- Research Article
- Open Access
A-Harmonic Equations and the Dirac Operator
© Craig A. Nolder. 2010
- Received: 2 January 2010
- Accepted: 14 March 2010
- Published: 26 April 2010
We show how -harmonic equations arise as components of Dirac systems. We generalize -harmonic equations to -Dirac equations. Removability theorems are proved for solutions to -Dirac equations.
- Weak Solution
- Harmonic Function
- Dirac Equation
- Dirac Operator
- Clifford Algebra
This component is the scalar (real) part of the Dirac system, under appropriate identifications. Hence any real-valued solution to the Dirac system is an -harmonic function. As such, the class of -harmonic functions which are also solutions of the Dirac system are a special class of -harmonic functions. See Section 3 for a detailed discussion. As an application, we show that a result concerning removable singularities for -harmonic functions satisfying a Lipschitz condition or of bounded mean oscillation extends to Clifford valued solutions to corresponding Dirac equations. The result also holds for functions of a certain order of growth. This seems to be new even in the case of an -harmonic function.
In Section 2, we present preliminaries about Clifford algebra along with definitions and notations. In Section 3, we introduce -Dirac equations and show the correspondence with -harmonic equations. The Caccioppoli estimate for solutions to -Dirac equations appears in Section 4 and the removability theorems are in Section 5 along with references. For other recent work on nonlinear Dirac equations, see [1–6].
where is an orthonormal basis of with the relation We write for the identity. The dimension of is . We have an increasing tower The Clifford algebra is a graded algebra as , where are those elements whose reduced Clifford products have length
For , denotes the scalar part of , that is, the coefficient of the element .
Throughout, is a connected and open set with boundary . A Clifford-valued function can be written as where each is real-valued and are reduced products. The norm used here is given by This norm is submultiplicative, .
Also Here is the Laplace operator which operates only on coefficients. A function is monogenic when
Throughout, is a cube in with volume . We write for the cube with the same center as and with sidelength times that of . For , we write for the space of Clifford-valued functions in whose coefficients belong to the usual space. Also, is the space of Clifford valued functions in whose coefficients as well as their first distributional derivatives are in . We also write for , where the intersection is over all compactly contained in We similarly write Moreover, we write for the space of monogenic functions in
The local space is similarly defined. Notice that if is monogenic, then if and only if Also it is immediate that
for some The exponent will represent this exponent throughout.
for all with compact support.
See  for the theory of -harmonic equations.
This motivates the following definition for Dirac systems of higher-order Clifford valued functions. We use the Clifford conjugation The product defines a Clifford-valued inner product.
Moreover, the scalar part of this Clifford inner product is the usual inner product in , when and are identified as vectors. We continue to use the Clifford notation for this scalar product. This conjugation compensates for the minus sign in (3.5) and in higher-order Clifford products. Moreover, the conjugation can be incorporated into an integration by parts formula.
These equations were introduced and their conformal invariance was studied in .
In the case of the -Dirac equation, , when , and , , are solutions to (3.13). Notice that a monogenic function is trivially a solution to (3.13) and if is a solution to (3.13), then so is for any monogenic function
Hence when is a function, (3.13) implies that is a harmonic field and locally there exists a harmonic function such that If is invertible, then Hence regularity of implies regularity of the solution . Notice that if , then , where
In general, -harmonic functions do not have such regularity. This suggests the study of the scalar part of the system equation (3.13) in general. Indeed a Caccioppoli estimate holds for solutions to the scalar part of (3.13). This is the topic of the next section.
Hence we see a special class of -harmonic functions, namely, real-valued solutions to the system (3.13). This class should have special properties.
On the other hand, results about -harmonic functions suggest possible properties of general solutions to (3.13). In this paper, we present one such extension, that of removable sets. The essential ingredient in the proof is the Caccioppoli estimate.
Next is a Caccioppoli estimate for solutions to (3.13). This result appears in . We give the short proof here for completeness.
Choose , , in and .
For monogenic functions with modulus of continuity , sets of -Hausdorff measure are removable . For Hölder continuous analytic functions, see . Sets satisfying a certain geometric condition related to Minkowski dimension are shown to be removable for -harmonic functions in Hölder and bounded mean oscillation classes in . In the case of Hölder continuity, this was sharped in  to a precise condition for removable sets for -harmonic functions in terms of Hausdorff dimension.
We show here that sets satisfying a generalized Minkowski-type inequality, similar to that in , are removable for solutions to the -Dirac equation which satisfy a certain oscillation condition. The following definition is motivated by the fact that real-valued functions satisfying various regularity properties are characterized by this definition. We explain below.
Assume that , and that
The infimum over monogenic functions is natural since they are trivially solutions to an -Dirac equation just as constants are solutions to an -harmonic equation. If is a function and , then (5.1) is equivalent to the usual definition of the bounded mean oscillation when and (5.1) is equivalent to the usual local Lipschitz condition when . Moreover, at least when is a solution to an -harmonic equation, (5.1) is equivalent to a local order of growth condition when ; see [9, 14]. In these cases, the supremum is finite if we choose to be the average value of the function over the cube . It is easy to see that in condition (5.1) the expansion factor "2" can be replaced by any factor greater than 1.
If the coefficients of an -Dirac solution are of bounded mean oscillation, local Hölder continuous, or of a certain local order of growth, then is in an appropriate oscillation class; see .
Notice that monogenic functions satisfy (5.1) just as the space of constants is a subspace of the bounded mean oscillation and Lipschitz spaces of real-valued functions.
We remark that it follows from Hölder's inequality that if and if is of -oscillation, then is of -oscillation.
The following lemma shows that Definition 5.1 is independent of the expansion factor of the cube.
We use a Whitney decomposition of . The decomposition consists of closed dyadic cubes with disjoint interiors which satisfy
(c) when is not empty.
Here is the Euclidean distance between and the boundary of ; see .
We now state the removability result.
then extends to a solution of the A-Dirac equation in .
Next let and assume that . Also let be those cubes with
Since is a solution in ,
Since and as it follows that as
Since and as , we have that as Hence .
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