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A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions. (English) Zbl 1362.35197

The authors derive upper estimates for the eigenvalue counting function \(N(\lambda)\) of the Krein-von Neumann extension and the Friedrichs extension of a symmetric, closed, and strictly positive partial differential operator of order \(2m\) acting on \(L^2(\Omega)\), where \(\Omega\) is a bounded domain of \(\mathbb{R}^n\) without any regularity assumptions on its boundary (the operator is an \(m\)-th power of a second-order operator). Apart from its interesting results, the paper is also well written and educational.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators
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