Prescribed Mean Curvature Research Articles

Positive solutions of the prescribed mean curvature equation with exponential critical growth

In this paper, we are concerned with the problem -div∇u1+|∇u|2=f(u)inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\text{ div } \\left( \\displaystyle \\frac{\\nabla u}{\\sqrt{1+|\\nabla u|^2}}\\right) = f(u) \\ \\text{ in } \\ \\Omega , \\ \\ u=0 \\ \\text{ on } \\ \\ \\partial \\Omega , \\end{aligned}$$\\end{document}where Omega subset {mathbb {R}}^{2} is a bounded smooth domain and f:{mathbb {R}}rightarrow {mathbb {R}} is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.

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.

MULTIPLE POSITIVE SOLUTIONS OF THE DISCRETE DIRICHLET PROBLEM WITH ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE OPERATOR

<abstract> We shall discuss the existence and multiplicity of positive solutions for the discrete Dirichlet problem with one-dimensional prescribed mean curvature operator. Based on the critical point theory, we shall show the existence of either one, or two, or three, or infinity many positive solutions depending on the asymptotic behavior of nonlinearity near zero. </abstract>

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.

On the Multiplicity One Conjecture in min-max theory

We prove that in a closed manifold of dimension between $3$ and $7$ with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.

Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

The influence of the geometry of the domain on the behavior of generalized solutions of Dirichlet problems for elliptic partial differential equations has been an important subject for over a century. We investigate the boundary behavior of variational solutions $f$ of Dirichlet problems for prescribed mean curvature equations in a domain $\Omega \subset \mathbb{R}^{2}$ near a point $\mathcal{O} \in \partial \Omega$ under different assumptions about the curvature of $\partial \Omega$ on each side of $\mathcal{O}$. We prove that the radial limits at $\mathcal{O}$ of $f$ exist under different assumptions about the Dirichlet boundary data $\phi$, depending on the curvature properties of $\partial \Omega$ near $\mathcal{O}$.

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.

On the fill-in of nonnegative scalar curvature metrics

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

Research in the Field of Geometric Analysis at Volgograd State University

This article discusses the main directions of research in geometric analysis, which were conducted and are being carried out by the scientific mathematical school of Volgograd State University. The results of the founder of the scientific school, Doctor of Physics and Mathematics, Professor Vladimir Mikhailovich Miklyukov and his students are summarized. These results concern the solution of a number of problems in the field of quasiconformal flat mappings and mappings with bounded distortion of surfaces and Riemannian manifolds, the theory of minimal surfaces and surfaces of prescribed mean curvature, surfaces of zero mean curvature in Lorentz spaces, as well as problems associated with the study of the stability of such surfaces. In addition, the results of the study of various classes of triangulations — an object that appears at the junction of research in the field of geometric analysis and computational mathematics — are noted. Besides, this review discusses papers that use the Fourier decomposition method for solutions of the Laplace — Beltrami equations and the stationary Schr¨odinger equation with respect to the eigenfunctions of the corresponding boundary value problems. In particular, the authors give the results on finding capacitive characteristics that allowed for the first time to formulate and prove the criteria for the fulfillment of various theorems of Liouville type and the solvability of boundary value problems on model and quasimodel Riemannian manifolds. The role of the equivalent function method is also indicated in the study of such problems on manifolds of a fairly general form. In addition to this, the article gives an overview of the results concerning estimates of calculating error integral functionals and convergence of piecewise polynomial solutions of nonlinear variational type equations: minimal surface equations, equilibrium equations capillary surface and equations of biharmonic functions.

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.

Boundary Behavior of Rotationally Symmetric Prescribed Mean Curvature Hypersurfaces in $$\pmb {\varvec{{\mathbb {R}}}}^{4}$$

In a domain $$\Omega \subset {\mathbb {R}}^{n}$$ whose boundary is smooth except on a set $${\mathcal {C}}$$ of codimension (in $$\partial \Omega $$ ) k, the behavior of a nonparametric prescribed mean curvature hypersurface $$z=f\left( \mathbf{x}\right) $$ in the vertical cylinder $$\Omega \times {\mathbb {R}}$$ at a point $${\mathcal {O}}\in {\mathcal {C}}$$ can be largely unknown when $$n\ge 3,$$ depending on the type of boundary condition f satisfies. In a previous note, the authors considered an example in which $$n=3$$ and $$k=2;$$ that is, when $${\mathcal {C}}$$ is a (nonconvex) conical point on $$\partial \Omega .$$ Here, we consider prescribed mean curvature boundary value problems which are rotationally symmetric and investigate the behavior of variational solutions near a point $$P\in {\mathcal {C}}$$ when $$n=3$$ and $$k=1.$$

On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }\Omega\\ v=\varphi \text{ on }\partial\Omega. \end{cases} $$ where $\Omega$ is a domain contained in a complete Riemannian manifold $M,$ $f:\Omega\times\mathbb{R\rightarrow R}$ is a fixed function and $\varphi$ is a given continuous function on $\partial\Omega$. This is done in three parts. In the first one we consider this problem in the most general form, proving the existence of solutions when $\Omega$ is a bounded $C^{2,\alpha}$ domain, under suitable conditions on $f$, with no restrictions on $M$ besides completeness. In the second part we study the asymptotic Dirichlet problem when $M$ is the hyperbolic space $\mathbb{H}^n$ and $\Omega$ is the whole space. This part uses in an essential way the geometric structure of $\mathbb{H}^n$ to construct special barriers which resemble the Scherk type solutions of the minimal surface PDE. In the third part one uses these Scherk type graphs to prove the non existence of isolated asymptotic boundary singularities for global solutions of this Dirichlet problem.

Periodic solution for prescribed mean curvature Rayleigh equation with a singularity

In this paper, we consider the existence of a periodic solution for a prescribed mean curvature Rayleigh equation with singularity (weak and strong singularities of attractive type or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.

Positive periodic solution for prescribed mean curvature generalized Li\xe9nard equation with a singularity

The main purpose of this paper is to investigate the existence of a positive periodic solution for a prescribed mean curvature generalized Liénard equation with a singularity (weak and strong singularities of attractive type, or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.

Existence of positive solutions for prescribed mean curvature problems with nonlocal term via sub‐ and supersolution method

In this paper, we are concerned with the existence of solution to the class of nonlocal quasilinear problem of the type where Ω is a smooth bounded domain on , , , and are functions whose hypotheses will be detailed later. We use sub‐ and supersolution method to find solutions to problem (P). Further, we apply our result to some specific nonlocal prescribed mean curvature problems.

Rotationally symmetric spacelike translating solitons for the mean curvature flow in Minkowski space

The rotationally symmetric spacelike translating solitons for the MCF in the Minkowski space are considered in this paper. To be specific, the complete classification for such translating solitons according to their rotation axes, which are timelike, spacelike and lightlike, is provided. In particular, the explicit parametrization of the rotationally symmetric spacelike hypersurface of the lightlike rotation axis with the prescribed mean curvature is obtained.

Invariant hypersurfaces with linear prescribed mean curvature

Our aim is to study invariant hypersurfaces immersed in the Euclidean space Rn+1, whose mean curvature is given as a linear function in the unit sphere Sn depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.

The global geometry of surfaces with prescribed mean curvature in ℝ³

We develop a global theory for complete hypersurfaces inRn+1\mathbb {R}^{n+1}whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces inRn+1\mathbb {R}^{n+1}, and also that of self-translating solitons of the mean curvature flow. For the particular casen=2n=2, we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end.

Periodic Solutions for a Class of n-dimensional Prescribed Mean Curvature Equations

In this paper, by using an extension of Mawhin’s continuation theorem and some analysis methods, we study the existence of periodic solutions for the following prescribed mean curvature system $$\frac{d}{{dt}}\varphi \left( {x'} \right) + \nabla W\left( x \right) = p\left( t \right),$$ where x ∊ ℝn, W ∊ C1(ℝn, ℝ), p ∊ C(ℝ, ℝn) is T-periodic and $$\varphi \left( x \right)\frac{x}{{\sqrt {1 + |x{|^2}} }}.$$

Rigidity and trace properties of divergence-measure vector fields

Abstract We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ ⁢ ( t ) {\varphi(t)} is a non-negative convex function vanishing only at t = 0 {t=0} . We show that this property is always satisfied in dimension n = 2 {n=2} , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ ⁢ ( t ) = c ⁢ t 2 {\varphi(t)=ct^{2}} ) in dimension n ≥ 4 {n\geq 4} . The validity of the quadratic rigidity, which we prove in dimension n = 2 {n=2} , implies the existence of the trace of a divergence-measure vector field ξ on an ℋ 1 {\mathcal{H}^{1}} -rectifiable set S, as soon as its weak normal trace [ ξ ⋅ ν S ] {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.