Potential theory on lipschitz domains in riemannian manifolds: hölder continuous metric tensors

1999 ◽  
Vol 25 (7-8) ◽  
pp. 1487-1536 ◽  
Author(s):  
Marius Mtrea ◽  
Michael Taylor
Keyword(s):  
Potential Theory ◽  
Riemannian Manifolds ◽  
Lipschitz Domains ◽  
Hölder Continuous ◽  
Metric Tensors

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS

2007 ◽  
Vol 18 (09) ◽  
pp. 1071-1111 ◽  
Author(s):  
JÉRÔME VÉTOIS
Keyword(s):  
Continuous Function ◽  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Beltrami Operator ◽  
Hölder Continuous ◽  
Nonlinear Elliptic

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holdër continuous function on M. We prove multiplicity of changing sign solutions for equations like Δg u + hu = |u|2* - 2 u, where Δg is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint.

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.

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