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Monotonicity Formulas in the Calculus of Variation

Luca Spolaor

Communicated by Notices Associate Editor Daniela De Silva

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1. Introduction

Calculus of variation is a branch of mathematics at the intersection between analysis and geometry that focuses on techniques to find minima (and more generally critical points) of functionals, that usually represent physical or geometrical energies, and to study their regularity. Some famous examples are the Dirichlet energy, used to model the shape of an elastic membrane, and the Plateau problem, named after the physicist Joseph Plateau and introduced by Lagrange to model the shape of soap films.

A basic question in partial differential equations (PDEs) is the existence of solutions given certain initial/boundary conditions. Calculus of variations is a collection of techniques that allows to answer this type of questions. The starting point is a basic fact in real analysis: the first derivative of a smooth function at an interior minimum is zero. Thus, by analogy, in order to find solutions to certain elliptic PDEs, one can look for minima of suitably defined energies, whose first derivative, often called first variation or Euler-Lagrange equation, is precisely the desired PDE. Again in analogy with the classical theory of differentiable functions, in order to guarantee the existence of such minima, compactness of the domain of the functional is often needed, and this leads to the introduction of new spaces of functions (or geometrical objects), such as Sobolev spaces, whose elements are not in general differentiable functions, and so will not solve any PDE in the classical sense. Thus, the problem of regularity of minimizers arises. In the study of such regularity questions, a very useful tool is given by the so-called monotonicity formulas. They allow to study the infinitesimal behavior of minimizers at a given point by reducing it to the classification of global homogeneous minimizers. Once again the underlying idea is based on a simple fact that I usually ask my undergraduate analysis students as part of their midterm problems:

Theorem 1.1.

Let be a differentiable function and suppose that for every , then there exists .

The goal of this note is to describe a set of techniques, inspired by basic concepts in analysis, such as Theorem 1.1, that have been of central importance to prove existence and regularity of minimizers in a variety of problems in the calculus of variation, such as harmonic maps, minimal surfaces and free-boundary problems, to name a few. Our toy model will be the Dirichlet energy. In particular we will see in the next sections how this energy can be used to prove the existence of classical harmonic functions, which are in fact analytic. This will be achieved using tools that are general enough to be applied to a variety of variational problems, such as the one mentioned above and many others.⁠Footnote1 In order to describe the full depth of these tools and how they are used in more complicated situations involving certain non-linear elliptic PDEs, we will also briefly address two of these problems: harmonic maps and minimal surfaces.

1

In fact, some of the techniques that we will discuss, such as monotonicity formulas, appear also in parabolic problems, such as Mean Curvature flow and Ricci flow. We will not discuss any of them here for lack of space and, more importantly, of expertise.

This note should be used as a roadmap for students interested in calculus of variations, to understand some fundamental ideas which lay at the foundation of many major results in the field. In what follows, I will only use symbols and formulas that any student who has taken a graduate analysis sequence should be able to understand. However, I often find certain formulas clearer than their descriptions, and for this reason I will insert them when needed.

2. Existence of Harmonic Functions: The Dirichlet’s Principle

Given an open, bounded, connected domain , an harmonic function in is a function which solves the so-called Laplace equation, that is

where denotes the second partial derivative in direction , for every , …, . The Laplace equation was introduce in 1799 by Pierre-Simon Laplace to prove that the solar system is stable over astronomical timescales, and since then it has been used to construct models for surface tension (soap bubbles and minimal surfaces), heat, electromagnetism, fluid mechanics, and many other physical phenomena.

Roughly speaking, the Laplace equation requires that the sum of the eigenvalues of the Hessian matrix of at every point of is equal to zero, that is from any point of you can go up as much as you can go down. This property is known as maximum principle, and, stated formally, it says that the maximum and the minimum of are achieved only on , unless is constant. Another key feature of this equation is its linearity, that is

for every , , , . As a simple exercise, one can combine these two facts to obtain uniqueness of solutions for the so-called Dirichlet problem, that is given a smooth function any solution of the following problem is unique

There are several methods to prove existence of solutions of 2.1 (see for instance 10). One of them is the Dirichlet principle, introduced by Riemann, and then formally justified by Weierstrass and finally Hilbert, who in 1900 developed the so-called direct method in the calculus of variations. The basic observation is that if minimizes the energy

in the space of functions , then solves the Dirichlet problem 2.1, and vice versa. This follows from a standard fact from real analysis, that is for any variation with zero boundary condition on , the differentiable function

has a minimum at , and so . An integration by parts and the convexity of the energy then give the desired correspondence.

After observing this equivalence, Riemann took for granted the existence of minimizers of , and it was only 50 years later that Hilbert, following an observation of Weierstrass, gave a rigorous proof of this fact by turning once again to real analysis. Indeed, a sufficient condition for a function to have a minimum is that is compact and is lower semicontinuous in the same topology in which is compact. In order to use this observation, one introduces the so-called Sobolev space of, roughly speaking, functions with gradient in (see 10) and whose suitably defined trace on is . Notice that, since functions are only defined almost everywhere, this is non-trivial and requires some assumptions on the regularity of and , which we will not discuss here. It is then possible to equip with a suitable topology that makes it compact and with respect to which is lower semicontinuous, thus obtaining existence of a minimizer in . This is also one of the starting points in the study of functional analysis, that is the study of spaces of functions, of which is but one of many examples.

Since minimizers a priori only belong to the space , the remaining question to prove the existence of solutions to 2.1 is the regularity of such minimizers, that is if they are and achieve their boundary datum smoothly, so to solve 2.1 in the classical sense. An affirmative answer to this question has been known for a long time in the case of harmonic functions. For more general minimization problems, such as harmonic maps or the Plateau problem, however, answering this regularity question has been one of the major focuses of research in analysis in the last 100 years. We will explain why in the next sections.

3. Regularity of Sobolev Functions: Decay and Growth

We want to discuss the interior regularity of minimizers of the energy , that is their differentiability at interior points of . It is easy to see that, if is a minimizer of in then it is a local minimizer of , that is

for every where we assume without loss of generality that . The advantage of this formulation is that it highlights how neither the boundary of nor the Dirichlet datum play any role in the study of interior regularity questions.

For local minimizers we want to understand the following questions:

(1)

Is a local minimizer of differentiable? Is it , so that we can conclude existence of harmonic functions? Is it ?

(2)

Is a local minimizer an analytic function?

The reason why we state the two questions separately is that, while the first is essentially a question about the decay of some suitably chosen norm of , the second requires also some form of non-degeneracy, that is polynomial growth, of . To get an understanding of this, one can consider the classical example of non-analytic function defined by

This function decays at faster than any polynomial, and in fact it is there, but in doing so it fails to have a polynomial growth, and so to be analytic.

Let us discuss question 1 first. As hinted above, in order to prove regularity of a Sobolev function (or even just ), one tries to prove some form of decay of its Sobolev norm at any given point. A celebrated such criterion for regularity is Campanato’s lemma, which states:

Lemma 3.1 (Campanato’s lemma (see 15).

Suppose , , , with , and

for every ball . Then there is a Hölder continuous representative for the -class of with

for every , where depends only on .

The intuition is the following. If is Hölder continuous in a ball , that is if 3.4 holds in , then we can write it as a constant (possibly zero), plus a remainder that does not exceed times a constant (say ). Vice versa, if this holds in every ball, then is . Inequality 3.3 is an integral version of this property, that holds for functions that are merely in . The reason we divide the integral by is to make the quantity on the left-hand side of the inequality independent of the measure of the ball over which we are integrating, that is to consider a scale invariant quantity. It is not difficult to check that 3.4 holds for the function defined in 3.2, with (the value of at ), and any .

Campanato’s lemma holds also for the gradients of functions, since they are functions themselves, and in this case gives regularity. For the gradient statement, 3.3 needs to be replaced by

where one should think of as the linear function that is the gradient of at the point . Notice also that the power of in the left-hand side of the inequality changed to keep the quantity scale invariant. Combining this with the Poincaré inequality, it is possible to prove the following -regularity theorem.

Theorem 3.2 (-regularity 15).

There exists , depending only on the dimension , such that if is a local minimizer of in and

then , for every .

Roughly speaking, the smallness of the Sobolev norm in 3.6, combined with the minimality of in , is enough to guarantee the decay assumptions of Lemma 3.1 in , and thus regularity. Since derivatives of harmonic functions are still harmonic, it is possible to bootstrap the above result to obtain regularity at points where 3.6 is satisfied. This is true for more general elliptic PDEs, under suitable regularity assumptions on their coefficients, and is known as Schauder theory.

Let us now discuss briefly a property strictly related to question 2., that is unique continuation. Let us consider a function , where is an open bounded connected domain. Then we say that satisfies

the unique continuation property if in a non-empty open subset implies that in ;

the strong unique continuation property if the condition that vanishes of infinite order at a point , i.e., if

implies that in .

It is easy to see that an analytic function in satisfies both of the above properties. On the other hand the function in 3.2 does not satisfy either of them. In particular, strong uniqueness continuation makes precise the statement that analytic functions cannot decay too fast at a point, in an sense.

Of course, there are much simpler techniques to prove the regularity and analyticity of local minimizers of than Theorem 3.2 (see 10). However, differently from the ones described in this note, many of them are not flexible enough to be applied to more complicated problems, such as harmonic maps or minimal surfaces.

4. Monotonicity Formulas: Energy Density and Frequency Function

We will see two monotone quantities associated to local minimizers of : the energy density and the frequency function. The name energy density requires no explanation, while the name frequency function is probably due to the fact that its value at is related to the first non-trivial term in the analytic expansion of an harmonic function, or analogously to the first non-trivial term in its Fourier expansion. The starting point is that, by choosing the variation properly in the definition of local minimizers 3.1, it is possible to prove the following two integral identities, valid for a local minimizer and :

where is the radial derivative of and is the spherical measure. It is then a simple exercise in calculus to show that the energy density, defined by

and the frequency function, defined by

for , are monotone non-decreasing functions in . For example, using 4.1, we have

where we drop the dependence on the point . We can then use these two monotone quantities to answer questions 1 and 2 from the previous section.

To answer question 1 we need to show that so that Theorem 3.2 can be applied. By Example 1.1, we know that there exists

A standard technique to compute it, common to many variational problems where monotonicity formulas are available, is the so-called blow-up procedure, that, as hinted in the introduction, together with Theorem 3.2, allows to reduce the study of regularity to the understanding of global homogeneous minimizers.

Theorem 4.1 (Blow-up procedure).

Let be a local minimizer and let be the sequence of function . There exist a subsequence , with as , and a functions such that

(1)

is a -homogeneous local minimizer of in , that is for every ;

(2)

for every .

The key ideas in the proof of this theorem are a compactness result for sequences of minimizers, that is such sequences converge strongly, up to subsequence, to a minimizer, and the use of 4.1 to prove homogeneity of the limit of the . The last step is the classification of -homogeneous solutions:

Theorem 4.2 (Liouville’s theorem).

Every -homogeneous minimizer of in is a constant function.

It follows that , as desired, thus giving regularity of minimizers at any interior point.

To answer question 2, one can proceed in a similar way, that is

Let .

Consider the sequence and prove a blow-up theorem to obtain a global minimizer which is -homogeneous.

Classify -homogeneous minimizers, to show that they are harmonic polynomials of degree .

Intuitively, is the order of the first non-trivial term in the analytic expansion of at and is the first non-trivial term. It is then possible to apply the same procedure with , instead of , and reasoning inductively in this way one can write the analytic expansion of at . Of course there are several technical steps to check, such as the uniqueness of and the convergence of the series. As an example, consider the harmonic function given in polar coordinates by , then and , as in the following picture.

At the next step and , as in the following picture.

Notice that the number of oscillations increases at every step, thus the name frequency function.

Using the monotonicity of the frequency function it is possible to prove unique and strong unique continuation in a straightforward manner, and in fact this is the reason why Almgren introduced it in his celebrated Big regularity paper (1). The idea is that

and integrating this simple ODE in and using the monotonicity of the frequency one obtains a so-called doubling condition, that is

from which both unique continuation properties follow easily (see for instance 13).

5. Harmonic Maps and Energy Density

We fix a smooth manifold and we consider locally minimizing harmonic maps, that is functions such that

for every such that for every . That is, is a vector-valued function that minimizes the Dirichlet energy among all functions taking values in the manifold . The Euler-Lagrange equation associated to this problem is a system of non-linear equations, where the non-linearity is given by a suitable expression involving the second fundamental form of .

It is possible to show that if is a locally minimizing harmonic map, then analogous versions of Theorem 3.2 and 4.1 hold, and the energy density is monotone. However, Theorem 4.2 fails. As an example one can take and defined by . This function is a -homogeneous, local minimizer in , is not constant, and in fact it is not at the origin, where Theorem 3.2 cannot be applied, since the energy density is not .

For this reason, one defines the regular and singular sets, and , of a local minimizer in to be respectively the collection of points in such that is smooth in a neighborhood of and its complement. At difference from the case of harmonic functions, might be non-empty, as shown by the previous example. In fact, singular points can be characterized precisely as those points at which the energy density does not converge to , that is

A typical question to ask at this point is how big is the singular set? Usually one looks at how big the singular set of blow-up functions is, that is of global homogeneous minimizers, and then tries to extend the same bound to general local minimizers by using once again the monotonicity formulas. In our example above, we have a point of singularity in , and in fact it is possible to prove that

Theorem 5.1 (Dimension of singular set of blow-ups 15).

Let be a -homogeneous, locally minimizing harmonic map in , then the Hausdorff dimension of is at most .

We remark that, using Theorem 3.2, the scaling of the energy density and a simple covering argument, it is easy to show that the -volume of the singular set is , where is the same power of as in the definition of energy density. However, in order to get the optimal bound one needs to use a more refined method, called dimension reduction, introduced by Federer, which formalizes the idea that the maximal dimension of the singular set of possible blow-up functions should bound the dimension of the singular set of any local minimizer. A very recent result of Naber-Valtorta (see 14) actually improves this regularity to rectifiability of the singular set and finite -volume:

Theorem 5.2 (Dimension of singular set of harmonic maps).

Let be a locally minimizing harmonic map in , then has locally finite Hausdorff measure and it is rectifiable.

6. An Application of Frequency Function Monotonicity: Area Minimizing Currents

In the previous section we suggested that in those problems where a monotonicity formula is available and the blow-up procedure can be carried out, understanding the maximal dimension of the singular set of admissible blow-ups is enough to understand that of general local minimizers. This is not always the case. As an example we can consider the case of -dimensional area minimizing surfaces in . It is well known that every holomorphic curve in is locally area minimizing (see 11), so an example of area minimizing surface is given by

Notice that this surface is not regular at the origin, where it is not embedded but only immersed. In this setting it is also possibile to define an energy density, prove an -regularity theorem, and run the blow-up argument (by one-homogeneous rescalings). However, running this procedure for at the origin would yield as unique blow-up the plane counted with multiplicity , which is regular, even though the minimal surface itself is singular: that is the singular structure of a blow-up might give no information on that of the minimizer itself, the problem of course being the presence of multiplicity.

Without entering too much into details, in order to prove the existence of area minimizing surfaces, Federer and Fleming introduced a notion of generalized surfaces, called integral currents (see 12), which play the same role as Sobolev spaces in the previous sections. This generalized surfaces are not smooth in general, and even minimizers might not be smooth in light of the previous example. However, a major achievement of Almgren-Chang, later completed by De Lellis-Spadaro and myself, is the following theorem.

Theorem 6.1 (Almgren-Chang’ regularity theorem 26789).

Let be a -dimensional integral current, locally area minimizing in , . Then the singular set of is locally finite and at every singular point is locally the union of holomorphic curves intersecting only at the singularity.

Among the many ideas needed to prove this result, one of the most successful is the Almgren’s frequency function. Almgren’s insight was to compare the expansion of area minimizing surfaces in holomorphic pieces to the analytic expansion of harmonic functions, and then observing that this last expansion can be done using the frequency function, as explained in the previous sections. In fact, he used this idea first to prove a more general bound on the dimension of the singular set of area minimizing currents in the celebrated big regularity paper (see 1345).

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Credits

Opening graphic and figures are courtesy of Luca Spolaor.

Photo of Luca Spolaor is courtesy of Di Tan.