Remark on the anisotropic prescribed mean curvature equation on arbitrary domains

Marquardt, Thomas

doi:10.1007/s00209-009-0476-0

Cited by 19 publications

(23 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will be concerned with strictly spacelike solutions of (1.1), that is, weak, or strong, solutions u of (1.1) satisfying ∇u ∞ < 1; a non-exhaustive list of references about this problem includes [10,30,3,2,22,26,28] These equations may also arise as Euler-Lagrange equations of some weighted area functionals (cf. [26,29,8,9,27,13]), such as…”

Section: Introductionmentioning

confidence: 99%

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Corsato

Obersnel

Omari

2017

Georgian Mathematical Journal

View full text Add to dashboard Cite

We 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.

show abstract

Section: Introductionmentioning

confidence: 99%

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Corsato

Obersnel

Omari

2017

Georgian Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…Firstly, we suppose ϕ ∈ C 2,γ (Ω 0 ) and prove the Dirichlet problem (1.1)-(1.2) has a solution u ∈ C 2,γ (Ω 0 ). This was proved in [14] for the case of α 1. Next, we assume α ∈ (0, 1).…”

Section: Lemma 22mentioning

confidence: 69%

“…In Section 2, we prove the existence of Dirichlet problem (1.1)-(1.2) with α > 0 on bounded domains, extending the main result in [14] for the case of α 1. Note that when 0 < α < 1, the hypothesis (sc) of the corresponding theorem in [14] is not satisfied and the techniques in [14] cannot be applied directly. In Section 3, we construct a family of auxiliary functions which will be used as super-solutions.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

See 1 more Smart Citation

Existence of translating solutions to the flow by powers of mean curvature on unbounded domains

Jian¹,

Ju²

2011

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper, we prove the existence of classical solutions to the Dirichlet problem of a class of quasi-linear elliptic equations on an unbounded cone and a U-type domain in R n (n 2). This problem comes from the study of mean curvature flow or its generalization, the flow by powers of mean curvature. Our approach is a modified version of the classical Perron method, where the solutions to the minimal surface equation are used as sub-solutions and a family auxiliary functions are constructed as super-solutions.

show abstract

“…, x n ) of prescribed mean curvature H ∈ C 1 (S n ), then u satisfies (1.2). The next result follows from [Mar,Corollary 1], and gives general conditions for the existence of H-graphs in R n+1 .…”

Section: )mentioning

confidence: 94%

Rotational hypersurfaces of prescribed mean curvature

Bueno

Gálvez

Mira

2020

Journal of Differential Equations

View full text Add to dashboard Cite

We use a phase space analysis to give some classification results for rotational hypersurfaces in R n+1 whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in S n , we show that a Delaunay-type classification holds for this class of hypersurfaces. We also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed (non-constant) mean curvature is much richer than in the constant mean curvature case.Mathematics Subject Classification: 53A10, 53C42, 34C05, 34C40 arXiv:1902.09405v1 [math.DG]

show abstract

Remark on the anisotropic prescribed mean curvature equation on arbitrary domains

Cited by 19 publications

References 7 publications

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Existence of translating solutions to the flow by powers of mean curvature on unbounded domains

Rotational hypersurfaces of prescribed mean curvature

Contact Info

Product

Resources

About