[Paper Review] Krein resolvent formulas for elliptic boundary problems in nonsmooth domains

Gerd Grubb|ArXiv.org|Oct 15, 2008

Spectral Theory in Mathematical Physics15 references39 citations

TL;DR

This paper establishes Kre’in resolvent formulas for second-order strongly elliptic operators with non-self-adjoint boundary conditions on $C^{1,1}$-smooth domains by extending the $M$-function formalism using a nonsmooth pseudodifferential boundary operator calculus. The key result is that the $M$-function is a generalized pseudodifferential operator of order $-1$ with Hölder-smooth symbols, enabling a full Kre’in formula for non-self-adjoint extensions in nonsmooth settings.

ABSTRACT

The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with $C^{1,1}$ Hölder smoothness, by use of a recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols.

Motivation & Objective

• To extend the theory of $M$-functions and Kre’in resolvent formulas to non-self-adjoint elliptic boundary problems on nonsmooth domains.
• To address the lack of a systematic framework for $M$-functions in non-self-adjoint and nonsmooth settings, particularly for $C^{1,1}$-regular boundaries.
• To apply the recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols to construct $M$-functions as generalized pseudodifferential operators.
• To establish the validity of the Kre’in resolvent formula in nonsmooth domains, including exterior domains and perturbed half-spaces, under parameter-ellipticity and Hölder regularity assumptions.

Proposed method

• Utilizes the abstract framework of adjoint pairs of closed densely defined operators $A_{\min}$, $A'_{\min}$ and their extensions $\widetilde{A} \in \mathcal{M}$, with $A_\gamma$ invertible.
• Applies the abstract Green's formula to relate the resolvent difference to the $M$-function via the operator $T$ defined on the kernel spaces $Z$ and $Z'$.
• Employs the nonsmooth pseudodifferential boundary operator calculus of Abels [3] to handle $C^{1,1}$-regular boundaries and define $M_L(\lambda)$ as a generalized $\psi$do.
• Constructs a parametrix for the system $\mathcal{A}(\lambda) = \begin{pmatrix} A - \lambda \\ \nu_1 - C\gamma_0 \end{pmatrix}$ on a time-periodic extension $\widehat{\Omega} = \Omega \times S^1$, enabling $\lambda$-dependent analysis.
• Uses parameter-ellipticity of $L^\lambda = C - P^{\lambda}_{\gamma_0,\nu_1}$ on a spectral ray to ensure invertibility of $L^\lambda$ for large $\lambda$, leading to $M_L(\lambda) = -(L^\lambda)^{-1}$.
• Applies Agmon's principle and symbol calculus to show that $M_L(\lambda)$ is a generalized pseudodifferential operator of order $-1$ with Hölder-smooth symbols when $C$ is a first-order differential operator with $C^{0,1}$-regularity.

Experimental results

Research questions

• RQ1 Can the $M$-function be defined as a generalized pseudodifferential operator on $C^{1,1}$-smooth boundaries for non-self-adjoint extensions of strongly elliptic operators?
• RQ2 Does the Kre’in resolvent formula hold in nonsmooth domains when the boundary operator has Hölder-continuous coefficients?
• RQ3 What conditions on the boundary operator $C$ and the associated $L^\lambda$ ensure the invertibility and regularity of the $M$-function for large spectral parameters $\lambda$?
• RQ4 How does the nonsmooth pseudodifferential boundary operator calculus allow the extension of $M$-function theory beyond smooth domains?
• RQ5 To what extent can the $M$-function be represented as a sum of an elliptic $\psi$do of order $-1$ and a lower-order remainder in the nonsmooth setting?

Key findings

• The $M$-function $M_L(\lambda)$ is shown to be a generalized pseudodifferential operator of order $-1$ with Hölder-smooth symbols of class $C^{0,1}$ when the boundary operator $C$ is a first-order differential operator with $C^{0,1}$-regularity.
• For large $\lambda$ on a spectral ray $\lambda = -\mu^2 e^{i\theta}$, $M_L(\lambda) = -(L^\lambda)^{-1}$ is the sum of an elliptic $\psi$do of order $-1$ and a lower-order remainder, ensuring regularity and invertibility.
• The domain of the extension $\widetilde{A}$ satisfies $D(\widetilde{A}) \subset H^2(\Omega)$, and the adjoint $\widetilde{A}^*$ is similarly defined when $C^*$ has $C^{0,1}$-regularity.
• When $A$ and $L$ are formally selfadjoint, the extension $\widetilde{A}$ is selfadjoint, providing a complete realization of the Kre’in formula in the selfadjoint case on $C^{1,1}$ domains.
• The parametrix construction for $\mathcal{A}(\lambda)$ on the time-periodic domain $\widehat{\Omega}$ yields an $O(\langle\mu\rangle^{-\theta})$ error estimate, ensuring the existence of a true inverse for $L^\lambda$ and thus for $M_L(\lambda)$.
• The theory applies not only to bounded domains but also to exterior domains and perturbed half-spaces, extending the scope of Kre’in formula applications to broader geometric settings.