The<i>n</i>-dimensional analogue of the catenary: existence and nonexistence

Dierkes, Ulrich; Huisken, Gerhard

doi:10.2140/pjm.1990.141.47

Cited by 38 publications

(38 citation statements)

References 10 publications

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“…Then the leading coefficient comes from f 8 . Thus, k = 8m and this coefficient is B 8m = a 8 m g 2 g 6 . As a m = 0, gg = 0 on J.…”

Section: 6)mentioning

confidence: 99%

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Separation of variables in equations of mean-curvature type

López¹

2016

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Then the leading coefficient comes from f 8 . Thus, k = 8m and this coefficient is B 8m = a 8 m g 2 g 6 . As a m = 0, gg = 0 on J.…”

Section: 6)mentioning

confidence: 99%

“…The two-dimensional catenary equation: the equation of the shape of a surface of constant mass density in equilibrium which is suspended from a curve and hangs under its own weight is 2H(x, y, z) = N 3 /z, where N 3 is the z-coordinate of the unit normal vector field N of the surface [3,6]. Then the function u = u(x, y) satisfies div ∇u…”

Section: Introductionmentioning

confidence: 99%

Separation of variables in equations of mean-curvature type

López¹

2016

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Furthermore, in the scalar case, we will investigate nonexistence of positive entire solutions, radial or not, of the elliptic inequality div g(u)A |Du| Du − g (u)A |Du| f |x|, u , x ∈ R n . (1.2) For possible geometrical and physical models related to (1.2) we refer to [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

Entire radial solutions of elliptic systems and inequalities of the mean curvature type

Filippucci

2007

Journal of Mathematical Analysis and Applications

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In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du) f (u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].

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“…In the case that u ∈ C 1 ( ), the surface energy term E = e −u 1 + |Du| 2 d x may be simplified by writing v = e −u , so E = v 2 + |Dv| 2 d x, which bears a close resemblance to the integral v + |Dv| 2 /4 d x investigated in [Bemelmans and Dierkes 1987]; see also [Dierkes and Huisken 1990].…”

Section: The Energy In the Isothermal Casementioning

confidence: 96%