scholarly journals A UNIQUENESS THEOREM ON TWO-DIMENSIONAL RIEMANNIAN MANIFOLDS WITH BOUNDARY

2001 ◽  
pp. 150-155
Author(s):  
Chuan-Chih Hsiung
Keyword(s):  
Uniqueness Theorem ◽  
Riemannian Manifolds ◽  
Two Dimensional ◽  
Manifolds With Boundary

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
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