Generalized Dirac Operators on Nonsmooth mitream/ Generalized Dirac Operators on Nonsmooth

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  • Generalized Dirac Operators on Nonsmooth

    Manifolds and Maxwell’s Equations

    Marius Mitrea ∗

    Abstract

    We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj- Calderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding Lp-decompositions. Our approach relies on an extension of the classical Calderón- Zygmund theory of singular integral operators which allows one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell’s equations, the main novelty is the treatment of the corresponding electro-magnetic bound- ary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.

    1 Introduction

    Since the introduction in 1928 by the physicist P. M. Dirac of a first-order linear differential operator whose square is the wave operator, Dirac type operators have become of central importance in many branches of mathematics such as PDE’s, differential geometry and topology. See, e.g., the monographs [6], [4], [17], [28], [44] and the references therein. At the heart of the matter lies the fact that elliptic systems of the first order generalizing the classical Cauchy-Riemann system give rise to a natural, rich function theory.

    The general aim of this paper is to develop such a function theory for a general Dirac operator D on a manifold M under minimal smoothness assumptions. A special emphasis is placed on studying Hardy type spaces associated with D,

    Hp(Ω, D) := {u; Du = 0 in Ω, N (u) ∈ Lp(∂Ω)}, (1.1)

    in an arbitrary Lipschitz subdomain Ω of M ; here N (u) stands for the nontangential maximal function of u (more precise definitions will be given shortly).

    When the underlying domain is the unit disk or even a more general but smooth domain in the complex plane, this topic is classical and a great deal of information is known; cf. the excellent accounts in [12], [18]. The study of Hardy spaces in nonsmooth subdomains of the complex plane originates in [25], based on conformal mapping techniques. Subsequent developments, emphasizing real methods, are in [13], [26], [11]. One suitable replacement for the Cauchy-Riemann operator ∂x + i∂y in higher dimensions is the Dirac operator

    ∑ ej∂j within the context of a Clifford algebra generated by the

    ∗Supported in part by NSF grant DMS #9870018 1991 Mathematics Subject Classification. Primary 31C12, 42B20, 35F15, 42B30; Secondary 58G20, 42B25, 78A25. Key words. Dirac operators, Hardy spaces, Maxwell’s equations, Lipschitz domains.

    1

  • (anticommuting) imaginary units {ej}j . Hardy spaces in Lipschitz domains of Rn associated with such operators have been studied in [29], [17], [35].

    In the present paper we continue this line of research and take the next natural step by considering generalized Dirac operators with variable coefficients in the context of Lipschitz domains on manifolds. These are first order, elliptic differential operators so that

    D and D∗ have the unique continuation property. (1.2)

    In fact, our entire theory of Hardy spaces in § 4 is developed based solely on ellipticity and this unique continuation property assumption. As is well known, Dirac operators naturally associated with Clifford algebra structures automatically satisfy (1.2). Thus, from this perspective, the primary role of Clifford algebras is to provide natural examples of operators D for which (1.2) holds. On the other hand, each operator D satisfying (1.2) as well as certain extra algebraic hypotheses arises precisely in this fashion; cf. the discussion in § 5 for a more precise statement.

    The highlights of the Hardy space theory we develop at this level of generality include two decom- position theorems which we now proceed to describe. Let Ω be a Lipschitz domain in the manifold M and set Ω+ := Ω, Ω− := M \ Ω̄. Also, consider the boundary Hardy spaces Hp±(∂Ω, D) := {u|∂Ω; u ∈ Hp(Ω±, D)}. As is well known, when D := ∂̄, the Cauchy-Riemann operator and M := C, the Plemelj- Calderón decomposition

    Lp(∂Ω) = Hp−(∂Ω, D)⊕H p +(∂Ω, D), 1 < p 0 so that

    Lp(∂Ω) = Hp(∂Ω, D)⊕ iσ(D∗; ν)Hp(∂Ω, D∗) (1.6)

    2

  • for any 2− � < p < 2 + �, where the sum is direct and topological. Moreover, when σ(DD∗; ξ) is scalar and ν ∈ vmo (∂Ω), the class of functions of vanishing mean oscillations on ∂Ω, then (1.6) is valid for any 1 < p

  • When all structures involved are smooth and solutions are sought in a sufficiently regular class of functions, such problems are regular elliptic and, hence, Fredholm solvable (cf. [44]). In this scenario, pseudodifferential operator techniques play a crucial role. The nature of the problem at hand changes when the smoothness assumptions are significantly relaxed. In particular, the method of layer potentials (which we employ in this paper) leads to considering singular integrals in place of pseudodifferential operators. One notable difficulty in the case we are interested, i.e. Lipschitz boundaries, metric tensors with a very limited amount of smoothness, is the absence of an algebra structure and the lack of a symbolic calculus within the class of general singular integral operators.

    Our approach utilizes an array of tools from harmonic analysis which have been successful in the treatment of second-order, constant coefficient elliptic boundary problems in Lipschitz domains of the Euclidean space. See [25] and the references cited there for a survey of the state of the art in this field up to early 1990’s. A more recent line of research, initiated in [39], [34] (and further developed in [40], [41], [42]), is the use of layer potentials in order to solve boundary problems for general second-order, variable coefficient, elliptic systems in non-smooth manifolds. The present paper, dealing with variable first-order elliptic systems, is a natural continuation of this work. A basic goal of this program is to achieve an “elliptization” of (the non-coercive) Maxwell’s equations

    (Maxwell)

    { dE − ikH = 0 in Ω,

    δH + ikE = 0 in Ω, (1.12)

    with N (E), N (H) ∈ Lp(∂Ω) and Etan or Enor prescribed in Lp(∂Ω), by embedding (1.12) into a larger, “half-Dirichlet” problem for a suitable Dirac type operator. Indeed, as explained in last part of our paper, (1.12) can be understood as a particular manifestation of the half-Dirichlet problem for Maxwell-Dirac operator (1.10). In this case, under the identification u = H − i dt · E, (1.11) becomes

    ( Generalized

    Maxwell

    )  E,H ∈ C0(Ω,GCM ),

    δE + dE − ikH = 0 in Ω,

    δH + dH + ikE = 0 in Ω, (1.13)

    with N (E), N (H) ∈ Lp(∂Ω) and Etan, Htan or Enor, Hnor prescribed in Lp(∂Ω), and we give necessary and sufficient conditions on the boundary data guaranteeing that (1.11) and (1.13) are equivalent. Of course, for this result to have practical value, we first need to give a thorough solution to (1.11) to begin with. See Theorem 7.1 for this.

    Following the work in the three-dimensional case in [37], the boundary problem (1.12) has been first solved on Lipschitz domains in [36], [20], [34] via integral equation methods. The philosophy of the approach in these papers is to reduce (1.12) to boundary problems for the (perturbed) Hodge-Laplacian with absolute and relative boundary conditions. In particular, one transforms it into a second order PDE. Here we provide an alternative approach based on working directly with the more general, first order system (1.13).

    Ultimately, implementing this idea requires understanding connections between Dirac operators and Maxwell’s equations in non-smooth domains. In the flat, Euclidean setting and for constant coefficient operators, this direction of research has been initiated in [31] and [30]. A basic ingredient in [31] is a Rellich type estimate to the effect that

    ‖utan‖L2(∂Ω) ≈ ‖unor‖L2(∂Ω), (1.14)

    for two-sided monogenic functions in Ω, i.e. elements in H2(Ω, IDk) which are also annihilated by the action of IDk to the right. A step forward in the direction of dealing with one-sided Clifford modules

    4

  • (as in the case of manifolds) was taken in [30] where the monogenicity assumptions were relaxed. In the present paper we continue this program by showing that (1.14) holds for any u ∈ H2(Ω, IDk) and Ω arbitrary Lipschitz subdomain of a Riemannian manifold.

    In somewhat greater detail, the organization of the paper is as follows. Section 2 contains a discussion of the global invertibility properties of D and Laplacians associated with D. A function theory associated with a general, first-order, variable coefficient elliptic system, with a special emphasis on Hardy type spaces and Cauchy like operators in Lipschitz domains is developed in §3-4. How Dirac type operators fit in this general framework makes the subject of Section 5. Boundary value problems for Hodge-Dirac operators of the form d+αδ, α ∈ R, in Lipschitz subdomains of Riemannian manifolds are studied in Section 6. The first part of Section 7 is reserved for a similar dis