[Paper Review] Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems

Helmut Abels, Gerd Grubb+1 more|University of Regensburg Publication Server (University of Regensburg)|Jan 1, 2010

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes a pseudodifferential boundary operator calculus for nonselfadjoint second-order elliptic operators on domains with $B^{rac{3}{2}}_{p,2}$-smooth boundaries ($p>2(n-1)$) and coefficients in $H^1_q$ spaces ($q>n$), providing a full characterization of selfadjoint extensions and deriving Kreïn-type resolvent formulas. The method leverages symbol smoothing and parametrix constructions to achieve regularity and mapping properties in Sobolev spaces of negative order, extending classical results to nonsmooth settings.

ABSTRACT

For a strongly elliptic second-order operator $A$ on a bounded domain $Ω\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(Ω)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of Hölder $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show how it can be done for domains with $B^\frac32_{p,2}$-smoothness and operators with $H^1_q$-coefficients, for suitable $p>2(n-1)$ and $q>n$. In particular, Kre\uın-type resolvent formulas are established in such nonsmooth cases. Some unbounded domains are allowed.

Motivation & Objective

• To extend extension theory and Kreïn-type resolvent formulas to nonsmooth elliptic boundary value problems beyond smooth or $C^{1,1}$ domains.
• To address the challenge of mapping properties in low-regularity Sobolev spaces, including negative order spaces, on domains with limited boundary smoothness.
• To develop a framework using pseudodifferential calculus that preserves ellipticity through principal symbol analysis and handles lower-order remainders.
• To characterize realizations of nonselfadjoint second-order operators on domains with $B^{rac{3}{2}}_{p,2}$-smooth boundaries and $H^1_q$-coefficients.
• To generalize existing results on the Laplacian to general second-order elliptic operators under weaker regularity assumptions on both domain and coefficients.

Proposed method

• Utilizes pseudodifferential boundary operator calculus to decompose operators into principal parts and lower-order remainders, enabling precise regularity analysis.
• Applies symbol smoothing techniques to handle $C^{rac{3}{2}+rac{1}{p}}$-regularity in the boundary parametrization, ensuring compatibility with ellipticity and mapping properties.
• Employs parametrix constructions for the Dirichlet-to-Neumann operator and Poisson solution operators, establishing continuity in Sobolev spaces of negative order.
• Uses order-reducing operators $\Lambda_{-,+}^r$ and $\Lambda_0^s$ to extend class concepts to negative orders, allowing analysis of trace and boundary operators in low-regularity settings.
• Applies the theory of $x$-forms and boundary symbol calculus to derive mapping properties for operators acting on $H^s(\mathbb{R}^n_+)$ and $H^s(\mathbb{R}^{n-1})$.
• Establishes continuity of the residual operator $\mathcal{R} = \mathcal{A}\mathcal{B}^0 - I$ in Sobolev spaces under specified regularity and order constraints.

Experimental results

Research questions

• RQ1 Can Kreïn-type resolvent formulas be derived for nonselfadjoint second-order elliptic operators on domains with $B^{rac{3}{2}}_{p,2}$-smooth boundaries?
• RQ2 How can pseudodifferential calculus be adapted to maintain ellipticity and regularity when the boundary is not $C^{1,1}$ or $C^{\frac{3}{2}+\varepsilon}$?
• RQ3 What mapping properties do resolvents, Poisson operators, and Dirichlet-to-Neumann maps possess in Sobolev spaces of negative order on such domains?
• RQ4 How can the theory of boundary triples and $m$-functions be extended to nonsmooth domains with $B^{rac{3}{2}}_{p,2}$-regularity and $H^1_q$-coefficients?
• RQ5 To what extent can the parametrix method be used to characterize extensions and resolvents in low-regularity function spaces?

Key findings

• The paper establishes a full characterization of selfadjoint extensions for nonselfadjoint second-order elliptic operators on $C^{rac{3}{2}+\varepsilon}$-type domains with $B^{rac{3}{2}}_{p,2}$-smooth boundaries, where $p>2(n-1)$.
• A pseudodifferential boundary operator calculus is developed that allows decomposition into principal parts and lower-order remainders, ensuring ellipticity and regularity in Sobolev spaces.
• The resolvent difference between a general realization and a reference operator is expressed via boundary operators, yielding a Kreïn-type resolvent formula in the nonsmooth setting.
• Mapping properties of the resolvent, Poisson solution operators, and Dirichlet-to-Neumann maps are established in Sobolev spaces of negative order, extending classical results.
• The theory applies to unbounded domains and allows for $H^1_q$-coefficients with $q>n$, ensuring sufficient regularity for multiplication and trace theorems.
• The use of order-reducing operators extends the class concept to negative orders, enabling analysis of trace and boundary operators in low-regularity function spaces.