Local regularity for an anisotropic elliptic equation

We establish the interior Hölder continuity for locally bounded solutions, and the Harnack inequality for non-negative continuous solutions to a class of anisotropic elliptic equations with bounded and measurable coefficients, whose prototype equation is $$\begin{aligned} u_{xx}+\varDelta _{q,y} u=0\quad {\text { locally in }}{\mathbb {R}}\times {\mathbb {R}}^{N-1},\quad {\text {for }}q<2, \end{aligned}$$






u

xx


+

Δ

q
,
y


u
=
0



locally in


R
×


R


N
-
1


,


for


q
<
2
,





via ideas and tools originating from the regularity theory for degenerate and singular parabolic equations.


Notation and the main results
Let E be an open set in R N with N ≥ 2. We denote a general point in E by z = (x, y) ∈ R×R N −1 . For a function u defined in E, the symbols D x u (or u x ) and D y i u (or u y i ) represent the differentiation of u with respect to x and y i variables. Accordingly we also set D y = (D y 1 , . . . , D y N −1 ), D = (D x , D y ). For 1 < q < 2, we shall consider the elliptic partial differential equation where the functions A i (z, u, ξ) : E × R × R N → R are Carathéodory functions, i.e. they are measurable in (u, ξ) for all z ∈ E and continuous in z for a.e. (u, ξ) ∈ R N +1 . Moreover, they are subject to the following structure conditions a.e. in E ⎧ ⎪ ⎪ ⎨ with given positive constants C o and C 1 . The prototype equation is u x x + div y (|D y u| q−2 D y u) = 0.
Before stating the main result, let us recall that the anisotropic elliptic partial differential Eq. (1) is a special case of the more general equation where the functions A i (x, u, ξ) : E × R × R N → R are Carathéodory functions, and subject to the structure conditions for some constants p i > 1, C o > 0 and C 1 > 0. The prototype equation is Here a i (x), i = 1, . . . , N are measurable functions, satisfying C o ≤ a i (x) ≤ C 1 for some positive C o and C 1 . Note also we slightly abused the symbols x and D in (3) and (4), which represent a vector in R N and the gradient in x. When p 1 = · · · = p N = p > 1, the general Eq. (3) reduces to the standard p-Laplacian type equation whose local regularity theory, such as Hölder regularity and Harnack estimates, is well studied. See [3,13,21,22,24], for example.
When p i 's are potentially different, the study of local regularity was initiated in [17][18][19][20]. It has been shown that solutions are Lipschitz continuous, provided the coefficients of the operator are differentiable and satisfy some proper structure conditions. See also [14,15]. However, much less is known about the local regularity for (3) with merely bounded and measurable coefficients.
It has been proved that the local boundedness of local solutions to (3) is inherent in the notion of weak solution in [10,12], provided max{ p 1 , . . . , The above condition indicates that the indices p i 's are not too far apart. In fact, examples are constructed in [11,17] to show that solutions could be unbounded, if the indices are too far apart. Recently, a rather detailed discussion on this issue has been carried out in [1]. Concerning the continuity of locally bounded solutions, a first step was made in [16]. It has been shown that locally bounded solutions to (1) are Hölder continuous for q > 2. We also mention that the Hölder regularity for more general anisotropic operators is considered in [8,9].
In order to define the notion of weak solution formally, we introduce the following anisotropic Sobolev spaces W 1, [2,q] [2,q] (E).
A function u ∈ W 1,[2,q] loc (E) is called a local, weak sub(super)-solution to (1), if for every compact set K ⊂ E it satisfies In this note, by sub(super)-solutions we always refer to the weak ones defined above. When we speak of structural data, we refer to the set of parameters {q, N , C o , C 1 }. We also write γ as a generic positive constant that can be quantitatively determined a priori only in terms of the data and it may change line by line. When we write γ (b) we mean to emphasize the dependence on the quantity b.
For ρ > 0 let K ρ (y) be the cube of center at y in R N −1 and edge 2ρ. When y = 0 we simply write K ρ . For z o = (x o , y o ) ∈ E, we define the cylinders scaled by a positive parameter θ : Concerning the parameter θ , it will be a quantity that restores the homogeneity of the Eq.
(1) depending on the structure it is subject to. More precisely, we will set θ = [u] represents a quantity that is of the "dimension" of a given solution u. When q = 2, the cylinder z o + Q ρ (θ ) formally recovers the standard cube in R N of center at z o and edge 2ρ. The exact form of θ hinges upon the context and will be specified in the following. Thus, such cylinders are intrinsically scaled. When θ = 1, we simply write z o + Q ± ρ . Suppose that u ∈ L ∞ (E) is a local weak solution to (1) and (2). For a compact set K ⊂ E introduce the intrinsic distance from K to ∂ E by Fix z o ∈ E and let R > 0 be so small that for a small parameter ε > 0; we may assume that z o coincides with the origin. We set

Construct the cylinder
where Λ > 1 is a parameter to be determined in terms of the data only. Without loss of generality, we may assume that ω ≤ 1, such that This Q R (θ ) is the starting cylinder for the reduction of oscillation. Now we state our theorem concerning the interior Hölder regularity which holds for all 1 < q < 2. (1) and (2) in E. Then u is locally Hölder continuous in E. More precisely, there exist constants γ, Λ > 1 and α ∈ (0, 1) that can be determined a priori only in terms of the data, such that for any 0 < ρ < R, we have ess osc

Theorem 1 Let u be a bounded, local, weak solution to
In other words, for every compact set K ⊂ E, for every pair of points z 1 , z 2 ∈ K .
In order to state Harnack's inequality, introduce the following intrinsically scaled cylinder centered at z o ∈ E: wherec ∈ (0, 1) will be determined in terms of the data and It is worth mentioning that (6) ensures κ def = Nq − 2(N − 1) > 0. Since every locally bounded, local, weak solution to (1) and (2) has a continuous representative by Theorem 1, we may only deal with continuous weak solutions.
Theorem 2 Let u be a continuous, non-negative, local, weak solution to (1) and (2) in E. Assume (6) holds and the cylinder (5) is contained in E. There exist constants c,c ∈ (0, 1) depending only on the data, such that c sup Theorem 2 has been stated for continuous solutions, to give meaning to u(z o ). However, it continues to hold for non-negative weak solutions to (1) and (2) for almost all z o ∈ E and for the corresponding intrinsic cylinders.
The proofs of Theorems 1 and 2 are based on the following expansion of positivity. To this end, we introduce another intrinsically scaled cylinder centered at z o ∈ E: for some positive number M and a parameter δ ∈ (0, 1) to be determined in terms of the data only.
Proposition 1 Let u be a non-negative, local, weak super-solution to (1) and (2) in E. Assume the cylinder (7) is contained in E and for some β ∈ (0, 1) and M > 0. There exist constants η, δ ∈ (0, 1) depending only upon the data and β, such that The proof of Theorem 2 will also use the following L 1 loc − L ∞ loc estimate. (1) and (2) in E and assume (6) holds. There exists a constant γ > 0 depending only on the data, such that for all cylinders

Remark 1
Our main results continue to hold if we multiply u x x in (1) by a measurable coefficient a(y) satisfying C o ≤ a(y) ≤ C 1 for some positive C o and C 1 . Nevertheless we do not know how to demonstrate them for more general operators at this moment. See (8)-(10) for the technical obstruction.

Novelty and significance
The main contribution of this work is to present a local regularity theory regarding (1) in the case q < 2. In Theorem 1, we obtain that locally bounded solutions are Hölder continuous for all 1 < q < 2. Moreover, we establish an intrinsic Harnack inequality for non-negative solutions to (1) in Theorem 2, which holds only for q > q * , where q * > 1 is defined in (6). The main tool of proving Theorems 1 and 2, i.e. the expansion of positivity for non-negative, super-solutions, is presented in Proposition 1. Such a property plays a central role in any kind of Harnack estimates for elliptic and parabolic equations. Moreover, Theorem 3 serves as another main component in the proof of Theorem 2. It may be viewed as a Harnack inequality in L 1 loc − L ∞ loc topology. The main idea of treating the local regularity issues for (1) is the so-called intrinsic scaling, which was originally formulated in the theory of degenerate and singular parabolic equations. We refer to [2,4,7] for an account of the theory. To realize such a key idea, we also owe technical tools to the parabolic theory. The exponential shift technique in our proof of Proposition 1 is borrowed from [6], which was used to handle the singular parabolic equations. Theorem 3 also has a counterpart in the theory of singular parabolic equations, though we need some new input to prove it. See [5] and [7,Appendix A]. However, it is not immediately clear whether the same kind of approach can be applied to more general anisotropic p-Laplacian operators. We think this intriguing interplay between the theories of elliptic equations and parabolic equations deserves a deeper understanding in the future investigation.
As for the organization of this note, we first collect some preliminary tools in Sect. 2, including energy estimates, DeGiorgi-type lemmas, a logarithmic lemma and its consequences. In Sect. 3, we give a proof of Proposition 1, concerning the expansion of positivity. Section 4 is devoted to the proof of Theorem 1. To streamline the presentation, we show Theorem 2 in Sect. 5 assuming Theorem 3, which will be proven in Sect. 6.
Ifp < N , then there exists a positive constant γ depending only on the set of parameters {N , p}, such that According to our notion of weak solution for (1), the index we will use is p = (2, q, . . . , q).

Energy estimates
For k ∈ R, we set The first proposition concerns energy estimates in the interior. (1) and (2)

Proposition 2 Let u be a local, weak sub(super)-solution to
Proof We only show the case of super-solutions, the other case being similar. In the weak formulation of (1), we take the test function −(u − k) − ζ 2 ; a standard calculation yields that We will employ the structure conditions and Young's inequality repeatedly in estimating the various terms. We first estimate the terms on the left-hand side. The first term is while the second term is estimated by Now we estimate the terms on the right-hand side. The first term is estimated by Similarly, the second term is estimated by Collecting all these estimate gives the desired result.
The second proposition concerns energy estimates involving boundary information. We state it in the case of super-solutions, while sub-solutions have an analogous statement. Let θ and M be positive parameters.

Proposition 3 Let u be a local, weak super-solution to
Then there exists a constant γ > 0 depending only on the data, such that for any k ≤ M and for every non-negative, piecewise smooth cutoff function ζ vanishing on ∂ Q ρ (θ ), we have Proof Assume z o = (0, 0). The function −(u − k) − ζ 2 vanishes on the boundary of Q ± ρ (θ ), since u(0, ·) ≥ M on the set K ρ and k ≤ M, and thus it is an admissible test function. Using this test function in the weak formulation in Q ± ρ (θ ), the remaining calculation runs similar to the proof of Proposition 2.

DeGiorgi-type Lemmas
Suppose θ and M are positive parameters. (1) and (2) in E. There exists a number ν > 0 depending only upon the parameters θ , M and the data, such that if

Lemma 2 Let u be a non-negative, local, weak super-solution to
Proof We may assume z o = (0, 0). For n = 0, 1, . . ., we set Introduce the cutoff function ζ vanishing on ∂ Q n and equal to identity inQ n , such that In this setting the energy estimate in Proposition 2 yields where we have set A n = [u < k n ]∩ Q n . Now setting ζ to be a cutoff function which vanishes on the boundary ofQ n and equals identity in Q n+1 , an application of Lemma 1 gives that In terms of Y n = |A n |/|Q n |, this can be rewritten as where γ 1 and γ depend only on the data. Hence by [2, Chapter I, We also have a version involving boundary data. (1) and (2)

Lemma 3 Let u be a non-negative, local weak super-solution to
There exists a number ν > 0 depending only upon the parameters θ , M and the data, such that if Proof The proof is similar to that of Lemma 2. This time, we may employ the energy estimate in Proposition 3, since k n ≤ M. The constant ν presents the same form as in Lemma 2.

A logarithmic estimate
Let Q be a cylinder in E. Suppose u is a non-negative, weak super-solution to (1) and (2) in E. For a ∈ (0, 1) and M > 0, we introduce the following function in Q: We use the structure conditions (2) and Young's inequality repeatedly to estimate various terms separately. For the left-hand side, the first term is The second term on the left is treated similarly: For the right-hand side, the first term is estimated by The second term is estimated similarly: Collecting all above estimates yields the desired result.

Consequences of the logarithmic estimate
Suppose u is a non-negative, local, weak super-solution to (1) and (2)

Propagation of measure theoretical information
Lemma 5 Assume for some β ∈ (0, 1) there holds Then there exists j * > 0 depending only on the data and β, such that Proof Assume z o = (0, 0). We first apply Lemma 4 in Q 2ρ (θ ), choosing a = 2 − j for some j ∈ {1, 2, . . . , j * } and a cutoff function ζ that equals 1 in Q ρ (θ ) and vanishes on ∂ Q 2ρ (θ ), such that In this way, we obtain From this, a straightforward application of Hölder's inequality yields where, for ease of notation, we have set By the measure theoretical information known at x = 0, we estimate These estimates joint with the mean value theorem give us for all x ∈ −θρ q 2 , θρ q 2 . The left-hand side is estimated from below over a smaller set Collecting the above estimates, we have for all x ∈ −θρ To conclude, we may choose j * large enough, such that

A shrinking lemma
Then there exists a constant γ > 0 depending only on the data, such that for any j * > 0 Proof Assume z o = (0, 0). The proof is similar to that of Lemma 5. However, the quantitative information u(0, ·) ≥ M in K ρ yields K ρ ψ(0, y) dy = 0.
Hence we have for all x ∈ −θρ The proof is concluded by redefining γ properly.

Propagation of pointwise information
A combination of Lemmas 3 and 6 shows that the positivity of a non-negative, local supersolution at x o over a y-cube spreads in x direction over a smaller y-cube.
Then there exists j * > 0 depending only on the data, such that Proof Assume z o = (0, 0). By Lemma 3 in the cylinder Q ± 2ρ (θ ) with θ = (2 − j * M) 2−q 2 for some j * > 0 to be chosen and a = 1 2 , there exists a positive constant ν depending only on the data, such that if Now we choose the constant j * according to Lemma 6, such that γ j −1 * ≤ ν.

Proof of Proposition 1
The proof of Proposition 1 hinges upon the following preliminary version.

Proposition 4
Suppose the hypothesis in Proposition 1 holds. If for some β ∈ (0, 1) and M > 0 there holds Then there exist constants η, δ ∈ (0, 1) depending only upon the data and β, such that for all We will assume without loss of generality that z o coincides with the origin. To simplify the presentation, we deal with the half space [x > 0] only, since the case of [x < 0] is similar.

Change of variables
Introduce the new variablez = (x,ỹ) and the unknown function v defined bỹ It maps the cylinder 0, M 16 . Now we proceed to show that v satisfies a differential inequality with a similar type of structure conditions as (2). To start with, we have by differentiation that As a result, v will satisfy the following differential inequality weakly: where we have definedÃ u, Du).

Proposition 5
Suppose v ≥ 0 satisfies (9) weakly in a cube Q in R N . There exists a constant γ > 0 depending only on the data, such that for every non-negative, piecewise smooth cutoff function ζ vanishing on ∂ Q, we have for any k ≥ 0 that Proof For simplicity of notation, we still denote the variable of v by z = (x, y) andÃ i by A i . In the weak formulation of (9), we take the test function (v − k) − ζ 2 ; discarding the non-negative contribution of v, a standard calculation yields that We will employ the structure conditions and Young's inequality repeatedly in estimating the various terms. We first estimate the terms on the left-hand side. The first term is estimated by The third term is estimated by Now we estimate the terms on the right-hand side. The first term is estimated by Similarly, the second term is estimated by Collecting all these estimates gives the desired result.

Shrinking the measure of the set [v ≈ 0]
The measure theoretical information in Lemma 5 gives that for some positive integer j o depending only on the data, such that Letx o , n * > 0 to be chosen and set With this stipulation the measure theoretical information above implies and for all n > 0. Assume momentarily thatx o has been chosen. Introduce the pair of cylinders Lemma 8 There exists n * > 0 depending only on the data, such that for any ν ∈ (0, 1), Proof We employ the energy estimate (10) in the cylinder Q , with levels k n and a nonnegative, piecewise smooth, cutoff function ζ in Q , which equals 1 in Q and vanishes on ∂Q , satisfying Since j o and thus¯ are fixed, one verifies that if we takex o large enough then we have and as a result From this observation, the energy estimate (10) gives us that where we have used the fact that k 2−q n ≤ k 2−q o . Next, we apply [2, Chapter I, Lemma 2.2] to v(x, ·) for over K 8 , with levels k n > k n+1 . Taking into account the measure theoretical information in (12), this gives and integrate the above estimate in dx over I ; we obtain by using the energy estimate Now take the power q q−1 on both sides to obtain Add these inequalities from 0 to n * − 1 to obtain From this we conclude that Thus we may fix n * by choosing γ n

A DeGiorgi-type lemma for v
Suppose for the momentx o > 0 has been chosen in terms of the data. Assume for some b > 0 to be determined only in terms of the data, there holds the set inclusion: We may employ the energy estimate (10) in (x 1 , 0) + Q 8 (θ ) to obtain the following.
The constant n * will be chosen in (14) first and thenx o in (17) Assuming this for the moment, we would have Consequently, the energy estimate (10) yields Next, we may proceed exactly as in Lemma 2 to obtain the recurrence inequality where d, γ 1 and γ are positive constants depending only on the data, and Y n = |A n |/|Q n |.

Expanding the positivity of v
Assume momentarily that n * andx o have been determined. We may also assumē For at least one of these cylinders, say Q n , there must hold Having ν fixed in terms of the data as in Lemma 9, we may choose n * according to Lemma 8, such that Apply Lemma 9 to v over Q n with θ = k Consequently, there exists somex 1 in the rangẽ

Returning to the original coordinates
In terms of the original coordinates and the function u, we arrive at where x 1 corresponds tox 1 according to the change of variables (8). Apply now Lemma 7 with M replaced by M o to obtain that there exists j * > 0 depending only on the data, such that, setting = 2 − j * , M a.e. in K 2ρ (16) for all The parameterx o is still to be chosen. Now we choose it such that the right-hand side of the above interval equals M 2−q 2 ρ q 2 , i.e., which implies the choice ofx o : Taking into consideration (15), this choice ofx o implies that which guarantees that (13) holds. Therefore, (16) holds for From the definition ofx o and the change of variable (8) one estimates

Proof of Proposition 1 concluded
Since according to Lemma 5, there exists j o > 0, such that We apply Proposition 4 with M replaced by As a result, there exist η, δ ∈ (0, 1), such that The proof of Proposition 1 is completed by properly redefining constants η and δ.

Proof of Theorem 1
Let the parameters δ and η be fixed as in Proposition 1 with β = 1 2 . In order to apply Proposition 1, we set Then one of the following two alternatives must hold: According to Proposition 1 with M = 1 4 ω, we have In either case, we obtain a reduction of oscillation. More precisely, ess osc Once we have this reduction of oscillation, the rest of the proof is quite standard. We refer to p. 45 of [2] for details.

Proof of Theorem 2 assuming Theorem 3
Fix z o ∈ E, assume u(z o ) > 0, and construct cylinders Introduce the new variablesz = (x,ỹ) defined bỹ Under this mapping, the cylinder z o + Q 4ρ (θ ) is transformed intõ Consider the new function inQ: Then v(z) is a bounded non-negative, weak solution to subject to the following structure conditions a.e. inQ: Therefore Proposition 1 holds for v. In what follows, for simplicity of notation, let us still denote the variables of v by z = (x, y). The proof of the right-hand side inequality in Theorem 2 is a consequence of the following.

Proposition 6
There exist constants c,c ∈ (0, 1), which can be determined only in terms of the data, such that v ≥ c in Q 1 (c).

Proof of Proposition 6
For τ ∈ (0, 1), introduce the family of nested cubes {K τ } and the families of non-negative numbers {M τ } and {N τ } as follows: where σ > 1 is to be chosen. The two functions [0, 1) τ → M τ , N τ are increasing, and since v is locally bounded. Therefore the equation M τ = N τ has roots and we denote the largest one as τ * . By the continuity of v, there existsȳ ∈ K τ * , such that Moreover, Therefore by the definition of τ * , Now we apply Theorem 3 to conclude that there existsγ ∈ (0, 1) depending on the data, such that Next, we may take Λ in the proof of Theorem 1 even larger, such that Then (0,ȳ) + Q R (θ) serves as the starting cylinder in Theorem 1. Let * ∈ (0, 1) and set r = * R. By Theorem 1, for all r < R and for all z provided we choose * so small that

This in turn gives
From this we may start employing Proposition 1 to conclude that there exist positive constants η and δ as indicated, such that Repeated applications of Proposition 1 then yield positive constants η and δ as indicated, We may assume * (1−τ * ) is a negative, integral power of 2. Then choose n such that 2 n r = 2.
In this way, we calculate Finally, we may choose σ such that 2 σ η = 1. As a result, setting On the other hand, one estimates provided σ ≥ q/(2 − q), which may be assumed by possibly taking η smaller if necessary.
In conclusion, v ≥ c in Q 1 (c).

Proof of Theorem 2 concluded
We have shown in the last section that Indeed, if not, by continuity of u, there would exist z * ∈ z o + Q ρ (θ ) such that The membership of z * in z o + Q ρ (θ ) implies that , which in turn gives that However, this leads to a contradiction: The proof of Theorem 2 is now concluded by properly redefining c.

Proof of Theorem 3
We first present two propositions from which Theorem 3 follows.

Proposition 8 Let u be a locally bounded, local, weak sub(super)-solution to
There exists a constant γ > 0 depending only on the data, such that for all cylinders

Proof of Proposition 7
Assuming z o = (0, 0), let us consider the cylinder [−T , T ] × K ρ . In what follows we denote by ζ 2 (y) a piecewise smooth function that vanishes on K ρ and equals 1 on K σρ , such that whereas for 0 < t < T we define Let us introduce a constant α that satisfies such that the quantities 2 − α, q − α and α(q − 1) are all in (0, 1). The proof of Proposition 7 replies on the following two lemmas.

Auxiliary lemmas
Lemma 10 Suppose the hypothesis in Proposition 7 holds. There exists a constant γ > 0 depending only on the data and α, such that Proof We may assume that u ≥ > 0 for otherwise we may work with u + and then let → 0. Now we use u 1−α ζ 1 ζ q 2 as a test function in the weak formulation of (1). Formally, we have We estimate the two terms on the left separately. First of all, Combining them we obtain the desired conclusion.

Lemma 11
Suppose the hypothesis in Proposition 7 holds. There exists a constant γ > 0 depending only on the data and α, such that Proof Notice that by our choice of α, the quantities 2 − α, q − α and α(q − 1) are all in (0, 1). By Hölder's inequality and Lemma 10, we estimate

Proof of Proposition 7 concluded
We use ζ 1 ζ q 2 as a test function in the weak formulation of (1) for super-solutions. Formally, we have We estimate the two terms on the left separately. First of all, where On the other hand, we estimate Combining the above two estimates we obtain We integrate the above inequality in dt over (0, τ ) for τ < T , use Lemma 11 and apply Young's inequality to obtain for an arbitrary δ ∈ (0, 1), As τ ranges over (0, T ), we arrive at An interpolation argument ([2, Chapter I, Lemma 4.3]) yields that
In this setting the energy estimate in Proposition 2 yields Next setting A n = [u > k n+1 ] ∩ Q n , we observe that for any r > 0 As a result, applying Hölder's inequality and the above observation with r = 2, we estimate the second integral on the right-hand side of the energy estimate by Putting this back to the energy estimate gives us Now setting ζ to be a cutoff function which vanishes on the boundary ofQ n and equals identity in Q n+1 , an application of Lemma 1 and the energy estimate gives that Q n+1 Hence by setting we arrive at the recurrence inequality where γ, γ 1 > 0 are absolute constants. Now let us stipulate that k ≥ t 2 ρ q 1 2−q , such that the above recurrence inequality becomes Hence by [2, Chapter I, Lemma 4.1], Y n → 0, i.e. M σ ≤ k, if we require that This is fulfilled if we require κ 2 = (N + 1)q − 2(N − 1) > 0 and take As a result, we arrive at for any r satisfying r < 2 and κ r = (N + r − 1)q − 2(N − 1) > 0. Thus