Prescribed Mean Curvature Equation Research Articles

Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of {{,mathrm{mathbb {R}},}}^n with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product M^n times {{,mathrm{mathbb {R}},}}. Precisely, given a mathscr {C}^2 bounded domain Omega in M and a function H = H (x, z) continuous in overline{Omega }times {{,mathrm{mathbb {R}},}} and non-decreasing in the variable z, we prove that the strong Serrin condition(n-1)mathcal {H}_{partial Omega }(y)ge nsup limits _{zin {{,mathrm{mathbb {R}},}}}left| H(y,z) right| forall yin partial Omega , is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins–Serrin and Serrin type sharp solvability criteria.

Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space

In this paper, bifurcation diagrams and exact multiplicity of positive solution are obtained for the one-dimensional prescribed mean curvature equation in Minkowski space in the form of [Formula: see text] where [Formula: see text] is a bifurcation parameter, [Formula: see text], the radius of the one-dimensional ball [Formula: see text], is an evolution parameter. Moreover, we make a comparison between the bifurcation diagram of one-dimensional prescribed mean curvature equation in Euclid space and Minkowski space. Our methods are based on a detailed analysis of time maps.

Boundary Continuity of Nonparametric Prescribed Mean Curvature Surfaces

We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $\phi$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.

Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is f, when f possesses certain reflection or rotation symmetry.

A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry

This paper deals with the quasilinear elliptic problem − div∇u∕1+|∇u|2+a(x)u=b(x)∕1+|∇u|2inB,u=0on∂B,where B is an open ball in RN, with N≥2, and a,b∈C1(B¯) are given radially symmetric functions, with a(x)≥0 in B. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a,b are not necessarily constant and no sign condition is assumed on b.

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

Ground state solutions for the Hénon prescribed mean curvature equation

Abstract In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in {{\mathbb{R}^{N}}} , both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space.

Two examples of minimal Cheeger sets in the plane

We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of Cheeger sets, as its perimeter does not coincide with the $1$-dimensional Hausdorff measure of its topological boundary. The second one is a kind of porous set, whose boundary is not locally a graph at many of its points, yet it is a weakly regular open set admitting a unique (up to vertical translations) non--parametric solution to the prescribed mean curvature equation, in the extremal case corresponding to the capillarity for perfectly wetting fluids in zero gravity.

The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.

Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.

Isolated singularities of the prescribed mean curvature equation in Minkowski 3-space

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space.

Bifurcation results for a class of prescribed mean curvature equations in bounded domains

Bifurcation for prescribed mean curvature equations in one dimension has been intensively investigated in recent years, and a striking phenomenon discovered is that the length of the interval may affect the shapes of bifurcation curves. However, to our best knowledge, no such results are known in higher dimensions. In this paper, we study a class of prescribed mean curvature equations in bounded domains of RN with general nonlinearities f satisfying f(0)=0 and f′(0)>0. We establish some formulas for directions of bifurcation at simple eigenvalues, which lead to a sufficient and necessary condition to ensure that the directions depend on the size of the domain. In contrast, this phenomenon does not occur for the semilinear case. Some interesting examples, including logistic and perturbed exponential nonlinearities, are also investigated.

Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases

This paper deals with the prescribed mean curvature equations \begin{document}$ - {\text{div}}\left( {\frac{{\nabla u}}{{\sqrt {1 \pm |\nabla u{|^2}} }}} \right) = g(u){\text{ }}\;\;\;\;{\text{in }}{\mathbb{R}^N},$ \end{document} both in the Euclidean case, with the sign +, and in the Lorentz-Minkowski case, with the sign -, for N ≥ 1 under the assumption g'(0)>0. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N = 1, while they are radial symmetric and decay to zero at infinity with their derivatives, if N≥ 2.

Gradient estimates of mean curvature equation with Neumann boundary condition in domains of Riemannian manifold

We study the prescribed mean curvature equation with Neumann boundary conditions in domains of Riemannian manifold. The main goal is to establish the gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

New positive periodic solutions to singular Rayleigh prescribed mean curvature equations

This paper is concerned with the existence of positive periodic solutions for the prescribed mean curvature Rayleigh equations with a singularity. Our results are based on the Manásevich-Mawhin continuation theorem. The results to be obtained here extend the existing ones in the literature. Moreover, an example is given to illustrate the applicability of our results.

On cusp solutions to a prescribed mean curvature equation

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Optimal Control of a Degenerate PDE for Surface Shape

Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape in this paper. Specifically, we consider an objective (tracking) functional for surface shape with the prescribed mean curvature equation in graph form as a state constraint. The control variable is the prescribed curvature. We prove existence of an optimal control, and using improved regularity estimates, we show sufficient differentiability to make sense of the first order optimality conditions. This allows us to rigorously compute the gradient of the objective functional for both the continuous and discrete (finite element) formulations of the problem. Numerical results are shown to illustrate the minimizers and optimal controls on different domains.

Gradient estimates of mean curvature equations with semi-linear oblique boundary value problems

In this paper, we consider the semi-linear oblique boundary value problem for the prescribed mean curvature equation. We find a suitable auxiliary function and use the maximum principle to get the gradient estimate. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations with semi-linear oblique derivative problems.

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

Abstract 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 { - div ⁡ ( ∇ ⁡ u 1 - | ∇ ⁡ u | 2 ) = f ⁢ ( x , u , ∇ ⁡ u ) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω . $\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$ 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.