scholarly journals Global Second Derivative Estimates for the Second Boundary Value Problem of the Prescribed Affine Mean Curvature and Abreu's Equations

2012 ◽  
Vol 2013 (11) ◽  
pp. 2421-2438 ◽  
Author(s):  
Nam Q. Le
Keyword(s):  
Boundary Value Problem ◽  
Mean Curvature ◽  
Boundary Value ◽  
Second Derivative ◽  
Affine Mean Curvature

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

2018 ◽  
Vol 29 (10) ◽  
pp. 1850069
Author(s):  
Hong Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Manifold With Boundary ◽  
Prescribed Mean Curvature Equation

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.

scholarly journals On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator

2021 ◽  
Vol 11 (1) ◽  
pp. 198-211
Author(s):  
Sijia Du ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Mean Curvature ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Curvature Operator ◽  
Dirichlet Boundary Value Problem ◽  
The Maximum Principle ◽  
Mean Curvature Operator

Abstract Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory. We further separately determine open intervals of the parameter to attain at least two positive solutions and an unbounded sequence of positive solutions with the help of the maximum principle.

scholarly journals S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

2019 ◽  
Vol 17 (1) ◽  
pp. 929-941
Author(s):  
Man Xu ◽  
Ruyun Ma
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Mean Curvature ◽  
Boundary Value ◽  
Normal Derivative ◽  
Euclidean Norm ◽  
Positive Parameter ◽  
Annular Domain ◽  
Connected Component ◽  
Curvature Problem

Abstract In this paper, we show the existence of an S-shaped connected component in the set of radial positive solutions of boundary value problem $$\begin{array}{} \displaystyle \left\{\,\begin{array}{} -\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\ \end{array} \right. \end{array} $$ where R2 ∈ (0, ∞) and R1 ∈ (0, R2) is a given constant, 𝓐 = {x ∈ ℝN : R1 < ∣x∣ < R2}, Γ1 = {x ∈ ℝN : ∣x∣ = R1}, Γ2 = {x ∈ ℝN : ∣x∣ = R2}, $\begin{array}{} \phi_N(s)=\frac{s}{\sqrt{1-|s|^2}}, \end{array} $ s ∈ ℝN, λ is a positive parameter, a ∈ C[R1, R2], f ∈ C[0, ∞), $\begin{array}{} \frac{\partial y}{\partial \nu} \end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝN. The proof of main result is based upon bifurcation techniques.

scholarly journals On a free boundary value problem for the anisotropic N-Laplace operator on an N−dimensional ring domain

2020 ◽  
Vol 28 (2) ◽  
pp. 195-208
Author(s):  
A. E. Nicolescu ◽  
S. Vlase
Keyword(s):  
Maximum Principle ◽  
Free Boundary ◽  
Boundary Value Problem ◽  
Laplace Operator ◽  
Mean Curvature ◽  
Boundary Value ◽  
Weak Sense ◽  
Ring Domain ◽  
Free Boundary Value Problem ◽  
Anisotropic Mean Curvature

AbstractIn this paper we are going to investigate a free boundary value problem for the anisotropic N-Laplace operator on a ring domain \Omega : = {\Omega _0}\backslash {\bar \Omega _1} \subset {\mathbb{R}^N}, N ≥ 2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff shaped ring. The proof makes use of a maximum principle for an appropriate P-function, in the sense of L.E. Payne and some geometric arguments involving the anisotropic mean curvature of the free boundary.

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