Prescribed Mean Curvature Equation Research Articles

New approach for the existence and uniqueness of periodic solutions to p-Laplacian prescribed mean curvature equations

Using a generalized Mawhin continuation theorem, we obtain some sufficient conditions which guarantee the existence and uniqueness of periodic solutions for two types of prescribed mean curvature p-Laplacian equations.

Existence and Uniqueness of Anti-Periodic Solutions for Prescribed Mean Curvature Rayleigh p-Laplacian Equations

In this paper, the prescribed mean curvature Rayleigh p-Laplacian equation is studied. By means of the Leray–Schauder degree theory and some analysis methods, we have established new results of existence and uniqueness of anti-periodic solutions. Our research enriches the contents of prescribed mean curvature equations and generalizes known results.

Radial limits of bounded nonparametric prescribed mean curvature surfaces

Consider a solution $f\in C^{2}(\Omega)$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ \Omega, \] where $\Omega\subset \Real^{2}$ is a domain whose boundary has a corner at ${\cal O}=(0,0)\in\partial\Omega.$ If $\sup_{x\in\Omega} |f(x)|$ and $\sup_{x\in\Omega} |H(x,f(x))|$ are both finite and $\Omega$ has a reentrant corner at ${\cal O},$ then the radial limits of $f$ at ${\cal O},$ \[ Rf(\theta) \myeq \lim_{r\downarrow 0} f(r\cos(\theta),r\sin(\theta)), \] are shown to exist and to have a specific type of behavior, independent of the boundary behavior of $f$ on $\partial\Omega.$ If $\sup_{x\in\Omega} |f(x)|$ and $\sup_{x\in\Omega} |H(x,f(x))|$ are both finite and the trace of $f$ on one side has a limit at ${\cal O},$ then the radial limits of $f$ at ${\cal O}$ exist and have a specific type of behavior.

On a prescribed mean curvature equation in Lorentz–Minkowski space

We are interested in providing new results on the following prescribed mean curvature equation in Lorentz–Minkowski space∇⋅[∇u1−|∇u|2]+up=0, set in the whole RN, with N⩾3.We study both existence and multiplicity of radial ground state solutions (namely positive and vanishing at infinity) for p>1, emphasizing the fundamental difference between the subcritical and the supercritical case.We also study speed decay at infinity of ground states, and give some decay estimates.Finally we provide a multiplicity result on the existence of sign-changing bound state solutions for any p>1.

Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space

In this article, by using the Leggett–Williams’ fixed point theorem, we prove the existence of at least three positive radial solutions of the singular Dirichlet problem for the prescribed mean curvature equation in Minkowski space {div(∇v1−|∇v|2)+f(|x|,v)=0inΩ;v=0on∂Ω, and the corresponding one-parameter problem. Here Ω is a unit ball in RN.

Dirichlet problem for anisotropic prescribed mean curvature equation on unbounded domains

In this paper, we consider the Dirichlet problem for hypersurfaces M=graphu of anisotropic prescribed mean curvature H=H(x,u,N) on unbounded domain Ω, where N is the unit normal to M at (x,u). As a corollary of the result, we obtain the existence of translating solutions to the mean curvature flow with a forcing term on unbounded domains. The approach used here is a modified version of classical Perron's method, where the solutions to minimal surface equation are used as supersolutions and a family of auxiliary functions is constructed as local subsolutions.

On the relationship of continuity and boundary regularity in prescribed mean curvature Dirichlet problems

In 1976, Leon Simon showed that if a compact subset of the boundary of a domain is smooth and has negative mean curvature, then the non-parametric least area problem with Lipschitz continuous Dirichlet boundary data has a generalized solution which is continuous on the union of the domain and this compact subset of the boundary, even if the generalized solution does not take on the prescribed boundary data. Simon's result has been extended to boundary value problems for prescribed mean curvature equations by other authors. In this note, we construct Dirichlet problems in domains with corners and demonstrate that the variational solutions of these Dirichlet problems are discontinuous at the corner, showing that Simon's assumption of regularity of the boundary of the domain is essential.

Gradient estimates of mean curvature equations with Neumann boundary value problems

In this paper, we use the maximum principle to get the gradient estimate for the solutions of the prescribed mean curvature equation with Neumann boundary value problem, which gives a positive answer for the question raised by Lieberman in 2013. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations.

Three solutions for a two-point boundary value problem with the prescribed mean curvature equation

The existence of at least three classical solutions for a parametric ordinary Dirichlet problem involving the mean curvature operator are established. In particular, a variational approach is proposed and the main results are obtained simply requiring the sublinearity at zero of the considered nonlinearity.

Homoclinic solutions for n-dimensional prescribed mean curvature p-Laplacian equations

In this paper, a n-dimensional prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument, $(\varphi_{p}(\frac{u'(t)}{\sqrt{1+|u'(t)|^{2}}}))'+F(t,u'(t))+G(t,u(t-\tau(t)))=e(t)$ , is studied. By means of Mawhin’s continuation theorem and some analysis methods, a new result on the existence of homoclinic solutions for the equation is obtained. Our research enriches the contents of prescribed mean curvature equations.

Constructing Graphs over ℝ n $\mathbb {R}^{n}$ with Small Prescribed Mean-Curvature

In this paper a convergent series expansion is constructed to solve the prescribed mean curvature equation for n-dimensional hypersurfaces in n+1 dimensional Euclidean or Minkowskian space(time) which are graphs of a smooth real function u, and whose mean curvature function H is not too large in Hoelder norm, and integrable. Our approach is inspired by the Maxwell-Born-Infeld theory of electromagnetism in Minkowski spacetime, for which our method yields the first systematic way of explicitly computing the electrostatic potential u for regular charge densities proportional to H and small Born parameter.

The Mean Curvature Equation with Oscillating Nonlinearity

Abstract We find a solution of the Dirichlet problem for the prescribed mean curvature equation in Ω with u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝN, N ≥ 1 and f : Ω × [0,∞) → ℝ is an unbounded continuous function with oscillatory behavior near the origin.

Radial Solutions of the Dirichlet Problem for the Prescribed Mean Curvature Equation in a Robertson-Walker Spacetime

Abstract We consider the prescribed mean curvature problem of spacelike graphs in Robertson- Walker spacetimes of flat fiber with homogeneous Dirichlet conditions on an Euclidean ball. Under reasonable assumptions, it is shown that every possible solution must be radially symmetric. Besides, an existence result for a singular nonlinear equation is proved by making use of the classical Schauder fixed point Theorem. The results are applied to realistic examples of Robertson-Walker spacetimes.

Existence results for parametric boundary value problems involving the mean curvature operator

In this note we propose a variational approach to a parametric differential problem where a prescribed mean curvature equation is considered. In particular, without asymptotic assumptions at zero and at infinity on the potential, we obtain an explicit positive interval of parameters for which the problem under examination has at least one nontrivial and nonnegative solution.

Time-map analysis to establish the exact number of positive solutions of one-dimensional prescribed mean curvature equations

We investigate the one-dimensional prescribed mean curvature equation with concave-convex nonlinearities in the form of − ( u ′ 1 + u ′ 2 ) ′ =λ( u p + u q ), u(x)>0, 0<x<1, u(0)=u(1)=0, where λ>0 is a parameter and p, q satisfy −1<p<q<+∞, and we obtain new exact results of positive solutions. Our methods are based on a detailed analysis of time maps.

Infinite time gradient blow-up for parabolic prescribed mean curvature equations

We prove some results on the existence of infinite time gradient blow-up phenomena for parabolic prescribed mean curvature equations over bounded, mean-convex domains in Rn.

Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations

Abstract In this work, we establish the existence and multiplicity results of positive solutions for one-dimensional prescribed mean curvature equations. Our approach is based on fixed point index theory for completely continuous operators which leave invariant a suitable cone in a Banach space of continuous functions. MSC:34B10, 34B18.

Magnetic Dirac-harmonic maps

We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physics. In two dimensions it is conformally invariant. We call critical points of this functional magnetic Dirac-harmonic maps. We study geometric and analytic properties of magnetic Dirac-harmonic maps including their regularity and the removal of isolated singularities.

Existence of Positive Solutions of One-Dimensional Prescribed Mean Curvature Equation

We consider the existence of positive solutions of one-dimensional prescribed mean curvature equation−(u′/1+u′2)′=λf(u),0&lt;t&lt;1,u(t)&gt;0,t∈(0,1),u(0)=u(1)=0whereλ&gt;0is a parameter, andf:[0,∞)→[0,∞)is continuous. Further, whenfsatisfiesmax{up,uq}≤f(u)≤up+uq,0&lt;p≤q&lt;+∞, we obtain the exact number of positive solutions. The main results are based upon quadrature method.

Periodic Solutions for a Prescribed Mean Curvature Equation with Multiple Delays

We study the existence of periodic solutions for the one-dimensional prescribed mean curvature delay equation(d/dt)(x'(t)/1+x't2) +∑i=1naitgxt-τit=pt. By using Mawhin's continuation theorem, a new result is obtained. Furthermore, the nonexistence of periodic solution for the equation is investigated as well.