Mean Curvature Equation Research Articles

The Dirichlet problem for the Lagrangian mean curvature equation

The Dirichlet problem for the Lagrangian mean curvature equation

Local bifurcation structure and stability of the mean curvature equation in the static spacetime

We consider the curvature equation in the static spacetime, $$ \text{div} \Big(\frac{f(x)\nabla u}{\sqrt{1-f^2(x)| \nabla u|^2}}\Big) +\frac{\nabla u \nabla f(x)}{\sqrt{1-f^2(x)| \nabla u|^2}}=\lambda NH \quad\text{in }\Omega, $$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N \geq 1\); the function \(H\) gives the mean curvature. We investigate the local bifurcation structure and stability of the solutions to this equation. For more information see https://ejde.math.txstate.edu/Volumes/2024/48/abstr.html

The prescribed mean curvature equation for t-graphs in the sub-Finsler Heisenberg group [formula omitted]

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K0⊆R2n, for t-graphs on a bounded domain Ω in the Heisenberg group Hn. When the prescribed datum H is constant and strictly smaller than the Finsler mean curvature of ∂Ω, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme.

Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $$\mathbb {R}^2$$

Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $$\mathbb {R}^2$$

Existence and nonexistence of solutions for the mean curvature equation with weights

In this paper we study existence and nonexistence of positive radial solutions of a Dirichlet problem for the prescribed mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem under consideration appears rather delicate, it requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity. In addition, sufficient conditions for global solutions to be oscillatory are given.

The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

In this paper we study existence and uniqueness of solutions to Dirichlet problems as g(u)-divDu1+|Du|2=finΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} g(u) \\displaystyle -{\ ext {div}}\\left( \\frac{D u}{\\sqrt{1+|D u|^2}}\\right) = f &{} \ ext {in}\\;\\Omega ,\\\\ u=0 &{} \ ext {on}\\;\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is an open bounded subset of RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\mathbb {R}}\\,}}^N$$\\end{document} (N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}) with Lipschitz boundary, g:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g:\\mathbb {R}\\rightarrow \\mathbb {R}$$\\end{document} is a continuous function and f belongs to some Lebesgue space. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get solutions for data f merely belonging to L1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\Omega )$$\\end{document} and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in LN(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{N}(\\Omega )$$\\end{document} as well as uniqueness if g is increasing.

The Dirichlet problem of translating mean curvature equations

In this paper, we study the Dirichlet problem of translating mean curvature equations over a domain via topological restrictions. Its main difficulty is the non-existence of a C^{0} a priori estimate for their classical solutions except few cases. Inspired by the work of Miranda and Giusti, we define a generalized solution of these Dirichlet problems and establish its general existence. We propose a non-closed-minimal (NCM) condition on the underlying domain. When the domain is mean convex and NCM, the generalized solution with continuous boundary data is the classical smooth solution. Moreover, the NCM assumption cannot be removed by a hemisphere example.

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

An iteration method for solving elliptic equations

In this paper, we use a natural iteration technique to prove the existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to unbounded domains, as we show with domains of the form Rn−1×(−d/2,d/2).

Solutions of the mean curvature equation with the Nehari manifold

Solutions of the mean curvature equation with the Nehari manifold

Regularity for convex viscosity solutions of Lagrangian mean curvature equation

Abstract We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Hölder continuous second derivatives.

Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi–Bernstein-type problems

In this article spacelike hypersurfaces immersed in twisted product spacetimes Itimes _f F with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, and a relation that needs to hold on each immersed spacelike hypersurface in Itimes _f F to be a simple warped product is also derived. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersurfaces and other for totally umbilic ones with a suitable restriction on their mean curvature are presented, and a full description of maximal hypersurfaces in twisted product spacetimes with a one-dimensional Lorentzian fiber is also included. Finally, the mean curvature equation for a spacelike graph on the fiber is computed and as an application, some Calabi–Bernstein-type results are proven.

Multiplicity of Solutions for Some Classes of Prescribed Mean Curvature Equations with Local Conditions

Multiplicity of Solutions for Some Classes of Prescribed Mean Curvature Equations with Local Conditions

Existence of heteroclinic solutions for the prescribed curvature equation

In this paper we use variational methods to establish the existence of heteroclinic solution for the prescribed mean curvature equation of the form−(q′1+(q′)2)′+a(t)V′(q)=0inR, where V is a double-well potential with minima at t=±α and a∈L∞(R) is an even non-negative function with 0<a0:=inft≥M⁡a(t) for some M>0.

Description and Exploration of Mean-Gauss Surfaces

In this paper we explore solving the prescribed mean curvature equation for surfaces meeting a new relation given by (H_S) = λ(K_S), where H_S and K_S are the mean and Gaussian curvatures, respectively. We prove several existence theorems for various families of surfaces and state a conjecture for surfaces of revolution. To conclude, we state a weak existence theorem, and a strong conjecture concerning possible solutions. The intention is that by using differential geometry tools which would have likely been seen at the undergraduate level, the paper and its results are more accessible. My hope is that these new theorems find applications in the classification of surfaces in the future, or at the very least serves as an interesting curiosity.

A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent

We study positive solutions to the fractional semi-linear elliptic equation $${\left( { - \Delta } \right)^\sigma }u = K\left( x \right){u^{{{n + 2\sigma } \over {n - 2\sigma }}\,}}\,\,\,{\rm{in}}\,{B_2}\backslash \left\{ 0 \right\}$$ with an isolated singularity at the origin, where K is a positive function on B2, the punctured ball B2 {0} ⊂ ℝn with n ⩾ 2, σ ∈ (0, 1), and (−Δ)σ is the fractional Laplacian. In lower dimensions, we show that for any K ∈ C1(B2), a positive solution u always satisfies that u(x) ⩽ C∣x∣−(n−2σ)/2 near the origin. In contrast, we construct positive functions K ∈ C1(B2) in higher dimensions such that a positive solution u could be arbitrarily large near the origin. In particular, these results also apply to the prescribed boundary mean curvature equations on $${\mathbb{B}^{n + 1}}$$ .

On the second boundary value problem for Lagrangian mean curvature equation

Considering the second boundary value problem of the Lagrangian mean curvature equation, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle–Warren’s theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.

Multiplicity of Solution for Some Classes of Prescribed Mean Curvature Equation with Dirichlet Boundary Condition

Multiplicity of Solution for Some Classes of Prescribed Mean Curvature Equation with Dirichlet Boundary Condition

A note on the two-dimensional Lagrangian mean curvature equation

In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $\mathbb{R}^n$.

Global Structure of a Nodal Solutions Set of Mean Curvature Equation in Static Spacetime

Global Structure of a Nodal Solutions Set of Mean Curvature Equation in Static Spacetime