Esteemed colleagues may want to check the MR review done by Taira for this book.
https://www.researchgate.net/publication/264551800_MR1478921_%2898k35026%29_3-_Elliptic_boundary_value_problems_with_indefinite_weights_variational_formulations_of_the_principal_eigenvalue_and_applications._%28English_summary%29
Book Elliptic Boundary Value Problems with Indefinite Weights +
Data MR1478921 (98k:35026) 3- Elliptic boundary value problems wi...
Despite the fact that this book was written nearly 2 decades ago, it still contains the basic and essential theory of Elliptic Boundary Value Equations with Indefinite Weights. Of course since then many papers and books have been written, and perhaps it is time to write the sequel of the new related advances in this regard. Asking this question is in part a prepartaion towards writing that sequel. I hope you can contribute to it by adding your answer to the question if any.
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