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Register a new userA Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
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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, $Sgeq varepsilon I_{mathcal{H}}$ for some $varepsilon >0$ in a Hilbert space $mathcal{H}$ to an abstract buckling problem operator. 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 $Omegasubsetmathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.
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We study spectral properties for $H_{K,Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-Delta+V$ defined on $C^infty_0(Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Omegasubsetmathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula [ #{jinmathbb{N} | lambda_{K,Omega,j}leqlambda} = (2pi)^{-n} v_n |Omega| lambda^{n/2}+Obig(lambda^{(n-(1/2))/2}big) {as} lambdatoinfty, ] where $v_n=pi^{n/2}/ Gamma((n/2)+1)$ denotes the volume of the unit ball in $mathbb{R}^n$, and $lambda_{K,Omega,j}$, $jinmathbb{N}$, 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 Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.
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, $Sgeq epsilon I_{mathcal{H}}$ for some $epsilon >0$ in a Hilbert space $mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=bar{-Delta|_{C_0^infty(Omega)}}$ in $L^2(Omega; d^n x)$ for $Omegasubsetmathbb{R}^n$ an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, 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 eq 0, ] is in one-to-one correspondence with the problem of {em the buckling of a clamped plate}, [ (-Delta)^2u=lambda (-Delta) u text{in} Omega, quad lambda eq 0, quad uin 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.).
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $nin mathbb N$ infinitely many elements of the class have $n$-dimensional null space.
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
Let $Omega$ be a curvilinear polygon and $Q^gamma_{Omega}$ be the Laplacian in $L^2(Omega)$, $Q^gamma_{Omega}psi=-Delta psi$, with the Robin boundary condition $partial_ u psi=gamma psi$, where $partial_ u$ is the outer normal derivative and $gamma>0$. We are interested in the behavior of the eigenvalues of $Q^gamma_Omega$ as $gamma$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^gamma_Omega$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $partial Omega$. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of $Q^gamma_Omega$ for a threshold depending on $gamma$, and show that the leading term is the same as for smooth domains.
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