Critical Points for Elliptic Equations with Prescribed Boundary Conditions

Critical Points for Elliptic Equations with Prescribed Boundary Conditions This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $${\nabla\cdot \sigma(x)\nabla u=0}$$ ∇ · σ ( x ) ∇ u = 0 posed on a bounded domain X with prescribed boundary conditions. In spatial dimension n = 2, it is known that the number of critical points (where $${\nabla u=0}$$ ∇ u = 0 ) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $${\sigma}$$ σ . We show that the situation is different in dimension $${n\geq3}$$ n ≥ 3 . More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for u on $${\partial X}$$ ∂ X , there exists an open set of smooth coefficients $${\sigma(x)}$$ σ ( x ) such that $${\nabla u}$$ ∇ u vanishes at least at one point in X. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $${\nabla u}$$ ∇ u on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $${\sigma(x)}$$ σ ( x ) . These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $${\sigma(x)}$$ σ ( x ) for which the stability of the reconstructions will inevitably degrade. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
D.O.I.
10.1007/s00205-017-1130-3
Publisher site
See Article on Publisher Site

Abstract

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $${\nabla\cdot \sigma(x)\nabla u=0}$$ ∇ · σ ( x ) ∇ u = 0 posed on a bounded domain X with prescribed boundary conditions. In spatial dimension n = 2, it is known that the number of critical points (where $${\nabla u=0}$$ ∇ u = 0 ) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $${\sigma}$$ σ . We show that the situation is different in dimension $${n\geq3}$$ n ≥ 3 . More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for u on $${\partial X}$$ ∂ X , there exists an open set of smooth coefficients $${\sigma(x)}$$ σ ( x ) such that $${\nabla u}$$ ∇ u vanishes at least at one point in X. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $${\nabla u}$$ ∇ u on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $${\sigma(x)}$$ σ ( x ) . These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $${\sigma(x)}$$ σ ( x ) for which the stability of the reconstructions will inevitably degrade.

Journal

Archive for Rational Mechanics and AnalysisSpringer Journals

Published: May 27, 2017

References

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