scholarly journals The Steklov and Laplacian spectra of Riemannian manifolds with boundary

2020 ◽  
Vol 278 (6) ◽  
pp. 108409 ◽  
Author(s):  
Bruno Colbois ◽  
Alexandre Girouard ◽  
Asma Hassannezhad
Keyword(s):  
Riemannian Manifolds ◽  
Manifolds With Boundary ◽  
Laplacian Spectra

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 ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

scholarly journals On Yamabe-type problems on Riemannian manifolds with boundary

2016 ◽  
Vol 284 (1) ◽  
pp. 79-102 ◽  
Author(s):  
Marco Ghimenti ◽  
Anna Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifolds ◽  
Manifolds With Boundary

Harmonic Tensors on Riemannian Manifolds with Boundary

1952 ◽  
Vol 56 (1) ◽  
pp. 128 ◽  
Author(s):  
G. F. D. Duff ◽  
D. C. Spencer
Keyword(s):  
Riemannian Manifolds ◽  
Manifolds With Boundary

scholarly journals Curvature estimates for stable free boundary minimal hypersurfaces

2020 ◽  
Vol 2020 (759) ◽  
pp. 245-264 ◽  
Author(s):  
Qiang Guang ◽  
Martin Man-chun Li ◽  
Xin Zhou
Keyword(s):  
Free Boundary ◽  
Riemannian Manifolds ◽  
A Priori ◽  
Minimal Submanifolds ◽  
Manifolds With Boundary ◽  
Curvature Estimate ◽  
Minimal Hypersurfaces ◽  
Curvature Estimates ◽  
Minimal Laminations ◽  
Uniform Area

AbstractIn this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

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.

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