Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary

2010 ◽  
Vol 53 (4) ◽  
pp. 674-683 ◽  
Author(s):  
Alexandru Kristály ◽  
Nikolaos S. Papageorgiou ◽  
Csaba Varga
Keyword(s):  
Boundary Condition ◽  
Riemannian Manifolds ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Manifolds With Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Manifold With Boundary

AbstractWe study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

scholarly journals Graph approximations to the Laplacian spectra

2020 ◽  
pp. 1-35
Author(s):  
Jinpeng Lu
Keyword(s):  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Beltrami Operator ◽  
Metric Measure Spaces ◽  
Proximity Graphs ◽  
Manifold With Boundary ◽  
Graph Laplacians ◽  
Laplacian Spectra ◽  
Measure Spaces

I prove that the spectrum of the Laplace–Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.

EXTREMAL FUNCTIONS FOR MOSER–TRUDINGER INEQUALITIES ON 2-DIMENSIONAL COMPACT RIEMANNIAN MANIFOLDS WITH BOUNDARY

2006 ◽  
Vol 17 (03) ◽  
pp. 313-330 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Extremal Functions ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Blowing Up

Let (M,g) be a 2-dimensional compact Riemannian manifold with boundary. In this paper, we use the method of blowing up analysis to prove the existence of the extremal functions for some Moser–Trudinger inequalities on (M,g).

scholarly journals $W^{1,N}$ versus $C^1$ local minimizer for a singular functional with Neumann boundary condition

2017 ◽  
Vol 37 (1) ◽  
pp. 71
Author(s):  
Kamel Saoudi
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Local Minimum ◽  
Smooth Boundary ◽  
Lagrange Equation ◽  
Neumann Boundary ◽  
Local Minimizer ◽  
Multiplicity Of Positive Solutions

Let $\Omega\subset\R^N,$ be a bounded domain with smooth boundary. Let $g:\R^+\to\R^+$ be a continuous on $(0,+\infty)$ non-increasing and satisfying $$c_1=\liminf_{t\to 0^+}g(t)t^{\delta}\leq\underset{t\to 0^+}{\limsup} g(t)t^{\delta}=c_2,$$ for some $c_1,c_2>0$ and $0<\delta<1.$ Let $f(x,s) = h(x,s)e^{bs^{\frac{N}{N-1}}},$ $b>0$ is a constant.Consider the singular functional $I: W^{1,N}(\Omega)\to \R$ defined as \begin{eqnarray*}&&I(u)\eqdef\frac{1}{N}\|u\|^N_{W^{1,N}(\Omega)}-\int_{\Omega}G(u^+)\,{\rm d} x-\int_{\Omega}F(x,u^+) \,{\rm d} x\nonumber\\&& -\frac{1}{q+1}||u||^{q+1}_{L^{q+1}(\partial\Omega)}\nonumber\end{eqnarray*} where $F(x,u)=\int_0^sf(x,s)\,{\rm d}s$, $G(u)=\int_0^s g(s)\,{\rm d}s$. We show that if $u_0\in C^1(\overline{\Omega})$ satisfying $u_0\geq \eta \mbox{dist}(x,\partial\Omega)$, for some $0<\eta$, is a local minimum of $I$ in the $C^1(\overline{\Omega})\cap C_0(\overline{\Omega})$ topology, then it is also a local minimum in $W^{1,N}(\Omega)$ topology. This result is useful %for proving multiple solutions to the associated Euler-lagrange equation ${\rm (P)}$ defined below.to prove the multiplicity of positive solutions to critical growth problems with co-normalboundary conditions.

Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Agil K. Khanmamedov ◽  
Nigar F. Gafarova
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Inverse Spectral Problem ◽  
Anharmonic Oscillator ◽  
Neumann Boundary ◽  
Effective Algorithm ◽  
Inverse Spectral Problems ◽  
Main Equation ◽  
Inverse Spectral ◽  
Half Axis

AbstractAn anharmonic oscillator {T(q)=-\frac{d^{2}}{dx^{2}}+x^{2}+q(x)} on the half-axis {0\leq x<\infty} with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated.

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