Courant's nodal domain theorem and the residual nature of simple eigenvalues under perturbations both hold for the degenerate elliptic operator A = -div(w ∇·) with w > 0 inside Ω and w = 0 on part of ∂Ω.
Henrot,Extremum Problems for Eigenvalues of Elliptic Operators, Birkh¨ auser, Basel
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Proves well-posedness of degenerate parabolic PDEs with Dirichlet conditions, develops shape-design approximation by non-degenerate equations, and obtains boundary observability inequality as application.
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Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators
Courant's nodal domain theorem and the residual nature of simple eigenvalues under perturbations both hold for the degenerate elliptic operator A = -div(w ∇·) with w > 0 inside Ω and w = 0 on part of ∂Ω.
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Shape Design for Degenerate Parabolic Equations with Degenerate Boundaries and Its Application to Boundary Observability
Proves well-posedness of degenerate parabolic PDEs with Dirichlet conditions, develops shape-design approximation by non-degenerate equations, and obtains boundary observability inequality as application.