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A Perspective on the Regularity Theory of Degenerate Elliptic Equations

Héctor A. Chang-Lara

Communicated by Notices Associate Editor Daniela De Silva

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1. Introduction

Ellipticity is a well-studied characteristic of some partial differential equations which enforces regularity on its solutions. The prototype problem is the Laplace equation whose solutions are known as the harmonic functions. To illustrate its usefulness, consider an arbitrary sequence of uniformly bounded functions over a compact set. In contrast to numerical sequences, a bounded sequence of functions does not necessarily have a uniformly convergent subsequence, even if we assume that each function in such sequence is smooth. However, as soon as we assume that the functions are harmonic, then such uniform limits are always guaranteed.

The property we have just described is known as compactness. We will see that it can be derived from regularity estimates for harmonic functions. It is a powerful tool that can be used in many fundamental results, such as the existence theorem for harmonic functions with prescribed boundary values or the convergence of numerical schemes.

To fix some ideas, let us consider a general second-order partial differential equation (PDE) of the form

where determines the nonlinear operator on the left-hand side, and the second-order differentiable function is the unknown of the problem. Our focus lies on elliptic problems, which are defined by the requirement that is nondecreasing in the Hessian variable ().⁠Footnote1

1

We assume the following partial order on : if and only if for all .

Uniform ellipticity arises when we further assume some quantitative control on the monotonicity of with respect to . In the case that the function is differentiable, we say that the operator is uniformly elliptic if

for some fixed constants . In the previous expression, is the identity matrix and is the matrix of partial derivatives of with respect to the Hessian variable .

Operators that are elliptic, but not necessarily uniformly elliptic, are called degenerate elliptic. These find practical applications in diverse fields such as material sciences, fluid dynamics, finance, and image processing. Some famous examples of degenerate equations include the minimal surface equation

the -Laplace equation ()

and, in the time-dependent case, the porous medium equation ()

In general, it has been observed that solutions of degenerate elliptic equations are not always guaranteed to have the same regularity estimates as uniformly elliptic equations.

Motivated by the observation for harmonic functions, we may wonder the following:

1. Have solutions of uniformly elliptic PDEs a compactness property, similar to the harmonic functions?

2. Which additional hypotheses could complement the degenerate ellipticity in order to recover a compactness property for the solutions?

The analysis of PDEs relies significantly on bounds for the modulus of continuity of a solution and its derivatives, these estimates constitute the cornerstones of the regularity theory. The development of the regularity theory for elliptic equations has a rich history with numerous authors. This survey concerns the regularity theory of uniformly elliptic equations that originated during the 1980s and 1990s, with main contributions due to Krylov, Safonov, Evans, and Caffarelli, among many others. This regularity theory is now commonly known as the Krylov–Safonov theory.

We aim to provide a perspective on ongoing developments in the regularity theory of degenerate elliptic problems. First, we give an overview of the regularity theory for the Laplacian and uniformly elliptic equations. This discussion will shed light on the fundamental strategies employed in classical scenarios, enabling us to appreciate better the challenges posed by degenerate equations.

Afterward, we focus on the case of degenerate ellipticity, where the absence of uniform ellipticity is complemented with some additional hypotheses. These assumptions could be interpreted as some sort of alternative regularizing mechanism for the solution. It is the interplay of these phenomena what we find to be a quite attractive venue of current research.

We focus on three types of degeneracies which can be loosely described in the following way:

1.

Elliptic equations that hold only where the gradient is large: Either the solution obeys a uniformly elliptic equation or its gradient is bounded.

2.

Small perturbations: Uniform ellipticity holds in a neighborhood of a given profile.

3.

Quasi-Harnack: Uniformly ellipticity holds at macroscopic scales.

Each one of the previous problems will be presented in a different section, titled accordingly. These section titles make reference to the articles IS16, Sav07, and DSS21, where the respective results were originally studied.

2. The Laplacian

The Laplacian is the fundamental differential operator that describes the ellipticity phenomenon in . Solutions of the equation , also known as harmonic functions, are abundant in pure and applied mathematics.

Perhaps the most characteristic features of uniformly elliptic problems are the Harnack inequalities. In the case of the Laplacian, these arise as consequences of the divergence theorem through the mean value theorem (Figure 1).

Figure 1.

The mean value theorem: If in , then for any it holds that .

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The weak Harnack inequality states that for any satisfying in , and any value , it holds that

Figure 2 illustrates a geometric argument from where to establish this estimate from the mean value theorem.

Figure 2.

The key observation is that for any we always get the inclusions .

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This control on the distribution of the solution is quite powerful. In particular, it can be used to prove an interior Hölder estimate for harmonic functions in the following form

for some and depending only on the dimension.

Although we have assumed that the solution is -regular, the point of the estimate is that its own continuity gets controlled by its size rather than its derivatives. Notably, a uniformly bounded family of harmonic functions is automatically equicontinuous on any compact subset. By the Arzelá–Ascoli theorem, this family always contains a sequence that converges locally uniformly to a limit. Moreover, it can be shown that the limit is also a harmonic function.

2.1. The diminish of oscillation

The Estimate 2.2 can be derived through a strategy known as diminish of oscillation. This elegant argument exemplifies the geometric approach in elliptic PDEs. Let us explain it in detail:

1.

Assume that is a harmonic function taking values between and . For and , we aim to show that

This can be proved recursively if there exists some small such that for any Footnote2

2

The oscillation of a function measures the variation of the values that it takes in a given set:

Indeed, 2.3 implies that the oscillation of has a geometric decay in triadic balls

By conveniently fixing and we now get the desired estimate in the following form and for every and , such that

2.

To get the diminish of oscillation 2.3, we apply the weak Harnack inequality to a given translation of . Keep in mind that oscillates between and over , and consider as well , the level set that sits just in the middle. Hence, at least one of the following alternatives must be true:

or

In the first case we apply the weak Harnack inequality 2.1 to the positive harmonic function in to get that

This estimate raises the lower bound on from over the ball , to

over (Figure 3).

Figure 3.

Diminish of oscillation: The lower bound of the solution improves if the measure of is at least half of the measure of .

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For the other alternative, we apply the weak Harnack inequality to to get a similar improvement on the upper bound instead. In conclusion, either option implies the diminish of oscillation Estimate 2.3 for .

3. Uniformly Elliptic Equations

Let us now consider the second-order equation

Assuming that is differentiable and , we get by the fundamental theorem of calculus that satisfies a homogeneous linear equation of the following form⁠Footnote3

3

We assume the following inner product in

Indeed, we just need to integrate the derivative from to to notice that the coefficients and are given by

Even though these coefficients depend on the solution as well, under suitable hypotheses on the derivatives of we can overlook this dependence and understand the equation in a broad sense.

To start, we can just assume that the coefficients are uniformly bounded. In particular, , which would follow from a Lipschitz assumption on . Uniform ellipticity requires that for some constants , it holds that , which means that . We summarize our hypotheses on as

By considering the extreme cases in 3.2 given under these assumptions, we obtain that

where

are known as the Pucci extremal operators.

The main result of the Krylov–Safonov regularity theory established in KS79, states that solutions of the uniformly elliptic problem 3.3 have an interior Hölder estimate as in 2.2.

Theorem 3.1 (Interior Hölder estimate).

Given the parameters of uniform ellipticity and the dimension , there exist and such that the following holds:

Let satisfy

Then

In the same way as for the Laplacian, this estimate can be deduced by a diminish of oscillation argument from the weak Harnack inequality, also known in this case as the estimate. The only difference is that, in the general setting, the bound on the distribution becomes of order , for some exponent , perhaps small and depending on the parameters of uniform ellipticity and the dimension.

From now on, and to simplify the statements of the following lemmas and theorems, we will assume that any constants mentioned in these statements depend by default on the parameters of uniform ellipticity and the dimension.

Lemma 3.2 (Weak Harnack inequality).

There exist such that the following holds:

Let be nonnegative and satisfy

Then, for any

For a long time, the challenge to demonstrate this type of result was to find some connection between pointwise and measure quantities on the solution. For the Laplacian, this connection is naturally suggested by the divergence theorem (keep in mind that ). In the general case, this link was eventually established by the Alexandrov–Bakelman–Pucci maximum principle, often abbreviated as the ABP lemma.

3.1. The ABP lemma

Before stating the main result of this section we will need some preliminary notions. This presentation showcases constructions due to Cabré Cab97 and Savin Sav07.

Consider the family of functions given by translations of a fixed profile

For a given function , we say that a vertical translation of touches from below at if and only if . We define the lower contact set as varies in some set in the following way (Figure 4)

Figure 4.

The contact set for a function is the set of points in the domain that admit a supporting graph of the form from below.

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The set may be designed to capture important information about . For instance, if then . Indeed, for any

Consider now the mapping , such that if . This transformation can be computed by solving for in the expression .

If is surjective, we get by the change of variable formula that

The Jacobian is given by

Hence, the measure of the contact set can be compared with the measure of , provided some bound on the Hessian of over .

At every contact point , we can use the second-derivative test to bound the eigenvalues of from below by⁠Footnote4 . On the other hand, if we now assume that satisfies

4

For , we denote . If is also symmetric then we also have that

then we can also bound the eigenvalues of from above.

To see this, we notice first that⁠Footnote5

5

For , we denote the positive and negative parts of .

Thanks to the lower bound on the eigenvalues of and the first-derivative test, we get that for any