Some Remarks on Energy inequalities for harmonic maps with potential

Some Remarks on Energy inequalities for harmonic maps with potential

Volker Branding TU Wien
Institut für diskrete Mathematik und Geometrie
Wiedner Hauptstraße 8–10, A-1040 Wien
[
July 27, 2019
Abstract.

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas and Liouville theorems under curvature and energy assumptions.

Key words and phrases:
Harmonic maps with potential; gradient estimates; monotonicity formulas; Liouville theorems
2010 Mathematics Subject Classification:
58E20, 53C43, 35J61

]volker@geometrie.tuwien.ac.at

1. Introduction and Results

One of the most studied partial differential equations for a scalar function is the Poisson equation, that is

where is some given function. If one allows the function to also depend on , that is

one calls the equation nonlinear Poisson equation. Following the terminology from the literature we call a solution of the nonlinear Poisson equation entire. For entire solutions of the nonlinear Poisson equation the following results characterizing its qualitative behavior have been obtained:

  1. Suppose that is a nonnegative function that is a potential for , that is . Let be an entire, bounded solution of the nonlinear Poisson equation. Then the following energy inequality holds [16]

    (1.1)

    Such kind of inequalities became known as Modica-type estimates.

  2. Making use of the Modica-type estimate (1.1) the following Liouville theorem was given [16, Theorem 1]: Suppose is a nonnegative function that is a potential for and an entire, bounded solution of the nonlinear Poisson equation. If for some , then must be constant.

  3. Again, making use of the Modica-type estimate, the following monotonicity formula has been established in [1]. Let be an entire, bounded solution of the nonlinear Poisson equation. Then the following monotonicity formula holds

    (1.2)

    where denotes the ball around the point with radius .

  4. Another kind of Liouville theorem was achieved in [19]: Suppose that is an entire solution of

    If and both and are bounded, then must be constant.

  5. Recently, a maximum principle has been established for solutions of in the vector valued-case [3, 2], that is , where is some domain. Here it is assumed that the potential vanishes at the boundary of a closed convex set.

In this note we focus on the study of a geometric generalization of the nonlinear Poisson equation, which leads to the notion of harmonic maps with potential. To this end let and be two Riemannian manifolds, where we set . For a smooth map we consider the Dirichlet energy of the map, that is . In addition, let be a smooth scalar function. We consider the following energy functional

(1.3)

The Euler-Lagrange equation of the functional (1.3) is given by

(1.4)

where denotes the tension field of the map . Note that in contrast to the Laplacian acting on functions the tension field of a map between Riemannian manifolds is a nonlinear operator. Solutions of (1.4) are called harmonic maps with potential. We want to point out that motivated from the physical literature one defines (1.3) with a minus sign in front of the potential.

Harmonic maps with potential have been introduced in [9]. It is shown that due to the presence of the potential, harmonic maps with potential can have a qualitative behavior that differs from the one of harmonic maps. Existence results for harmonic maps with potential have been obtained by the heat flow method [10], [6] under the assumption that the target has negative curvature. In addition, an existence result for harmonic maps with potential from compact Riemannian manifolds with boundary was obtained in [5], where it is assumed that the image of the map lies inside a convex ball.

Besides the aforementioned existence results there also exist several Liouville theorems for harmonic maps with potential. For a compact domain manifold these were derived by the maximum principle under curvature assumptions in [9, Proposition 4]. A Liouville theorem for harmonic maps with potential from a complete noncompact Riemannian manifold and the assumption that the image of the map lies inside a geodesic ball is given in [4]. A monotonicity formula for harmonic maps with potential together with several Liouville theorems was derived in [14].

For functions on Riemannian manifolds several generalizations of the Modica-type estimate (1.1) have been established, see [15], [17]. These results hold under the assumption that the manifold has positive Ricci curvature.

However, is was also noted that estimates of the form (1.1) do not hold if we consider vector-valued functions [11], [13].

It is the aim of this article to discuss if the results obtained for the nonlinear Poisson equation stated in the introduction still hold when considering harmonic maps with potential.

This article is organized as follows: In Section 2 we discuss in which sense the Modica-type estimate (1.1) for solutions of the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. In the last section we will give a Liouville theorem for harmonic maps with potential under curvature and boundedness assumptions.

2. Energy inequalities for harmonic maps with potential

Before we turn to deriving energy inequalities let us make the following observation:

Remark 2.1.

If we want to model the trajectory of a point particle in a curved space, we can make use of harmonic maps with potential from a one-dimensional domain, which are just geodesics coupled to a potential. To this end we fix some interval and consider a curve that is a solution of (1.4), which in this case reads

Here represents the derivative with respect to the curve parameter, which we will denote by . For a curve satisfying this equation the total energy is conserved, that is

This can easily be seen by calculating

where we used the equation for harmonic maps with potential in the last step.

This fact is well-known in classical mechanics, that is the mechanics of point particles governed by Newton’s law. The total energy consists of the sum of the kinetic and the potential energy and it is conserved when the equations of motion are satisfied.

However, if the dimension of the domain is greater then one, we cannot expect that a statement about the conservation of the total energy will hold in full generality.

We will make of the following Bochner formula for a map , that is

(2.1)

Here is an orthonormal basis of . Throughout this article we make use of the Einstein summation convention, that is we sum over repeated indices. In addition, by the chain rule for composite maps we find

(2.2)

where we used that is a solution of (1.4) in the second step.

In order to obtain the Modica-type estimate (1.1) for solutions of the scalar nonlinear Poisson equation one makes use of the so-called P-function technique, which heavily makes use of the maximum principle. The generalization of the -function to harmonic maps with potential is given by

(2.3)

Unfortunately, it turns out that the P-function does not satisfy a “nice” inequality in the case of harmonic maps with potential.

Lemma 2.2.

Let be a smooth harmonic map with potential. Then the -function (2.3) satisfies the following inequality

(2.4)
Proof.

Using the Bochner-formulas (2.1), (2.2) a direct calculation yields

In addition, we apply the Kato-inequality and find

yielding the result. ∎

Let us make some comments about (2.4):

Remark 2.3.
  1. If the target has dimension one, then the last two terms on the right hand side in (2.4) cancel each other. In this case one can successfully apply the maximum principle under the assumption that the domain has positive Ricci curvature giving rise to the Modica-type estimate (1.1).

  2. If , then the last two terms in (2.4) will no longer cancel each other. Moreover, it is well known by counterexamples, see [20, Section 2] and references therein, that one cannot expect to obtain a Modica-type estimate in the case that .

Since we cannot derive energy inequalities by making use of the techniques that were developed for solutions of the scalar nonlinear Poisson equation, we will apply ideas that were used to derive gradient estimates and Liouville theorems for harmonic maps between complete Riemannian manifolds [8]. Here, one assumes that the image of the map lies inside a geodesic ball in the target.

2.1. Gradient estimates for harmonic maps with potential

In the following we will make use of the following

Lemma 2.4.

Let be a smooth harmonic map with potential. Suppose that the Ricci curvature of and the Hessian of the potential satisfy and that the sectional curvature of satisfies . Then the following inequality holds

(2.5)
Proof.

This follows from the Bochner formula (2.1) and the identity . ∎

Now we fix a point in and by we denote the Riemannian distance from the point . Let be a positive function. On the geodesic ball in we define the function

(2.6)

Clearly, the function vanishes on the boundary , hence attains its maximum at an interior point . We can assume that the Riemannian distance function is smooth near the point , see [7, Section 2].

In the following we will apply the Laplacian comparison theorem, see [12, p. 20], that is

with some positive constant . Moreover, we make use of the Gauss Lemma, that is .

Lemma 2.5.

Let be a smooth harmonic map with potential. Suppose that the Ricci curvature of and the Hessian of the potential satisfy and that the sectional curvature of satisfies . Then the following inequality holds

(2.7)
Proof.

At the maximum the first derivative of (2.6) vanishes, yielding

(2.8)

Applying the Laplacian to (2.6) at gives

(2.9)

Squaring (2.8) we find

(2.10)

Inserting (2.5) and (2.10) into (2.9) and using the Gauss Lemma we get the claim. ∎

To obtain a gradient estimate from (2.7) for noncompact manifolds and we have to specify the function .

First, we choose a function that is adapted to the geometry of the target manifold motivated by a similar calculation for harmonic maps between complete manifolds [8]. Let be the Riemannian distance function from the point in the target manifold . We define

(2.11)

with some positive number to be fixed later, where denotes the geodesic ball of radius around the point in . We will assume that , thus on the ball .

Lemma 2.6.

On the geodesic ball we have the following estimate

(2.12)

where denotes the Riemannian metric on .

Proof.

This follows from the Hessian Comparison theorem, see [12, Proposition 2.20] and [8, p. 93]. ∎

We will also make use of the following fact: If for , then the following inequality holds

(2.13)

At this point we can give the following two results similar to [5, Theorem 3.2]:

Theorem 2.7.

Let be a smooth harmonic map with potential. Suppose that the Ricci curvature of and the Hessian of the potential satisfy and that the sectional curvature of satisfies . Moreover, assume that , where is the geodesic ball of radius around in with . Then the following estimate holds

(2.14)

where the positive constant depends on the geometry of .

Proof.

We choose the function defined in (2.11) and insert it for in (2.7). By the Hessian comparison theorem (2.12) we find

where we also used that is a harmonic map with potential. In addition, making use of the assumption on the image of , there exists a positive constant such that

holds. Inserting this into (2.7) we find

The claim then follows from (2.13). ∎

Corollary 2.8.

Under the assumptions of Theorem 2.7 we can take the limit while keeping the point in fixed and obtain the estimate

If has positive Ricci curvature and if the potential is concave, then the following inequality holds

which can be interpreted as a Modica-type estimate for harmonic maps with potential.

There is another way how we can obtain a gradient estimate from (2.7), by assuming that the potential has a special structure. More precisely, we have the following

Theorem 2.9.

Let be a smooth harmonic map with potential. Suppose that the Ricci curvature of satisfies and that the sectional curvature of satisfies . Moreover, assume that the potential satisfies

Then the following energy estimate holds

(2.15)

The constant depends on the geometry of .

Proof.

We make use of the formula (2.7), where we now choose . Making use of the assumptions on the potential we note that

In addition, again by the assumptions on the potential , we get

for some positive constant . Inserting into (2.7) then yields

The statement follows from applying (2.13) again. ∎

Corollary 2.10.

Under the assumptions of Theorem 2.9 we can take the limit while keeping the point in fixed and obtain the estimate

If has nonnegative Ricci curvature then is trivial.

2.2. Generalized Monotonicity formulas

In the following we will make use of the stress-energy-tensor for harmonic maps with potential, which is locally given by

(2.16)

The stress-energy-tensor is divergence-free, when is a smooth harmonic map with potential [14], that is

Let us recall the following facts: A vector field is called conformal if

where denotes the Lie-derivative of the metric with respect to and is a smooth function.

Lemma 2.11.

Let be a symmetric 2-tensor. For a conformal vector field the following formula holds

(2.17)

By integrating over a compact region and making use of Stokes theorem, we obtain:

Lemma 2.12.

Let be a Riemannian manifold and be a compact region with smooth boundary. Then, for any symmetric -tensor and a conformal vector field the following formula holds

where denotes the normal to .

We now derive a type of monotonicity formula for smooth solutions of (1.4) for the domain being .

Lemma 2.13.

Let be a smooth harmonic map with potential. Let be a ball with radius in . Then the following formula holds

(2.18)
Proof.

For we choose the conformal vector field with . Note that . The statement then follows from (2.12) applied to (2.16). ∎

Making use of the coarea formula we obtain the following

Theorem 2.14.

Let be a smooth harmonic map with potential. Let be a ball with radius in . Then the following formula holds

Corollary 2.15.

Let be a smooth harmonic map with potential. Suppose that . Then we have the following monotonicity formula

Note that this monotonicity formula is different from the one for solutions of the nonlinear Poisson equation (1.2) since we do not have a Modica-type estimate for harmonic maps with potential.

Monotonicity formulas for harmonic maps with potential with the domain being a Riemannian manifold have been established in [14].

The results presented above also hold for harmonic maps with potential that have lower regularity. To this end we need the notion of stationary harmonic maps with potential.

Definition 2.16.

A weak harmonic map with potential is called stationary harmonic map with potential if it is also a critical point of the energy functional with respect to variations of the metric on the domain , that is

(2.19)

Here is a smooth symmetric 2-tensor.

Every smooth harmonic map with potential is stationary, which is due to the fact that the associated stress-energy-tensor is conserved. However, a stationary harmonic map with potential can have lower regularity.

For stationary harmonic maps with potential we have the following result generalizing [1, Theorem 3.1]:

Theorem 2.17.

Let be a harmonic map with potential. Suppose that with and

then the following inequality holds

In particular, this implies that is constant when .

Proof.

We will prove the result for the case that . Let be a smooth cut-off function satisfying for , for and . In addition, we choose with . Hence, we find

Inserting this choice into (2.19) we obtain

We can bound the right-hand side as follows

Making use of the properties of the cut-off function we obtain

Taking the limit and making use of the assumptions we find

which finishes the proof for the case that . Making use of the Theorem of Cartan-Hadamard the proof carries over to hyperbolic space. ∎

Remark 2.18.

The last Theorem can be interpreted as an integral version of (1.1) for bounded harmonic maps with potential.

Now we derive a generalized monotonicity formula for harmonic maps with potential, where we take into account the pointwise gradient estimate (2.8).

Theorem 2.19.

Let be a smooth harmonic map with potential. Suppose that the Hessian of the potential satisfies and that the sectional curvature of satisfies . Moreover, assume that , where is the geodesic ball of radius around in with .

Then the following monotonicity-type formula holds

(2.20)

where the positive constant depends on .

Proof.

Throughout the proof we set

Making use of the coarea formula and rewriting (2.18) we find

Applying (2.8) we obtain the following inequality

from which we get the claim. ∎

Let us make several comments on Theorem 2.19:

Remark 2.20.
  1. The monotonicity type-formula (2.20) can be interpreted as the generalization of (1.2) to harmonic maps with potential.

  2. It is straightforward to generalize (2.18) to the case of the domain being a Riemannian manifold.

3. A Liouville theorem

In this section we derive a Liouville theorem for harmonic maps with potential from complete noncompact manifolds with positive Ricci curvature. Our result is motivated from a similar result for harmonic maps, see [18, Theorem 1]. In addition, this result also generalizes the Liouville theorem for solutions of the nonlinear Poisson equation [19], which is stated in detail in the introduction.

Theorem 3.1.

Let be a complete noncompact Riemannian manifold with nonnegative Ricci curvature, and a manifold with nonpositive sectional curvature. Let be a smooth harmonic map with potential with finite energy, that is . If the potential is concave then is a constant map.

Proof.

We follow the presentation in [21, pp. 26] For a solution of (1.4) and by the standard Bochner formula (2.1) we find

(3.1)

Making use of the curvature assumptions and the fact that the potential is a concave function, (3.1) yields

(3.2)

In addition, by the Cauchy-Schwarz inequality we find

(3.3)

We fix a positive number and calculate

where we used (3.2) and (3.3). Let be an arbitrary function on with compact support. We obtain

Now let be a point in and let be geodesic balls centered at with radii and . We choose a cutoff function satisfying

In addition, we choose such that

for a positive constant . Then, we find

We therefore obtain

We set and find

Letting we get

Now, letting and under the assumption that the energy is finite, we have

hence the energy has to be constant. If , then the volume of would have to be finite. However, by [22, Theorem 7] the volume of a complete and noncompact Riemannian manifold with nonnegative Ricci curvature is infinite. Hence , which yields the result. ∎

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