Prescribed Mean Curvature Problem Research Articles

A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem

Abstract In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space: (P) u ′ 1 + ∣ u ′ ∣ 2 ′ + h ( x ) f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , \left\{\begin{array}{l}{\left(\frac{u^{\prime} }{\sqrt{1+{| u^{\prime} | }^{2}}}\right)}^{^{\prime} }+h\left(x)f\left(u)=0,\hspace{1em}-1\lt x\lt 1,\hspace{1.0em}\\ u\left(-1)=u\left(1)=0,\hspace{1.0em}\end{array}\right. where h ∈ C 1 ( [ − 1 , 1 ] ) h\in {C}^{1}\left(\left[-1,1]) and f ∈ C 1 ( [ 0 , ∞ ) ; [ 0 , ∞ ) ) f\in {C}^{1}\left(\left[0,\infty );\hspace{0.33em}\left[0,\infty )) . Under suitable conditions on h h and monotone condition on f ( s ) s \frac{f\left(s)}{s} , by introducing a modified Picone-type identity, we prove that the problem has at most one positive symmetric solution.

On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.

New existence results for prescribed mean curvature problem on balls under pinching conditions

We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball Bn and on the boundary Sn−1 the mean curvature is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results in higher dimensional case n≥5, under suitable pinching conditions on the mean curvature function.

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.

The Dirichlet problem for a prescribed mean curvature equation

We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.

A diffused interface with the advection term in a Sobolev space

We study the asymptotic limit of diffused surface energy in the van der Waals–Cahn–Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min–max method.

Positive solutions of superlinear indefinite prescribed mean curvature problems

This paper analyzes the superlinear indefinite prescribed mean curvature problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with a regular boundary [Formula: see text], [Formula: see text] satisfies [Formula: see text], as [Formula: see text], [Formula: see text] being an exponent with [Formula: see text] if [Formula: see text], [Formula: see text] represents a parameter, and [Formula: see text] is a sign-changing function. The main result establishes the existence of positive regular solutions when [Formula: see text] is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for [Formula: see text] small is further discussed assuming that [Formula: see text] satisfies [Formula: see text] as [Formula: see text], [Formula: see text] being such that [Formula: see text] if [Formula: see text]; thus, in dimension [Formula: see text], the function [Formula: see text] is not superlinear at [Formula: see text], although its potential [Formula: see text] is. Imposing such different degrees of homogeneity of [Formula: see text] at [Formula: see text] and at [Formula: see text] is dictated by the specific features of the mean curvature operator.

Bifurcation of nodal radial solutions for a prescribed mean curvature problem on an exterior domain

We investigate the existence and multiplicity of nodal radial solutions for a prescribed mean curvature problem on the exterior domain of a ball using the global bifurcation theory. Moreover, we establish the asymptotic behavior of the solutions on the subcontinua, which bifurcate from a trivial branch. All results are obtained by considering a radially equivalent second order problem with a singular weight.

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.

Continuity results for TV-minimizers

This paper deals with continuity preservation when minimizing generalized total variation with a $L^2$ fidelity term or a Dirichlet boundary condition. We extend several recent results in the two cases, mainly by showing comparison principles for the prescribed mean curvature problem satisfied by the level-sets of such minimizers.

Classification and evolution of bifurcation curves for a one-dimensional prescribed mean curvature problem

We study the classification and evolution of bifurcation curves of positive solutions $u\in C^{2}(-L,L)\cap C[-L,L]$ for the one-dimensional prescribed mean curvature problem \begin{equation*} \left\{ \begin{array}{l} - \Big ( \dfrac{u^{\prime }(x)}{\sqrt{1+ ( {u^{\prime }(x)} ) ^{2}}} \Big ) ^{\prime }=\lambda [ \exp (\frac{au}{a+u})-1 ] , \ \ -L < x < L,\\ u(-L)=u(L)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is a bifurcation parameter, and $L,a>0$ are two evolution parameters. We prove that, on $(\lambda , \Vert u \Vert _{\infty })$ -plane, the bifurcation curve of this problem is (i) $\supset $-shaped when $ 0 < a < 2$; (ii) $\supset $-shaped or monotone decreasing when $a=2$; (iii) reversed $S$-like shaped or monotone decreasing when $a>2$. Moreover, for $ a>2,$ the bifurcation curve is reversed $S$-like shaped if $L>0$ is large enough and is monotone decreasing if $L>0$ is small enough.

Global structure of radial positive solutions for a prescribed mean curvature problem in a ball

In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem div ( ϕ N ( ∇ v ) ) + λ f ( | x | , v ) = 0 in B ( R ) , v = 0 on ∂ B ( R ) , where ϕ N ( y ) = y 1 − | y | 2 , y ∈ R N , λ is a positive parameter, B ( R ) = { x ∈ R N : | x | < R } , and | ⋅ | denotes the Euclidean norm in R N . All results, depending on the behavior of nonlinear term f near 0, are obtained by using global bifurcation techniques.

Conformal metrics on the unit ball with prescribed mean curvature

This paper focuses on the study of the prescribed mean curvature problem on the unit ball. If the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small, then the existence of positive solutions of conformal mean curvature equation has been known. The purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method.

Double grow-up and global convergence of solutions for a parabolic prescribed mean curvature problem

The paper is concerned with a parabolic mean curvature type problem with a varying parameter λ. We study large time behavior of global solutions for different ranges of λ and initial data and present some new asymptotic results about global convergence and infinite time blow-up. In particular, it is shown that for suitable ranges of parameter λ and initial data, there exists a double grow-up phenomenon: the solution itself blows up at every interior point and its gradient blows up at the boundary of the domain as t→+∞. We also establish an interesting connection between global convergence and the non-classical solution of the associated stationary problem: if initial data are smaller than the non-classical solution, then the solutions must decay to zero in C1 norm as t→∞.

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.

Exact multiplicity and bifurcation diagrams of positive solutions of a one-dimensional multiparameter prescribed mean curvature problem

We study the exact multiplicity and bifurcation diagrams of positive solutions u∈C2(−L,L)∩C[−L,L] of the one-dimensional multiparameter prescribed mean curvature problem{−(u′(x)1+(u′(x))2)′=λ(up+uq),−L<x<L,u(−L)=u(L)=0, where λ>0 is a bifurcation parameter, L>0, the radius of the one-dimensional ball (−L,L), is an evolution parameter, and 0≤p<q<∞ are two constants. We prove that the problem has at most two positive solutions for any 0≤p<q<∞ and λ,L>0. In addition, if 0≤p<q≤q˜(p)=p+1+2p+1, we give a classification of totally three qualitatively different bifurcation diagrams on the (λ,‖u‖∞)-plane for any L>0. For any fixed p≥0 and q≥q¯(p)=p+2+22p+3, we prove that there exist positive L⁎<L⁎ such that the bifurcation diagrams on the (λ,‖u‖∞)-plane are individually qualitatively different for the cases (i) 0<L<L⁎ or L>L⁎, (ii) L=L⁎, (iii) L⁎≤L<L⁎.

Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS

We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u<1,−L<x<L,u(−L)=u(L)=0, where λ>0 is a bifurcation parameter, and p,L>0 are two evolution parameters. We determine the exact number of positive solutions by the values of p,L and λ. Moreover, for p≥1, the bifurcation diagram undergoes fold and splitting bifurcations. While for 0<p<1, the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity (1−u)−2 to (1−u)−p with general p∈(1,∞). We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p>0. Concerning this open question, we find and prove that global bifurcation diagrams for 0<p<1 are different to and more complicated than those for p≥1.

On the multiplicity of nodal solutions of a prescribed mean curvature problem

AbstractWe prove existence and multiplicity results for sign‐changing solutions of a prescribed mean curvature equation with Dirichlet boundary conditions. Our arguments involve a perturbation of the degenerate part \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$1/\sqrt{1+s}$\end{document}, which allows us to use classical variational techniques and to localize small regular solutions.

Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models

In this paper, we consider a one-dimensional prescribed mean curvature problem related to MEMS models. We obtain all exact multiplicity for classical solutions and non-classical solutions when the length L of the interval and the parameter λ change. As a by-product, it also provides a complete analytic proof for an interesting new phenomenon, which is first noticed by Brubaker and Pelesko. Our methods are based on a detailed analysis of time maps.

Volume-constrained minimizers for the prescribed curvature problem in periodic media

We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a suitable Wulff Shape, when the volume tends to infinity.