for positive integers m in the three-dimensional Euclidean space E 3 . We compute algebraic equation, degree and integral free representation of Enneper minimal surface for m = 1 , 2 , 3 . Finally, we give some results and relations for the family E m .

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23

Hwang, Jenn-Fang. "Growth property for the minimal surface equation in unbounded domains." Proceedings of the American Mathematical Society 121, no. 4 (April 1, 1994): 1027. http://dx.doi.org/10.1090/s0002-9939-1994-1204379-x.

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24

Spadaro, Emanuele, and Ulisse Stefanelli. "A variational view at the time-dependent minimal surface equation." Journal of Evolution Equations 11, no. 4 (May 31, 2011): 793–809. http://dx.doi.org/10.1007/s00028-011-0111-5.

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25

Li, Yibao, and Shimin Guo. "Triply periodic minimal surface using a modified Allen–Cahn equation." Applied Mathematics and Computation 295 (February 2017): 84–94. http://dx.doi.org/10.1016/j.amc.2016.10.005.

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26

Kurmanbayev, D., and K. Yesmakhanova. "SOLITON DEFORMATION OF INVERTED CATENOID." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (April 15, 2021): 24–32. http://dx.doi.org/10.32014/2021.2518-1726.17.

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The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known potentials of the Dirac operator and this algorithm is said to be Moutard transformations [5]. Firstly, the geometric meaning of these transformations which found in [6], [7], gives us the definition of the inversion of the minimal surface, further after finding the exact solutions of the mVN equation, we can represent the inverted surfaces. And these representations of the new potential determine the soliton deformation [8], [9]. In 2014, blowing-up solutions to the mVN equation were obtained using a rigid translation of the initial Enneper surface in [6]. Further results were obtained for the second-order Enneper surface [10]. Now the soliton deformation of an inverted catenoid is found by smooth translation along the second coordinate axis. In this paper, in order to determine catenoid inversions, it is proposed to find holomorphic objects as Gauss maps and height differential [11]; the soliton deformation of the inverted catenoid is obtained; particular solution of modified Karteweg-de Vries (KdV) equation is found that give some representation of KdV surface [12],[13].

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27

Dorfmeister, Josef F., Walter Freyn, Shimpei Kobayashi, and Erxiao Wang. "Survey on real forms of the complex A2(2)-Toda equation and surface theory." Complex Manifolds 6, no. 1 (January 1, 2019): 194–227. http://dx.doi.org/10.1515/coma-2019-0011.

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AbstractThe classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.

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28

Trukhlyaeva, Irina. "On Convergence of Polynomial Approximate Solutions of Minimal Surface Equations in Domains Satisfying the Cone Condition." Mathematical Physics and Computer Simulation, no. 4 (February 2021): 5–12. http://dx.doi.org/10.15688/mpcm.jvolsu.2020.4.1.

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In this paper we consider the polynomial approximate solutions of the minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation. In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions.The study of this issue is especially important for nonlinear equations since in this case there is a series of difficulties related with the impossibility of employing traditional methods and approaches used for linear equations. At present, a quite topical problem is to determine the conditions ensuring the uniform convergence of approximate solutions obtained by various methods for nonlinear equations and system of equations of variational kind (see, for instance, [1]). For nonlinear equations it is first necessary to establish some a priori estimates of the derivatives of approximate solutions.In this paper, we gave a substantiation of the variational method of solving the minimal surface equation in the case of multidimensional space.We use the same approach that we used in [3] for a two-dimensional equation. Note that such a convergence was established in [3] under the condition that a certain geometric characteristic Δ(Ω) in the domain Ω, in which the solutions are considered, is positive. In particular, domain with a smooth boundary satisfied this requirement. However, this characteristic is equal to zero for a fairly wide class of domains with piecewise-smooth boundaries and sufficiently "narrow" sections at the boundary. For example, such a section of the boundary is the vertex of a cone with an angle less than π/2. In this paper, we present another approach to determining the value of Δ(Ω) in terms of which it is possible to extend the results of the work [3] in domain satisfying cone condition.

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29

HE, QUN, and WEI YANG. "VOLUME FORMS AND MINIMAL SURFACES OF ROTATION IN FINSLER SPACES WITH (α, β)-METRICS." International Journal of Mathematics 21, no. 11 (November 2010): 1401–11. http://dx.doi.org/10.1142/s0129167x10006483.

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In this paper, the relation between the volume form of (α, β)-metric [Formula: see text] and that of Riemannian metric α is given. Minimal submanifolds in Finsler spaces with (α, β)-metrics are studied. Finally, an ordinary differential equation that characterizes minimal surfaces of revolution and an example of minimal surface of rotation are given.

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30

Čiupaila, Regimantas, Mifodijus Sapagovas, and Olga Štikonienė. "Numerical solution of nonlinear elliptic equation with nonlocal condition." Nonlinear Analysis: Modelling and Control 18, no. 4 (October 25, 2013): 412–26. http://dx.doi.org/10.15388/na.18.4.13970.

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Two iterative methods are considered for the system of difference equations approximating two-dimensional nonlinear elliptic equation with the nonlocal integral condition. Motivation and possible applications of the problem present in the paper coincide with the small volume problems in hydrodynamics. The differential problem considered in the article is some generalization of the boundary value problem for minimal surface equation.

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31

Alsaedi, Ramzi. "Perturbation effects for the minimal surface equation with multiple variable exponents." Discrete & Continuous Dynamical Systems - S 12, no. 2 (2019): 139–50. http://dx.doi.org/10.3934/dcdss.2019010.

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32

Jiang, Guosheng, Zhehui Wang, and Jintian Zhu. "Liouville type theorems for the minimal surface equation in half space." Journal of Differential Equations 305 (December 2021): 270–87. http://dx.doi.org/10.1016/j.jde.2021.10.019.

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33

Wong, Willie Wai Yeung. "Global existence for the minimal surface equation on $\mathbb {R}^{1,1}$." Proceedings of the American Mathematical Society, Series B 4, no. 5 (November 29, 2017): 47–52. http://dx.doi.org/10.1090/bproc/25.

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34

Gatsunaev, Mikhail Andreevich, and Alexey Alexandrovich Klyachin. "On uniform convergence of piecewise-linear solutions to minimal surface equation." Ufimskii Matematicheskii Zhurnal 6, no. 3 (2014): 3–16. http://dx.doi.org/10.13108/2014-6-3-3.

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35

Lewintan, Peter. "The “Wrong Minimal Surface Equation” does not have the Bernstein property." Analysis 31, no. 4 (November 2011): 299–303. http://dx.doi.org/10.1524/anly.2011.1137.

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36

López, Rafael. "The Dirichlet problem for the $$\alpha $$ α -singular minimal surface equation." Archiv der Mathematik 112, no. 2 (October 19, 2018): 213–22. http://dx.doi.org/10.1007/s00013-018-1255-0.

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37

HUFF, ROBERT, and JOHN MCCUAN. "MINIMAL GRAPHS WITH DISCONTINUOUS BOUNDARY VALUES." Journal of the Australian Mathematical Society 86, no. 1 (February 2009): 75–95. http://dx.doi.org/10.1017/s1446788708000335.

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AbstractWe construct global solutions of the minimal surface equation over certain smooth annular domains and over the domain exterior to certain smooth simple closed curves. Each resulting minimal graph has an isolated jump discontinuity on the inner boundary component which, at least in some cases, is shown to have nonvanishing curvature.

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38

DONTEN-BURY, MARIA. "COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES." Glasgow Mathematical Journal 58, no. 2 (July 21, 2015): 325–55. http://dx.doi.org/10.1017/s0017089515000221.

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AbstractWe investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

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39

BUKHTYAK, Mikhail S. "PSEUDO-MINIMALITY AND RULED SURFACES." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 67 (2020): 18–27. http://dx.doi.org/10.17223/19988621/67/2.

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This paper is a follow-up to the author's series of works about shape modeling for an orthotropic elastic material that takes an equilibrium form inside the area with the specified boundaries. V.M. Gryanik and V.I. Loman, based on thin shell equilibrium equations, solved about 30 years ago a similar problem for an isotropic mesh attached to rigid parabolic edges. With a view to extend modeling to orthotropic materials (and other boundary contours), the author in his publications of 2016–2017 proposed an approach to the problem based on the application of surfaces with a constant ratio of principal curvatures. These surfaces are called pseudo-minimal surfaces. A partial differential equation that defines (in the local sense) a class of pseudo-minimal surfaces is very complex for analysis. However, for some classes of surfaces, the analysis is greatly simplified, notably, the analysis can be performed without this inconvenient PDE, but with the method of moving frames. The author is referring to a class of ruled surfaces. This class is interesting not only due to the aforesaid but also due to an evident interest manifested by architects and builders. However, one should discuss not the pseudo-minimal ruled surfaces (they exist but are obviously trivial) but an invariant (principal curvatures ratio), which is not an identical constant on a given surface but its contour lines coincide with the lines of some invariant family. Roughly speaking, there are surfaces whose pseudo-minimal condition is satisfied identically, and surfaces that are pseudo-minimal "in a limited sense"—lengthways the lines of a certain family, internally connected with the surface. The article finds that the role of such a family can be obviously played by "equidistant" lines for the striction line of a skew ruled surface, and rays are the carriers of such a ruled surface, they form a regulus with constant Euclidean invariants.

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40

Li, Hengyan, and Weiping Yan. "Explicit self-similar solutions for a class of zero mean curvature equation and minimal surface equation." Nonlinear Analysis 197 (August 2020): 111814. http://dx.doi.org/10.1016/j.na.2020.111814.

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41

Williams, Graham. "The best modulus of continuity for solutions of the minimal surface equation." Pacific Journal of Mathematics 129, no. 1 (September 1, 1987): 193–208. http://dx.doi.org/10.2140/pjm.1987.129.193.

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42

Williams, Graham H. "Solutions of the minimal surface equation continuous and discontinuous at the boundary." Communications in Partial Differential Equations 11, no. 13 (January 1986): 1439–57. http://dx.doi.org/10.1080/03605308608820469.

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43

Kuwert, Ernst. "On solutions of the exterior Dirichlet problem for the minimal surface equation." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 10, no. 4 (July 1993): 445–51. http://dx.doi.org/10.1016/s0294-1449(16)30211-6.

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44

Rosales, Leobardo. "The geometric structure of solutions to the two-valued minimal surface equation." Calculus of Variations and Partial Differential Equations 39, no. 1-2 (December 23, 2009): 59–84. http://dx.doi.org/10.1007/s00526-009-0301-y.

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45

Maria da Silva, Rosângela, and Keti Tenenblat. "Helicoidal Minimal Surfaces in a Finsler Space of Randers Type." Canadian Mathematical Bulletin 57, no. 4 (December 1, 2014): 765–79. http://dx.doi.org/10.4153/cmb-2013-047-7.

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AbstractWe consider the Finsler space obtained by perturbing the Euclidean metric of ℝ3 by a rotation. It is the open region of ℝ3 bounded by a cylinder with a Randers metric. Using the Busemann–Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in . We prove that the helicoid is a minimal surface in only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space , the only minimal surfaces in the Bonnet family with fixed axis Ox̄3 are the catenoids and the helicoids.

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46

Williams, Graham H. "Global regularity for solutions of the minimal surface equation with continuous boundary values." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 3, no. 6 (November 1986): 411–29. http://dx.doi.org/10.1016/s0294-1449(16)30375-4.

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47

Rosales, Leobardo. "A Hölder estimate for entire solutions to the two-valued minimal surface equation." Proceedings of the American Mathematical Society 144, no. 3 (November 20, 2015): 1209–21. http://dx.doi.org/10.1090/proc/12774.

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48

Chalifour, V., and A. M. Grundland. "General solution of the exceptional Hermite differential equation and its minimal surface representation." Annales Henri Poincaré 21, no. 10 (August 10, 2020): 3341–84. http://dx.doi.org/10.1007/s00023-020-00945-x.

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49

López, Rafael. "Uniqueness of critical points and maximum principles of the singular minimal surface equation." Journal of Differential Equations 266, no. 7 (March 2019): 3927–41. http://dx.doi.org/10.1016/j.jde.2018.09.024.

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50

Carvalho, T. M. M., H. N. Moreira, and K. Tenenblat. "Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space." Canadian Journal of Mathematics 64, no. 1 (February 1, 2012): 44–80. http://dx.doi.org/10.4153/cjm-2011-047-4.

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AbstractWe consider the Randers space (Vn, Fb) obtained by perturbing the Euclidean metric by a translation, Fb = α + β, where α is the Euclidean metric and β is a 1-form with norm b, 0 ≤ b < 1. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces (V3, Fb) of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when b = 0. It also reduces to the equation that characterizes the minimal rotational surfaces in (V3, Fb) when H = 0, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for . Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical systemand by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

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Journal articles on the topic 'Minimal surface equation'

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