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**A survey on the Krein-von Neumann extension, the corresponding abstract buckling problem, and Weyl-type spectral asymptotics for perturbed Krein Laplacians in nonsmooth domains.**
*(English)*
Zbl 1283.47001

Demuth, Michael (ed.) et al., Mathematical physics, spectral theory and stochastic analysis. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0590-2/hbk; 978-3-0348-0591-9/ebook). Operator Theory: Advances and Applications 232. Advances in Partial Differential Equations, 1-106 (2013).

Authors’ abstract: In the first (and abstract) part of this survey, we prove the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, \(S\geq \varepsilon I_{\mathcal{H}}\) for some \(\varepsilon >0\) in a Hilbert space \(\mathcal{H}\), to an abstract buckling problem operator.

In the concrete case where \(S=\overline{-\Delta|_{C_0^\infty(\Omega)}}\) in \(L^2(\Omega; d^n x)\) for \(\Omega\subset\mathbb{R}^n\) an open, bounded (and sufficiently regular) set, this recovers, as a particular case of a general result due to G. Grubb [J. Oper. Theory 10, 9–20 (1983; Zbl 0559.47035)], that the eigenvalue problem for the Krein Laplacian \(S_K\) (i.e., the Krein-von Neumann extension of \(S\)), \[ S_K v = \lambda v, \quad \lambda \neq 0, \] is in one-to-one correspondence with the problem of the buckling of a clamped plate, \[ (-\Delta)^2u=\lambda (-\Delta) u \;\text{ in } \;\Omega, \quad \lambda \neq 0, \quad u\in H_0^2(\Omega), \] where \(u\) and \(v\) are related via the pair of formulas \[ u = S_F^{-1} (-\Delta) v, \quad v = \lambda^{-1}(-\Delta) u, \] with \(S_F\) the Friedrichs extension of \(S\).

This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).

In the second and principal part of this survey, we study spectral properties for \(H_{K,\Omega}\), the Krein-von Neumann extension of the perturbed Laplacian \(-\Delta+V\) (in short, the perturbed Krein Laplacian) defined on \(C^\infty_0(\Omega)\), where \(V\) is measurable, bounded and nonnegative, in a bounded open set \(\Omega\subset\mathbb{R}^n\) belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class \(C^{1,r}\), \(r>1/2\). (Contrary to other uses of the notion of “domain”, a domain in this survey denotes an open set without any connectivity hypotheses. In addition, by a “smooth domain” we mean a domain with a sufficiently smooth, typically, a \(C^\infty\)-smooth, boundary.) In particular, in the aforementioned context we establish the Weyl asymptotic formula \[ \#\{j\in\mathbb{N}\,|\,\lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega|\,\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) \, \text{ as }\, \lambda\to\infty, \] where \(v_n=\pi^{n/2}/ \Gamma((n/2)+1)\) denotes the volume of the unit ball in \(\mathbb{R}^n\), \(|\Omega|\) denotes the volume of \(\Omega\), and \(\lambda_{K,\Omega,j}\), \(j\in\mathbb{N}\), are the non-zero eigenvalues of \(H_{K,\Omega}\), listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein-von Neumann extension of \(-\Delta+V\) defined on \(C^\infty_0(\Omega)\)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of V. A. Kozlov [J. Sov. Math. 35, 2180–2193 (1986); translation from Probl. Mat. Anal. 9, 34–56 (1984; Zbl 0632.47014)]. Our work builds on that of Grubb [loc. cit.], who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by A. Alonso and B. Simon [J. Oper. Theory 4, 251–270 (1980; Zbl 0467.47017); addenda ibid. 6, 407 (1981; Zbl 0476.47019)] pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.

We also study certain exterior-type domains \(\Omega = \mathbb{R}^n\backslash K\), \(n\geq 3\), with \(K\subset \mathbb{R}^n\) compact and vanishing Bessel capacity \(B_{2,2} (K) = 0\), to prove equality of Friedrichs and Krein Laplacians in \(L^2(\Omega; d^n x)\), that is, \(-\Delta|_{C_0^\infty(\Omega)}\) has a unique nonnegative self-adjoint extension in \(L^2(\Omega; d^n x)\).

For the entire collection see [Zbl 1264.00036].

In the concrete case where \(S=\overline{-\Delta|_{C_0^\infty(\Omega)}}\) in \(L^2(\Omega; d^n x)\) for \(\Omega\subset\mathbb{R}^n\) an open, bounded (and sufficiently regular) set, this recovers, as a particular case of a general result due to G. Grubb [J. Oper. Theory 10, 9–20 (1983; Zbl 0559.47035)], that the eigenvalue problem for the Krein Laplacian \(S_K\) (i.e., the Krein-von Neumann extension of \(S\)), \[ S_K v = \lambda v, \quad \lambda \neq 0, \] is in one-to-one correspondence with the problem of the buckling of a clamped plate, \[ (-\Delta)^2u=\lambda (-\Delta) u \;\text{ in } \;\Omega, \quad \lambda \neq 0, \quad u\in H_0^2(\Omega), \] where \(u\) and \(v\) are related via the pair of formulas \[ u = S_F^{-1} (-\Delta) v, \quad v = \lambda^{-1}(-\Delta) u, \] with \(S_F\) the Friedrichs extension of \(S\).

This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.).

In the second and principal part of this survey, we study spectral properties for \(H_{K,\Omega}\), the Krein-von Neumann extension of the perturbed Laplacian \(-\Delta+V\) (in short, the perturbed Krein Laplacian) defined on \(C^\infty_0(\Omega)\), where \(V\) is measurable, bounded and nonnegative, in a bounded open set \(\Omega\subset\mathbb{R}^n\) belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class \(C^{1,r}\), \(r>1/2\). (Contrary to other uses of the notion of “domain”, a domain in this survey denotes an open set without any connectivity hypotheses. In addition, by a “smooth domain” we mean a domain with a sufficiently smooth, typically, a \(C^\infty\)-smooth, boundary.) In particular, in the aforementioned context we establish the Weyl asymptotic formula \[ \#\{j\in\mathbb{N}\,|\,\lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega|\,\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) \, \text{ as }\, \lambda\to\infty, \] where \(v_n=\pi^{n/2}/ \Gamma((n/2)+1)\) denotes the volume of the unit ball in \(\mathbb{R}^n\), \(|\Omega|\) denotes the volume of \(\Omega\), and \(\lambda_{K,\Omega,j}\), \(j\in\mathbb{N}\), are the non-zero eigenvalues of \(H_{K,\Omega}\), listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein-von Neumann extension of \(-\Delta+V\) defined on \(C^\infty_0(\Omega)\)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of V. A. Kozlov [J. Sov. Math. 35, 2180–2193 (1986); translation from Probl. Mat. Anal. 9, 34–56 (1984; Zbl 0632.47014)]. Our work builds on that of Grubb [loc. cit.], who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by A. Alonso and B. Simon [J. Oper. Theory 4, 251–270 (1980; Zbl 0467.47017); addenda ibid. 6, 407 (1981; Zbl 0476.47019)] pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.

We also study certain exterior-type domains \(\Omega = \mathbb{R}^n\backslash K\), \(n\geq 3\), with \(K\subset \mathbb{R}^n\) compact and vanishing Bessel capacity \(B_{2,2} (K) = 0\), to prove equality of Friedrichs and Krein Laplacians in \(L^2(\Omega; d^n x)\), that is, \(-\Delta|_{C_0^\infty(\Omega)}\) has a unique nonnegative self-adjoint extension in \(L^2(\Omega; d^n x)\).

For the entire collection see [Zbl 1264.00036].

Reviewer: Michael Perelmuter (Kyïv)

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47F05 | General theory of partial differential operators |

47A10 | Spectrum, resolvent |

35J25 | Boundary value problems for second-order elliptic equations |

35J40 | Boundary value problems for higher-order elliptic equations |

35P15 | Estimates of eigenvalues in context of PDEs |

35P05 | General topics in linear spectral theory for PDEs |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |