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Unique Continuation for First Order Systems of PDEs

Shiferaw Berhanu

Communicated by Notices Associate Editor Daniela De Silva

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1. Introduction

This article is a survey of some of the results on weak and strong unique continuation for systems of linear and nonlinear first order partial differential equations. The linear PDEs arise as sections of a vector subbundle of the complexified tangent bundle of a connected manifold . The bundle satisfies the involutivity condition

which means that for any sections of , the Lie bracket is also a section of . We will always assume that is locally integrable, that is, the orthogonal of in the complexified cotangent bundle is locally generated by exact forms. In this case, if the fiber dimension of over is , each point in has a neighborhood such that if is a basis of over , then there exist sections , …, which are solutions of

and the differentials , …, are linearly independent over at each point of . We will call a complete set of first integrals on .

When the bundle is locally integrable, then it satisfies the involutivity condition 1.1 and we will refer to the pair as a locally integrable structure.

In such a structure, given any point , there are local coordinates , …, , , …, vanishing at such that is generated locally by a basis of the form ( and are as before)

For the nonlinear systems

that we consider, local coordinates () can be found in which the equations take the form

2. The Weak and Strong Unique Continuation Properties

It will be instructive to consider some examples of locally integrable structures.

Example 2.1.

Let , …, be smooth, linearly independent real vector fields on a domain such that the Lie bracket is in the linear span of , …, . Let denote the subbundle of generated by , …, . By the Frobenius theorem, each is a center of local coordinates , …, in which the bundle is locally generated by , . Hence is locally integrable and, in these coordinates, any solution has the form .

Example 2.2.

Let be a domain, and let be the bundle generated by

The coordinate functions , are a complete set of first integrals and so is locally integrable. The solutions are holomorphic functions.

Example 2.3.

Let , …, be linearly independent vector fields with real analytic, complex-valued coefficients on a domain . Assume that the brackets are in the linear span of , …, , and let denote the subbundle generated by the . By the holomorphic version of the Frobenius theorem, is locally integrable (see 9).

Example 2.4 (Embedded CR submanifolds).

Let be a smooth real submanifold of dimension . is called a CR manifold if for each there are real-valued, smooth defining functions , …, defined on a neighborhood of such that the one-forms

are linearly independent on . Here, . Let be the subbundle of generated by , and set

Since the forms are linearly independent, is a subbundle of of fiber dimension . The restrictions of the coordinates to form a complete set of first integrals and so is locally integrable.

In general, the forms of locally defining functions may not be linearly independent and so may not be a bundle. However, when is a hypersurface, it is a bundle. When is a CR pair, the solutions are called CR functions or CR distributions. Examples of CR functions are provided by the restrictions to of holomorphic functions. However, CR functions may not even be continuous. For example, if is the graph

where , and smooth, then the function (with the principal branch of the square root) is a continuous CR function which is not differentiable at the point , where . If , then any function of is a CR function.

Example 2.5 (Tube structures).

Let and denote the coordinates in and , respectively. Let , be smooth functions on an open set and define on

The differentials , …, are linearly independent everywhere on and their orthogonal in is a bundle which has the basis

Thus is locally integrable, and it generalizes Example 2.2. Note that when the map is an immersion, then can be identified with a CR submanifold.

A particular case with and leads to the planar vector field

which is called the Mizohata vector field and is the simplest nonsolvable partial differential operator in the sense that there exist smooth functions in a neighborhood of the origin for which the equation has no solution (not even a distribution solution) in any neighborhood of the origin.

Remark 2.1.

There are involutive bundles which are not locally integrable. For example, in 19 Nirenberg constructed a vector field in the plane of the form

with smooth vanishing to infinite order at the origin with the property that for any solution of near the origin, .

Remark 2.2.

In the opening graphic which is taken from 18, defines a vector field in the plane with first integral and the picture illustrates the domain of extendability of any solution of in the unit disc.

Definition 2.1.

The locally integrable structure is said to satisfy the weak unique continuation property if any solution that vanishes on a nonempty open subset vanishes on .

Definition 2.2.

The locally integrable structure satisfies the strong unique continuation property if any solution that is flat at a point vanishes on .

The validity of the weak unique continuation property both for linear and nonlinear systems is connected with the notion of Sussmann’s orbits (20) which are very useful geometric objects associated with a given family of real vector fields. To describe them briefly, let be a set of locally defined smooth vector fields on a manifold . Each element of is defined on some open subset of and we assume that the union of the domains of the members of is . We will say two points and in are related if there is a curve with the properties:

(i) ,

(ii) there exist and such that for each , is an integral curve of or .

The equivalence classes of this relation are called the Sussmann orbits of . In 20, it was shown that these orbits can be equipped with a topological and differential structure that makes them submanifolds of .

For a simple example, consider the classical Frobenius foliation which arises from a manifold and a smooth real subbundle of which is closed under brackets. In this case, is a disjoint union of submanifolds each of dimension ( the fiber dimension of and they are integral manifolds of . These submanifolds are the Sussmann orbits of the family that consists of the smooth sections of . In general, even locally, the dimension of Sussmann’s orbits may not be constant. Moreover, the tangent space of an orbit may contain vectors that are not in the Lie algebra generated by the elements of (see 20, 9). Thus the concept of Sussmann’s orbits is a substantial generalization of the Frobenius foliation.

We can now state a result on the weak unique continuation property for linear systems. The theorem is due to F. Treves (21). We will present a proof that appeared in 12.

Theorem 2.1.

Let be a locally integrable structure, and set

If a solution on vanishes in a neighborhood of , then it vanishes in a neighborhood of the Sussmann orbit of through . Thus the support of is a union of orbits of . In particular, if is an orbit, then satisfies the weak unique continuation property.

Example 2.6.

Let be locally integrable and . Suppose at each , the linear span of all of the repeated brackets of sections of equals . Then is the only orbit of and so by Theorem 2.1, the weak unique continuation property holds for .

Remark 2.3.

There are examples of locally integrable structures where is the only orbit of although the hypothesis in Example 2.6 is not satisfied.

Example 2.7.

In the work 14 P. Cohen gave an example of a smooth vector field

in the plane with a smooth solution on of whose support . The support of such a solution is not a union of the orbits of . This example shows that Theorem 2.1 may not hold for involutive structures that are not locally integrable.

3. Proof of Theorem 2.1

The proof of Theorem 2.1 will use the uniqueness in the noncharacteristic Cauchy problem for locally integrable structures which in turn is a consequence of the celebrated approximation theorem of Baouendi and Treves (5). Given a locally integrable structure and a complete set of first integrals on an open subset , if is a holomorphic function on a neighborhood of , then the function is a solution on . In general, even locally, solutions may not have such a form. The approximation theorem states that any solution in a locally integrable structure can be locally approximated by solutions of the above form. Moreover, the theorem shows that a solution is determined by its restriction to certain -dimensional (, ) submanifolds called maximally real submanifolds. These submanifolds are generalizations of the totally real submanifolds of with maximal dimension.

Definition 3.1.

Let be a locally integrable structure, , . A submanifold is called maximally real if for each ,

Locally, there is a good and useful description of such submanifolds. Observe first that a submanifold is maximally real if and only if for any complete set of first integrals , …, on a neighborhood of in , the restrictions of the to have linearly independent differentials. If is small enough, we can find local coordinates , …, , , …, that vanish at and a complete set of first integrals , …, such that

and

The approximation theorem of Baouendi and Treves can be stated as follows.

Theorem 3.1 (5).

Let be a locally integrable structure where is open, and let , …, be a complete set of first integrals near a point . Then there exists a neighborhood of such that if is any continuous solution, there exists a sequence of entire functions that satisfy

Remark 3.1.

The convergence in the theorem also holds for functions or distributions in various function spaces (see 9).

The entire functions in the theorem have an explicit form. Given , let be a maximally real submanifold through . Choose local coordinates , …, , , …, vanishing at on a neighborhood of such that

is a complete set of first integrals on , with real-valued, , .

We may assume that . Let with for and for .

For , , …, define

where . Theorem 3.1 is proved by showing that converges uniformly to on for small enough, but independent of .

Theorem 3.1 has the following immediate consequences.

Corollary 3.1.

Let be a locally integrable structure and a maximally real submanifold. If is a continuous solution on and vanishes on , then it vanishes on a neighborhood of .

Corollary 3.2.

Let be a locally integrable structure and a noncharacteristic hypersurface in the sense that at each point . If is a continuous solution on and it vanishes on , then it vanishes on a neighborhood of .

Corollary 3.2 is proved using Corollary 3.1 and the fact that a noncharacteristic hypersurface contains a maximally real submanifold through each of its points.

We are now ready to present a proof of Theorem 2.1 from 12. We will use some concepts and a result of Bony (13).

Definition 3.2.

Let be an open set and a closed subset. A vector is said to be normal to at in if there is an open ball centered at such that and for some .

Definition 3.3.

Suppose and are as in Definition 3.2. A vector field is said to be tangent to if whenever is normal to at , the vector is orthogonal to .

Bony proved the following.

Theorem 3.2 (13).

Suppose is an open set and a closed subset. Let be a Lipschitz vector field in which is tangent to . If an integral curve of intersects at a point, then it is entirely contained in .

Let be as in the statement of Theorem 2.1. Suppose is a solution on and the support of . Let . Define to be the set of over points in such that there exists a real-valued, smooth function defined near satisfying , , and on near . Fix and suppose with . Let be as above with . Since , the zero set of is a smooth hypersurface near . Observe that on a side of this hypersurface. Since , by Corollary 3.2, has to be characteristic to at . Thus for any Lipschitz section of . By Theorem 3.2, the integral curve of through has to lie in .

4. The Strong Unique Continuation Property

There are some classes of locally integrable structures where the validity of the strong unique continuation is well understood. If is a smooth locally integrable structure where the dimension of is two and is one dimensional, then the strong unique continuation holds if and only if every local first integral is an open mapping into the complex plane. If is a tube structure as in Example , and the map is real analytic, then the strong unique continuation property holds if and only if for every , the function does not have a local extremum. This is a consequence of the fact that this latter condition is equivalent to the analyticity of all solutions (see 6).

In the rest of this section we focus on CR vector fields and present some of the known results on the strong unique continuation property for CR functions.

If is a continuous function on the closure of a domain , we will say it is flat at if

We begin with the case of holomorphic functions of one variable since boundary unique continuation results for such functions have important applications to the unique continuation problem for CR functions.

Let be the half ball in the plane. The following result provides a sufficient condition for the boundary unique continuation property to hold for holomorphic functions on .

Theorem 4.1 (16).

Let be a holomorphic function on , continuous up to . Assume that the real part of is nonnegative on . If is flat at , then .

In 1 H. Alexander proved a significant generalization of the preceding theorem: if is holomorphic on , continuous up to , and is a nonspiraling set, then whenever it is flat at . A good example of a nonspiraling set is the complement in of a curve emanating from the origin. For other results along this line, see 10.

Many well-known results on unique continuation for CR mappings follow from the one variable results described above. We will present next some of these results and indicate how they can be proved.

Let be a totally real submanifold of class of dimension . We may assume that in a neighborhood of the origin in , real-valued, .

A wedge with edge is a set of the form

where is a convex open cone.

Theorem 4.2 (2).

Let and be as above, with of class and continuous, holomorphic on satisfying:

(1) is flat at , and

(2) with a totally real submanifold of class .

Then in the connected component of in .

Observe that if and are real analytic, Theorem 4.2 follows from the classical edge of the wedge theorem.

In 16 the authors defined a hypersurface to be positive at a point if there is a holomorphic change of coordinates mapping to and in the new coordinates