Quasi-Harnack inequality
Daniela De Silva Ovidiu Savin





Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA
desilva@math.columbia.edu

Department of Mathematics, Columbia University, New York, NY 10027, USAsavin@math.columbia.edu
O. S. is supported by  NSF grant DMS-1200701.



In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some r_0>0 (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale r_0 they satisfy a Weak Harnack type estimate. 

We also give several applications of the main result in very different settings such as discrete difference equations, nonlocal equations, homogenization and the quasi-minimal surfaces of Almgren.  



Quasi-Harnack inequality
    O. Savin
    
========================



§ INTRODUCTION




 The Harnack inequality of Krylov and Safonov plays a fundamental role in the regularity theory of nonlinear second order elliptic equations. It states that a nonnegative solution u ≥ 0 to a second order equation satisfies

    a^ij(x)u_ij=0     ⟹    sup_B_1/2u ≤ C u(0),

 where C is a universal constant depending only on n and the ellipticity constants λ_min, λ_max of the coefficients a^ij(x). It is a quantified version of the maximum principle and it provides the C^α estimates and the compactness for the class of solutions to uniformly elliptic equations with measurable coefficients.

Harnack inequality has a different version for supersolutions (see Lemma 4.5 in <cit.>). It is a pointwise-to-measure estimate known as Weak Harnack inequality and it can be stated in the following way.

Weak Harnack inequality. Assume that u is a positive supersolution

    u ≥ 0,    a^ij(x)u_ij≤0          u(0)=1.

Given δ>0 small, there exists C(δ) large depending on δ and the universal constants such that

    |{u ≤ C(δ) }| ≥ (1-δ) |B_1|.




An interesting observation is that the class of supersolutions in the Weak Harnack inequality can be enlarged to include functions that might not necessarily satisfy any equation. The proof in <cit.> carries through if we require u to satisfy the comparison principle by below only with one quadratic polynomial

    P_Λ(x):= Λ/2 x_n^2 - 1/2 |x'|^2,

together with rotations and dilations of it, with Λ some large fixed constant. 

In this paper we investigate the situation when the comparison with the family of functions P_Λ is satisfied only up to a scale r_0 (small), and at scale C(Λ)r_0, a version of weak Harnack inequality holds. In our main result we show that the Harnack estimates still remain valid with constants independent of r_0. 

The second part of the paper is devoted to show how our results can be applied in several different situations and we discuss four representative examples. 

First, as a direct application, we recover the Harnack inequality of Hung-Ju and Trudinger <cit.> for discrete difference uniformly elliptic equations.   

A second motivating example comes from the theory of nonlocal equations. Solutions to integro-differential equations of order σ∈ (0,2), when σ is close to 2, belong to a family with a two-scale behavior. At small scales the integral behavior dominates whereas at large scales the equation becomes close to being a local equation. We deduce the nonlocal version of the Harnack inequality of Caffarelli and Silvestre <cit.> (with uniform constants as σ→ 2^-) from our results. There are many other problems involving nonlocal operators which fit the general framework given in this paper. We mention for example the uniform Hölder estimates of Caffarelli and Valdinoci for nonlocal s-minimal surfaces as s →1/2^-, or the boundary layer estimates for phase transitions from <cit.>.

A third application of our results comes from the homogenization of degenerate PDEs. 
A simple situation we consider is the case of linear equations with measurable coefficients with ellipticity constants λ_min(x/), λ_max( x/) for some functions λ_min, λ_max periodic of period 1. Notice that if λ_min, λ_max degenerate in the interior of the unit cube but they are bounded away from 0 and ∞ near the sides of the cube, then Harnack inequality holds at scale . It turns out that we end up in a setting as above and the uniform Holder estimates remain valid at all scales (see Theorem <ref>).

Finally, we discuss the case of quasi-minimizers for the perimeter functional for sets which are sufficiently close to half-spaces, and we show that quasi-minimizers exhibit a two-scale type behavior. Then we sketch a nonvariational proof of the C^1,α estimates of Almgren and Tamanini based on Harnack inequality.

    The paper is organized as follows. In Section 2 we introduce the classes of functions 𝒫^I_Λ(r) and 𝒲^a_M(ρ) and we state the main results of the paper Theorem <ref> and Proposition <ref>. Section 3 is devoted to the proof of the main results. In Section 4 we discuss some of the applications mentioned above, and finally in Section 5 we consider the case of quasi-minimizers. 
 


§ STATEMENT OF THE MAIN RESULTS

Let Λ>0 be fixed (large) and let

    P_Λ(x):= Λ/2 (x ·ξ)^2 - 1/2  |x|^2 + b · x + d,     ξ, b ∈^n,    d ∈,
 
be a quadratic polynomial with ξ a unit direction 

    |ξ|=1,      |b|, |d| ≤ 1.

Let r >0, and I ⊂ [0, ∞) be a closed interval. 

We introduce the following definition for 𝒫 _Λ^I(r) which can be thought as the class of supersolutions of size I at scale r.



 Let u: Ω→ be a continuous function on a domain Ω⊂^n. We say that

    u ∈𝒫_Λ^I(r)     ,
 if and only if u cannot be touched from below at any point x_0 in a neighborhood B_r(x_0) ⊂Ω by a polynomial a   P_Λ(x-x_0) as above, for some a ∈ I.


We recall that if u and φ are continuous functions in a domain Ω, we say that φ touches u from below (resp. above) at x_0 ∈Ω in a neighborhood B_r(x_0) whenever

    φ≤ u    (resp. φ≥ u)   in B_r(x_0),   φ(x_0)=u(x_0).


Notice that 

    𝒫_Λ_1^I(r) ⊂𝒫_Λ_2^I(r)     Λ_1 ≤Λ_2,

and in our analysis we assume without loss of generality that Λ is large.

Now, for parameters M, a, ρ>0 we introduce another class of functions 𝒲_M^a(ρ) which can be thought as the class of functions of size a which satisfy Weak Harnack inequality at scale ρ.

 Let u: Ω→ be a continuous function on a domain Ω⊂^n. We say that 

    u ∈𝒲_M^a(ρ)     ,
 if u ≥ 0 and whenever B_2ρ(x_0) ⊂Ω,
and 

    u(x_0) ≤ a,

then 

    |{u ≤ M  a}∩ B_ρ(x_0)|/|B_ρ(x_0)|≥ 1-δ,
 with δ(n)>0 a given constant depending only on n. 


The value of δ(n) in the definition above will be specified later, in the proof of Theorem <ref>.

Let Λ, M>0  be given. Our main theorem reads as follows. 

 Let u be a continuous function in B_1, with 0 ≤ u ≤ 1. There exist constants c_0, C̅, depending only on n and Λ and positive constants a(M),η(M) depending also on M such that if 

    1-u,u  ∈ 𝒫_Λ^[η, 1](r) ∩𝒲_M^a(C̅ r)

for some r ≤ c_0, then  

    osc_B_1/2 u ≤ 1-η.



The proof of Theorem <ref> is based on the following version of Weak-Harnack Inequality.

  Let u be a continuous function in B_1, with u ≥ 0 and assume that 
    u(x̅) ≤ 1   for some x̅∈ B_1/2.
 There exist constants c_0, C_0,C̅ >0 depending only on n and Λ such that 
if 

    u  ∈ 𝒫_Λ^[1,C_0](r) ∩𝒲_M^C_0(C̅ r)

for some r ≤ c_0, 
then  

    |{u ≤ 2 C_0 M }∩ B_1/2| ≥3/4 |B_1/2|.




Once Theorem <ref> is established, the proof of Theorem <ref> follows from standard arguments 
by choosing a=1/4M and η= 1/4C_0M.


Notice that the estimates of Theorems <ref> and <ref> do not depend on r. As r → 0 we recover the standard Harnack inequality for viscosity solutions since any continuous function u satisfies for a fixed a>0, u ∈𝒲^a_2(r) (with δ=0) for all r sufficiently small (depending on the modulus of continuity of u).

We also obtain a version of Theorem <ref> which applies to multivalued graphs or, more generally, to closed sets Γ⊂^n+1 (see Proposition <ref> and Remark <ref>).

In the second part of the paper we provide a variety of applications of our results. 
We may use slightly weaker definitions for the classes of function 𝒫 and 𝒲 which we make precise below. This will be useful whenever we deal with nonlocal problems for which the global behavior of u in B_1 is also relevant.

To this aim, first let us denote by

    P^a_y(x):= P^a(x-y) +c_y,    P^a(x):= -a/2|x|^2,      a>0,    c_y>0,

the class of paraboloids of opening -a and vertex y which have the additional property that

    P^a_y(x) ≤ 0   .

It is straightforward to check that this property implies that 

    y ∈ B_3/4,    P^a_y ≤ a,    |∇ P^a_y| ≤ a



 We say that a continuous function in B_1, u ≥ 0,  belongs to the class 
𝒫^I_Λ(r) whenever u cannot be touched from below at any point x_0 by

    P_y^a(x) + a Λ/2 ((x-x_0)·ξ)^2 χ_B_r(x_0)     B_1,

with a ∈ I. Here P_y^a is a quadratic polynomial as above and ξ is a unit direction.


The set of all contact points (from below) between a given continuous function u ≥ 0 and a paraboloid P_y^a of the class above will be denoted by A_a(u). Precisely,

    A_a(u):={z ∈ B_1: ∃ P^a_y  touching u from below at z in B_1},

and we have that A_a(u) is a compact set included in B_3/4, u ≤ a on A_a(u) and

    A_a_1(u) ⊂ A_a_2(u)







The difference between Definition <ref> and the original Definition <ref> is that above we require in addition that the comparison with a   P_Λ (x-x_0) should hold only at points x_0 in A_a(u).  
It is clear from (<ref>) that if a nonnegative function satisfies Definition <ref> in B_1 then it satisfies also Definition <ref>.





We modify similarly Definition <ref> to hold only at contact points in A_a(u). 


We say that u ∈𝒲_M^a(ρ) in B_1 if u ≥ 0 and whenever
 x_0 ∈ A_a(u) with touching polynomial P_y^a, then

    |{u ≤ P_y^a + Ma}∩ B_ρ(x_0)|/|B_ρ(x_0)|≥ 1- δ(n).

with δ(n) a small given constant depending only on n.

Clearly, 
    𝒲_M^a_1(ρ) ⊂𝒲_M^a_2(ρ)







§ PROOF OF THEOREMS <REF> AND <REF>.


We present here the proof of our main results.
We follow the lines of the standard proof from <cit.> and we use an Alexandrov-Bakelman-Pucci (ABP) type estimate combined with the Calderon-Zygmund cube decomposition. The difference in our case is that the estimates hold only at scale larger than r and we need to use some discrete versions of the arguments. We provide the details below.

 

Throughout this section, universal constants mean positive constants that depend only on n and Λ.

Our starting point is the study of the contact set A_a(u) defined in (<ref>) for a function u in the class 𝒫_Λ^a(r).

 Let z_1,z_2 ∈ A_a(u) ∩ B_5/8 and let y_1,y_2 be the corresponding vertices of the touching paraboloids.
If u ∈𝒫_Λ^2a(r) for some r>0, then

    |z_1-z_2| ≥ c_0 |y_1-y_2|
 with  c_0 >0 universal, as long as 
    |y_1 - y_2| ≥ C_0 r
 for a C_0 >0 universal.

 
After dividing by a we may assume that a=1.
Denote 
    |y_1-y_2|=:d,
 and assume by contradiction that the corresponding touching points z_1, z_2 
are at at distance less than c_0 d  with c_0 to be made precise later. 
Then, we show below that u can be touched by below 
at a point x_0 ∈ A_2(u) in a B_r(x_0) neighborhood 
by a polynomial of the form  2 P_Λ(x-x_0), 
and we contradict the fact that u ∈𝒫_Λ^2(r).

Indeed, let Π be the hyperplane determined by the intersection {P^1_y_1=P^1_y_2} 
of the touching paraboloids at z_1,z_2 respectively. 
Let us assume, without loss of generality that the line segment joining z_1 and z_2 
intersects Π at 0 and that y_2-y_1=d   e_n.
In particular, 
    Π: ={x_n=0},    z_1 ∈{x_n ≤ 0},    z_2 ∈{x_n ≥ 0}

and

    |z_i| ≤ c_0   d,    i=1,2.


Denote by P̃ the quadratic polynomial 

    P̃(x):= P^1_y_1(x) + d/2 x_n + Λ x_n^2 - 1/2 |x|^2 +  c_0 d^2,

and notice that P̃ can be written as

    P̃= P^2_(y_1+y_2)/4 + Λ x_n^2 + const.



Let 
    v(x):= u(x)-P̃(x),

and let D denote the annular region

    D:={c_1 d ≤ |x| ≤  4c_1 d },
 with c_1:=c_0^1/3≫ c_0 small, universal. We claim that 

    v ≥max_i=1,2 P^1_y_i - P̃ > 0   on D, and   v(z_1) <0.

This shows that P̃ + min_B_4c_1dv touches u from below 
at a point x_0 ∈ B_c_1 d in the neighborhood B_4c_1d⊃ B_3c_1d(x_0). 
We contradict the hypothesis u ∈𝒫^2_Λ(r) 
provided that we show x_0 ∈ A_2(u) and that 
we choose C_0 = (3c_1)^-1 in (<ref>) so that 3 c_1 d ≥ r.


Next we check that x_0 ∈ A_2(u). 
Let P^2_y̅ be the polynomial of opening -2 tangent to P̃ + min_B_4c_1dv at x_0. 
Then P^2_y̅≤ u in B_4c_1 d and P^2_y̅≤max_i=1,2 P^1_y_i outside B_4c_1 d. 
The last assertion follows from the fact that 
the difference between max P^1_y_i and P^2_y̅ is convex 
and by (<ref>) it is positive on D and negative at x_0, therefore it must be positive also outside B_4c_1 d. 
In conclusion P^2_y̅ touches u by below at x_0 in the whole domain B_1 
and also P^2_y̅≤ 0 outside B_3/4, hence x_0 ∈ A_2(u). 


It remains to prove the claim (<ref>). For the first inequality we restrict, say, to the region D ∩{x_n ≤ 0}, and we have that either

    x_n ≤ -c_1 d/2      |x'| ≥c_1 d/2.

In the first case,

    P_y_1^1 - P̃≥(c_1/4  - 16 Λ c_1^2 -  c_0)  d^2 >0,
 
by choosing c_1 sufficiently small. In the second case

    P_y_1^1 - P̃≥ -x_n (d/2 + Λ x_n ) + (c_1^2/8 -  c_0 )d^2 > 0.

For the second part of (<ref>) we compute

    v(z_1)=  P^1_y_1(z_1)- P̃(z_1)  ≤( c_0/2  + 1/2 c_0^2  - c_0 ) d^2 ≤  0.
 





In what follows, 
we work on cubes Q_ρ(x̃) of size ρ defined as:

    Q_ρ(x̃) := {x ∈^n : |x_i - x̃_i| < ρ/2,    i=1,…, n.}
 Sometimes, for simplicity of notation, we drop the dependence on the center x̃ when there is no possibility of confusion.



We decompose the space in dyadic cubes of size 2^-l, l=0,1,2,... 
and by abuse of notation, denote by Q_l a  cube of size 2^-l obtained from this decomposition. Set,

    A_a^l(u):= ⋃_Q_l∩ A_a(u) ≠∅ Q_l,

that is A_a^l(u) denotes the collection of the dyadic cubes of size 2^-l which intersect A_a(u).

 Then, with this notation
Lemma <ref> gives the following corollary.

 Assume Q_ρ=Q_ρ(x̃) ⊂ B_5/8 and that 
    A_a(u) ∩ Q_ρ/2(x̃) ≠∅.
 
If u ∈𝒫_Λ^4a(r) and r≤ 2^-l, then

    |A_2a^l(u) ∩ Q_ρ| ≥η |Q_ρ|,
 for a universal constant η.

 Let  x̅∈ A_a(u) ∩ Q_ρ/2 and let P^a_y̅ be a corresponding touching paraboloid at x̅. For a set of vertices y ∈ B_ρ/10(x̅), we consider the polynomial 

    P̃(x):= P_y̅^a(x) - a/2 |x-y|^2 + c_y,

 with c_y chosen so that P̃ becomes tangent to u from below. Notice that
P̃(x)=P_(y+y̅)/2^2a(x).
Since y ∈ B_ρ/10(x̅) we easily conclude that P̃≤ P^a_y̅ outside B_ρ/2(x̅). 
This means that the corresponding contact point between the polynomial P̃ and u 
lies in Q_ρ∩ A_2a(u).
 
 We now select vertices y in B_ρ/10(x̅), with x̅ among them, at distance larger than 2 C_0 2^-l from each other. By Lemma <ref>, the corresponding contact points will also be at distance larger than c_0 2^-l from each other, with c_0, C_0 as above. Since we can select  C ρ^n 2^ln such vertices, we have at least as many contact points at distance larger than c_0 2^-l. Then our statement follows immediately from the definition of A^l_2a(u).

Next, we obtain a generalized version of the corollary above, in which A_a(u) intersects 
Q_3 ρ  instead of Q_ρ/2.

 Assume B_6 √(n)ρ(x̃) ⊂ B_5/8 and that 

    A_a(u) ∩ Q_3 ρ(x̃) ≠∅.

If u ∈𝒫_Λ^[a, C_1 a](r) with r ≤min{c_1 ρ, 2^-l}, then

    |A^l_C_1 a(u) ∩ Q_ρ| ≥η |Q_ρ|,

with C_1, c_1 and η universal.

 After a dilation, translation and multiplication with a constant we may assume that x̃=0, ρ=1 and a=1.
In view of Corollary <ref>, it suffices to show that there exists a point 
x_0 ∈ A_C(u) ∩ Q_1/2, for C large universal.

To prove the existence of x_0, we define the following barrier:

    ϕ(x):= |x|^-γ-β,      x ≠ 0

with γ=4Λ,
β>0 chosen so that

    ϕ(x)=0   on B_6√(n).

Denote by P_σ the quadratic polynomial tangent to ϕ by below on the sphere of radius σ>0.
Using the homogeneity of |x|^-γ, we find  

    ϕ≥ P_σ=P^d_0

and if |x_0|=σ, then by the Taylor expansion of ϕ near x_0 we get

    (ϕ - P_x_0/2^2d)(x) ≥(γ+1) d/2  ((x-x_0)·ξ )^2,         x ∈ B_c̅σ(x_0),    ξ=x_0/|x_0|,

with c̅, sufficiently small universal.

Let 

    ϕ̃(x)= ϕ(x)   in |x| ≥1/4,   ϕ̃(x) = P_1/4,   on B_1/4.
 
For x̅∈ A_1(u) ∩ Q_3, we call P^1_y̅ a corresponding touching polynomial of vertex y̅.

We slide ϕ_t= P^1_y̅ + αϕ̃+ t, (for some constant α>0) from below till it touches u for the first time at some point x_0. Since ϕ≤ 0 outside B_6√(n) and u≥ P^1_y̅ with equality at x̅∈ Q_3 ⊂ B_6√(n), the first contact point x_0 occurs for some value of t ≤ 0 and x_0 must belong to B_6√(n).

We claim that x_0 ∈ B_1/2. Otherwise, at x_0 we have a tangent polynomial of opening a_0:=1+ 2 α d by below
with σ=|x_0| and d as in (<ref>), and 

    α (γ +1)d ≥Λ (1+2 α d)=Λ a_0,

provided that α is chosen sufficiently large universal. 
We use (<ref>)-(<ref>) together with the fact that |x_0| ≥1/2 to contradict that u ∈𝒫_Λ^[1,C_1](r),  as long as  r ≤ c_1 ≤c̅ / 2, and C_1 ≥ 1+ 2 αγ 2^γ+2≥ a_0.

Thus the touching occurs in the  ball B_1/2⊂ Q_1  and the desired point x_0 is obtained.



A standard iteration argument and the Calderon Zygmund cube decomposition give the following result.

  

Assume B_6 √(n)ρ(x̃) ⊂ B_5/8 and Q_ρ(x) is a dyadic cube such that 

    A_a(u) ∩ Q_3 ρ(x̃) ≠∅.

If u ∈𝒫_Λ^[a, C^k_1 a](r) with r ≤ c_1 2^-l, and 2^-l≤ρ, then

    |A^l_C^k_1 a(u) ∩ Q_ρ| ≥ (1-(1-η)^k) |Q_ρ|.





 After a dilation and a multiplication by a constant we may assume as above that ρ=1, x̃=0, a=1. The case k=1 holds by Lemma <ref>.

For k=2 we apply the Calderon Zygmund decomposition to the set A_C_1(u) in Q_1: we decompose dyadically the cube Q_1, up to scale 2^-l, and at each step j in our decomposition, we only split the cubes Q̃_j which have non-empty intersection with the set of contact points A_C_1(u). A remaining cube in the decomposition (which has empty intersection with A_C_1(u)) will be called “clean". Let Q̃_j be a clean cube of size 2^-j≥ 2^-l, and by construction the dilation of factor 3 around the center Q̃_j intersects A_C_1(u). Lemma <ref> applied to the cube Q̃_j with a=C_1 gives

    |A_C_1^2^l(u) ∩Q̃_j| ≥η |Q̃_j|.

Since this inequality holds for all clean cubes, we conclude that

    |A^l_C_1^2(u) ∩ (Q_1 ∖ A_C_1^l(u))| ≥η (|Q_1| - |A_C_1^l(u)|).,

which gives the desired conclusion. The general case follows by the argument above k-1 times.


Next we use a covering argument and obtain the following version of Lemma <ref> for functions in the general class 𝒫_Λ^I(r).


 Let u be a continuous function in B_1, with u ≥ 0 and assume that 
    u(x̅) ≤ 1   for some x̅∈ B_1/2.
 Given μ>0, there exist constants c̅, C̅ >0 depending only on n and Λ and C(μ) depending also on μ such that if

    u  ∈ 𝒫_Λ^I(r),        I=[1,C(μ)],

for some r ≤c̅, 
then  

    |A^l_C(μ)(u) ∩ B_1/2| ≥ (1-μ) |B_1/2|,       C̅ r ≤ 2^-l.



The proposition states that if we decompose the space in cubes of size C̅ r and then we consider the collection of those cubes which intersect the contact set 
A_C(μ)(u) then they cover all but a μ-fraction of B_1/2. We recall that A_C(μ)(u) represent the set of points where u admits a tangent polynomial of opening - C(μ) by below and which also are less than 0 outside B_3/4.

Clearly Proposition <ref> follows from Lemma <ref> since the existence of x̅ implies
A_C(u) ∩ B_1/16(x̅) ∅ for some large C, and then we can cover B_1/2 with a finite collection of dyadic cubes Q_ρ, with ρ universal, for which we can apply Lemma <ref>.


It is straightforward to generalize Definition <ref> to include also closed sets Γ⊂ B_1 ×_+ ⊂^n+1 (which above can be thought as of the graph of u), and to define the corresponding contact sets A_a(Γ) and A^l_a(Γ).
Then Proposition <ref> holds also in the setting of sets Γ∈𝒫^I_Λ(r) since we did not use so far the graph property of u.  

We are now ready for the proof of Proposition <ref>.


Proof of Proposition <ref>. We choose μ = 1/8 in Proposition <ref> and let C_2=C(μ) and l to be the largest l for which C̅ r ≤ 2^-l. Then

    |A_C_2^l(u)∩ B_1/2|≥7/8 |B_1/2|.

Now, let x̃∈ Q_l ∩ A_C_2(u) and assume u ∈𝒲_M^C_2(2 √(n)C̅ r) (see Definition <ref>). 
Then

    |{u ≤ 2 M C_2}∩ B_2 √(n)C̅ r(x̃)| ≥ (1-δ)|B_2 √(n)C̅ r(x̃)|,

which implies

    |{u ≤ 2 M C_2}∩ Q_l| ≥9/10|Q_l|,

provided that δ=δ(n) is chosen sufficiently small. Summing over all cubes Q_l included in A^l_C_2(u), together with (<ref>) and the fact that l is sufficiently large gives

    |{u ≤ 2 M C_2}∩ B_1/2| ≥3/4|B_1/2|.

The proposition is proved after relabeling the constants.




§ APPLICATIONS



 §.§ Discrete equations.
 Harnack inequality for discrete difference equations was obtained by Hung-Ju and Trudinger in <cit.>. Here we verify that the setting we developed for the classes 𝒫 and 𝒲 applies in this context. We consider
for simplicity the discrete operator involving only the neighbors along the coordinate axes (>0 small)

    Lu(x):= ∑_i=1^n λ_i(x) u^_ii(x),    x∈ B_1 ∩ℤ^n,


    u^_ii(x):= u(x+ e_i)+ u(x- e_i) -2 u(x)/^2,

with 

    0<λ_min≤λ_i(x) ≤λ_max,    x∈ B_1 ∩ℤ^n.

Below we show that Theorem <ref> gives the Hölder continuity for bounded solutions of Lu=0. 

 Assume that 

    Lu=0,    |u|≤ 1   in B_1.
 Then,

    |u(x)-u(y)| ≤ C |x-y|^γ,   in B_1/2∩ℤ^n,

with C, γ universal independent of .

Sketch of the Proof. By linearity, it suffices to show that, 

    u ∈𝒫_Λ^1(r) ∩𝒲_M^1(C̅ r),
 
for some appropriate Λ and M large independent of  (to be specified below) and with r ≤ c_0 and C̅ universal as given in Theorem <ref>.

Here we work with the graph of u (see Remark <ref>) and the inequality in the class 𝒲_M^a(ρ) is understood to hold everywhere (with δ=0) instead of in the ℋ^n-measure sense. Another alternative way is to think that u is extended from the lattice to the whole space by some linear interpolation.

It is straightforward to see that for  Λ large

    u ∈𝒫_Λ^1(2).

Indeed, if 
    P_Λ(x)=Λ/2 (x ·ξ)^2- 1/2 |x|^2+ l(x),     ,
 
touches u from below, say at 0 in B_2,
and we choose Λ such that 
    (Λ/n-1)λ_min>(n-1)λ_max,
 we obtain that Lu(0) ≥ LP_Λ(0)>0 and get a contradiction.

Also, if 

    u≥ 0   in B_ρ∩ℤ^n,   ρ≤ K
 and 
    u(0)≤ 1,
 it immediately follows from the fact that Lu(0)=0 that 
    u( e_i)≤ C(n, λ_min, λ_max).

Hence,

    u ≤ C(K)   in B_ρ.
 Choosing K=2C̅, M=C(K) and r= 2 we obtain that u ∈𝒲^1_M(C̅ r) as desired.





 §.§ Homogenization.
 Next we obtain Hölder estimates in the context of homogenization by considering linear equations with measurable coefficients whose ellipticity constants might degenerate in the interior of Q_ the cubes of size .  
Let L_ u be a second order operator 

    L_ u(x):= a^ij(x/) u_ij(x)

with A(x):=a^ij(x) continuous coefficients in ^n satisfying 

    λ̃_min(x) I ≤ A(x) ≤λ̃_max(x) I,    A(x) ≠ 0,

and 

    λ̃_min, λ̃_max≥ 0   periodic of period 1

with

    0<λ_min≤λ̃_min(x) ≤λ̃_max(x)≤λ_max,   in Q_1 ∖ D,

for some set D ⊂⊂ Q_1.

Similarly as in the previous application, we can obtain the following result.

 Assume that 

    L_ u=0,    |u|≤ 1   in B_1.
 Then,

    |u(x)-u(y)| ≤ C |x-y|^γ,   in B_1/2

with C, γ depending on n, λ_min, λ_max and dist(D̅,  Q_1).

Sketch of the Proof. It suffices to show that

    u ∈𝒫_Λ^1(r) ∩𝒲_M^1(C̅ r),
 
as long as ≤ c for some appropriate Λ, M, r.
We rescale u as

    ũ(x)= ^-2 u( x).

Then, 
    L ũ:= a^ij(x)ũ_ij(x)=0   in ^n.
 By assumption (<ref>), a polynomial of the form P_Λ cannot touch ũ in Q_1 ∖ D, as long as Λ is large enough (depending only on n, λ_min, λ_max) so that L P̃_Λ >0. Now, assume for simplicity that P_Λ=Λ/2 x_n^2 - 1/2 |x|^2 touches ũ from below say at 0 ∈ D in B_2√(n). Then, 
    ũ≥ - 2 n   in B_2√(n),

and for c>0 depending on dist(D,  Q_1)

    ũ(x̅) + 2 n≥ cΛ ,   at some x̅∈ Q_1.
 By Harnack inequality applied to ũ+ 2n in a neighborhood of Q_1 (depending on dist(D,  Q_1)) we get that,

    ũ + 2 n ≥c̃Λ  on Q_1.
 Choosing Λ large this gives

    ũ≥ c' >0   on Q_1.
 By the maximum principle the inequality holds also in Q_1 hence ũ (0)>0, a contradiction. Rescaling back this shows that

    u ∈𝒫_Λ^1(r),      r:=2√(n).
 

To prove the other inclusion, assume

    ũ (x_0)=1,
 for some x_0 ∈ Q_1. Again, by the maximum principle,
ũ (x̅) ≤ 1 at some x̅∈ Q_1, and by Harnack inequality,
ũ≤ C(K) on Q_K and therefore in Q_K as well. We take K=8 n C̅ and we obtain
ũ≤ M in B_2 √(n)C̅(x_0). Thus,

    u ∈𝒲^1_M(C̅r),

as desired.




Assume that u satisfies an oscillatory fully nonlinear equation

    F(D^2u, x/)=0,

with F an elliptic operator which is periodic in the second variable, and with ellipticity constants which might degenerate when the second variable belongs to a set D as above. Then Theorem <ref> can be applied to differences between translations of u, and then it follows that u satisfy a uniform C^1,α interior estimate up to scale .  





 §.§ Non-local operators
 We show that solutions to integro-differential equations with measurable kernels satisfy the hypotheses of Theorem <ref>.

Let σ∈ (0,2) and let 
K_x(y) be a symmetric, measurable kernel proportional to the kernel K_σ of ^σ/2

    λ_min   K_σ(y) ≤ K_x(y) ≤λ_max   K_σ(y),      K_x(y)=K_x(-y),

with λ_min, λ_max >0 and

    K_σ(y):=(2-σ)|y|^-n-σ.
 


Consider the integro-differential operator

    ℒu(x):= ∫_^n(u(x+y)-u(x))K_x(y)dy
 and let
u be a viscosity solution to 

    ℒu = f.

For definitions and properties of viscosity solutions to integro-differential equations as above, we refer the reader to <cit.>.

As an application of our Theorem <ref>, we recover the Hölder continuity property of u, with uniform estimates as σ→ 2, which is due to Caffarelli and Silvestre <cit.>.

 Let u be a viscosity solution to 

    ℒ u = f   .
 
Then, 
    u_C^α(B_1/2)≤ C  (f_L^∞ + u_L^∞(B_1) + (2-σ)∫_^n ∖ B_1|u(y)|/|y|^n+σ dy ).
 The constants α>0, and C depend only on n, λ_min, λ_max if σ is close to 2.
  
 
 Once we establish the following lemma, the proof follows easily by scaling and it is left to the reader. 
We assume that σ is sufficiently close to 2.
 
   Let u be a continuous function satisfying
 
    ℒu(x) ≤ 1,      u ≥ 0   in B_1

 and
 
    ∫_^n ∖ B_1 u^-     K_σ dy ≤ 1.

Then there exists Λ depending on n, λ_min, λ_max, such that 

    u ∈𝒫_Λ^a(r) ∩𝒲_2^a(C̅ r),     ∀ a ∈ [1,∞),

for some r small, and with C̅=C̅(n, Λ) the constant from Theorem <ref>.


 
 
 We remark that 

    ∫_B_1 |y|^2   K_σ dy =(2-σ)∫_0^1 t^2-n-σω_nt^n-1dt =ω_n,

where ω_n represents the surface area of the unit sphere in ^n.

Let us choose r so that

    ∫_B_r |y|^2   K_σ dy= 1/2∫_B_1 |y|^2   K_σ dy = ω_n/2  ⟹   r^2-σ= 1/2,

We check that 
    u ∈𝒫^a_Λ(r),     ∀ a ∈ [1, ∞),

for some large Λ. It suffices to consider the case a=1, since the general case follows after dividing by u by a. Assume by contradiction that (see Definition <ref>) 

    P(x) + Λ/2 ((x-x_0) ·ξ)^2 χ_B_r (x_0),

touches u from below at x_0 in B_1, where P=P_y_0^1 is a quadratic polynomial of opening -1 that is below 0 outside B_3/4. We reach a contradiction since

    ℒu(x_0)  ≥∫ _B_r(x_0)    (u-P)(y)K_x_0(y) dy + ∫_B_1/4 (x_0)  (P(y)- P(x_0))K_x_0(y)dy  
        + ∫_ℝ^n ∖ B_1/4(x_0) (u(y)-u(x_0))K_x_0(y)dy,
    ≥ c(n) Λ   λ_min - C(n) λ_max- C(n) λ_max >1

where we have used that u(x_0)=P(x_0) ≤ 1, and (<ref>), (<ref>), (<ref>) together with

    ∫ _ℝ^n ∖ B_1/4 K_σ dy ≤ C(n).


Next we check that 

    u ∈𝒲^a_2(ρ),     ∀ a ∈ [1, ∞),      (2-σ) ρ^-σ≥ C_0.

As above, it suffices to consider the case a=1 and assume that a polynomial P=P^1_y_0 touches u by below at x_0. Using the computation above, together with ℒu(x_0) ≤ 1 (and replacing r by ρ) we find 
 
    C ≥ (2-σ)∫_B_ρ(x_0)(u-P)(y)/|y|^n+σ dy,

for come C depending on n, λ_min, λ_max.
If 
    μ:=|{u-P > 2}∩ B_ρ(x_0)|/|B_ρ|,
 then we obtain

    C ≥ 2 (2-σ) μ |B_1| ρ^-σ,

and we obtain the desired conclusion μ≤δ(n) provided that we choose C_0 sufficiently large.

Finally, it is straightforward to check that ρ=C̅ r with r defined in (<ref>) satisfies the inequality in (<ref>) if σ is close to 2.




If σ is not close to 2 (or if we allow the constants to depend on σ) then the weak Harnack inequality is satisfied at scale 1, and there is no need to consider the class 𝒫. Indeed, the last argument gives u ∈𝒲_M^a(1/2) for an appropriate constant M and all a ≥ 1, and the conclusion of Theorem <ref> is already satisfied. 







§ QUASI-MINIMIZERS OF PERIMETER
 

Let E be a measurable set and Ω an open set in ^n. We denote by P(E, Ω) the so-called perimeter of E in Ω and refer to <cit.> for definitions and properties of the perimeter functional and Caccioppoli sets.

We say that E is a quasi-minimizer of the perimeter functional in Ω if there exist constants κ,α>0 such that

    P(E, B_ρ(x)) ≤ (1+κρ^α)P(F,B_ρ(x))

for all x ∈ E, and all measurable sets F which coincide with E outside B_ρ(x)⊂Ω.

Classical results of <cit.> give that the boundary of a quasi-minimizer E can be split into the union of a C^1,α/2
relatively open hypersurface and a closed singular set of Hausdorff dimension at most n-8. These results extend the well-know theory for minimizers, due to De Giorgi.



Here we want to use the Harnack inequality established in Section 2 to show that quasi-minimizers enjoy a flatness regularity theory, as in  the classical case of minimizers, using a non-variational approach. Precisely, the following theorem holds and can be obtained with a slight modification of our arguments from Section 2, combined with the technique developed in <cit.> for perimeter minimizers.

 Assume that the open set E with  

    {x_n ≤ -}∩𝒞⊂ E ⊂{x_n ≤ -}

is a quasi-minimizer in the cylinder 𝒞:= {|x'| < 1}×{|x_n|<1}. Then 
    E ∩{|x'|<1/2}
 is a C^1,γ surface if  is small enough depending only on n, κ, α. 

The proof of Theorem <ref> is based on an improvement of flatness lemma for E as in <cit.>.
Without loss of generality, after a dilation, we can assume κ=1. Constants depending only on n, α will be called universal. 

We remark that if E is a quasi-minimizer and 0 ∈ E then  

    P(E,B_r) ≤ 2 P(F,B_r),       ∀ r ≤ 1,

and by taking F=E ∖ B_r it can be shown by standard arguments that (see <cit.>) 

    P(E,B_r) ≤ C(n) r^n-1,      |E ∩ B_r| ≥ c(n) r^n.



  From now on E will be a quasi-minimizer to the perimeter functional in the unit cylinder 𝒞 with κ=1. 
Also, balls in ^n-1 will be denoted with B'. Finally, given r>0 (small) we denote by E_r the 1/r dilation of E
 
    E_r := 1/r E.

 
 The desired improvement of flatness lemma is stated below.
 
  There exist r_0, _0>0 small universal such that if 

    E_r ∩ B_1⊂{|x_n| ≤ r^_0} ,    0 ∈  E_r,    r ≤ r_0

then in some new system of coordinates (y', y_n) 

    E_r ∩ B_η⊂{|y_n| ≤η^1+_0r^_0}

for some η>0 small universal.
 

The proof of this lemma relies on a compactness argument, once the appropriate Harnack-type inequality is obtained. 
 The Harnack inequality reads as follows.

 There exist _0, c_0>0, universal such that if for r>0 small

    E_r ∩ (B'_1 × [-1,1]) ⊂{|x_n| ≤ a},    a ∈ [r^_0, c_0]

then 

    E_r ∩ (B'_1/2× [-1,1]) ⊂{ a' ≤ x_n ≤ a”},

with either a'=(η_0-1)a or a”=(1-η_0)a, for some η_0>0 small universal.



Before we proceed with the proof of Lemma <ref>, we show first that E belongs to an appropriate 𝒫_Λ^I class.

 Let E be a quasi-minimizer and ρ>0 small. Then E cannot be touched by below at 0 in the cylinder B'_ρ× [-ρ, ρ] with the graph

    x_n= γ   Q(x'),        Q(x'):=2n x_1^2- 1/2 |x'|^2 + b' · x',

if 
    γ∈ [ρ^-1+μ,  c_1ρ^-1],     γ |b'| ≤ c_1,
 
and c_1, μ>0 universal.





 
Assume that E lies above the graph of γ  Q. Consider the graph

    Γ:={   x_n= ψ(x')},     ψ(x'):=γ(Q + 1/4ρ^2- 1/2 |x'|^2),

with γ≤ c_1 ρ^-1, and define the competitor F 
 
    F:=E ∩{x_n >ψ(x'}.


 We claim that
 
    P(E, B_ρ) - P(F, B_ρ) ≥ c(n) γ   |A|,
 for a universal constant c(n), with
  
    A := E ∩{x_n < ψ(x')}⊂⊂ B_ρ.





To derive (<ref>) let us denote by d_Γ the signed distance function from Γ, which is positive in F and compute
 
    ∫_A Δ d_Γ  dx = ∫_ A (d_Γ)_ν dℋ^n-1≥ℋ^n-1(Γ∩ E) - P(E, {x_n < ψ}) =
 
    = P(F, B_ρ)- P(E, B_ρ).

 Since,
 
    Δ d_Γ(x)= H_Γ_p(x),    x∈ A

 where Γ_p is the parallel surface to Γ passing through x, it suffices to show that
 
    H_Γ_p≥ c(n)  γ  in A.
 
This follows from standard computations (see <cit.>) by using that 

a) |∇ψ|≤ C(n) γρ + c_1 ≤ c_1 C(n), 

b) the principal curvatures of Γ are bounded by C γ, 

c) |d_Γ| ≤ C ρ^2 ≪ (C γ)^-1 in A 

and

    H_Γ(x) = div∇ψ/√(1+|∇ψ|^2)≥ 2 Δψ - C(n) (max |D^2ψ|) |∇ψ|^2  ≥ c   γ,

provided that c_1 is chosen sufficiently small. Thus the desired bound (<ref>) is proven.


 
 
 Now, since E is a quasi-minimizer we get that 
 
    P(E, B_ρ)- P(F,B_ρ) ≤ρ^αP(F,B_ρ) ≤ρ^αP(E, B_ρ) ≤ C ρ^α+n-1

 where in the last inequality we used (<ref>) with r= ρ.
 Combining this estimate with (<ref>) we get that
 
    c  γ  |A| ≤ C ρ^α+n-1.

 By the second density estimate in (<ref>) we have

    |A| ≥ |E ∩ B_γρ^2/8| ≥ c_0 (γρ^2)^n.

Here we have used that by the Lipschitz continuity of Γ

    B_γρ^2/8⊂{x_n <ψ(x')}∩ B_ρ.

Then (<ref>) together with (<ref>) gives 
 
    γ≤ C ρ^-1 + α/n-1≤ρ^-1+μ.




 We remark that the proof above holds if we replace the quadratic part of Q by 1/2 (x')^T M x' with tr M>0 provided that the constant c_1(M) is chosen sufficiently small depending on M as well. In particular the estimate (<ref>) is valid with a constant c=c(M) as long as γρ≤ c_1(M).


We have the following corollary of Lemma <ref>.


Assume that E satisfies the hypothesis (<ref>) of Lemma <ref>. Then in B_1' we have

    E_r + a e_n   ∈ 𝒫^I_4n(a),      I=[a^2, c_1 ].




If x_n= σ P_4n(x') with σ∈ I touches from below E_r at some point, say on x'=0, in the cylinder |x'| ≤ a, then 
    x_n= σ   r   P_4n(x' r^-1) =: σ   r^-1 Q(x'),
 touches E from below in the cylinder |x'| ≤ ar=:ρ, with Q a polynomial as in Lemma <ref>. We contradict the conclusion of the lemma since it is straightforward to verify that

    σ r^-1∈ [ρ^μ-1,c_1 ρ^-1]     σ≤ c_1,

provided that _0 and c_0 are sufficiently small. 




Proof of Lemma <ref>.
If E_r does not lie above x_n=(η_0-1)a in B_1/2' then, by Corollary <ref>, we may apply Proposition <ref> to E_r + a e_n.
 
If [η_0 a, C η_0 a] ⊂ [a^2 , c_1] with C universal, then

    ℋ^n-1(A^l_C η_0 a( E_r) ∩ B_1/2') ≥3/4  ℋ^n-1(B_1/2'),

where 
    A^l_C η_0a( E_r)
 represents the collections of cubes Q_l' of size C̅ a (with C̅ universal) in the x' variable, in which the contact set A_C η_0a ( E_r) projects.

We choose η_0 such that C η_0 = 1/2, and then a ≤ c_0 small such that a^2 ≤η_0 a. Denote by 
A^-_l( E_r) the union of the cubes Q'_l of size C̅ a (small) with the property that 

    Q'_l × [-a, -1/2 a]   ∩ E_r ≠∅.

Since A^l_C η_0a( E_r) ⊂ A^-_l( E_r), the inequality above gives

    ℋ^n-1(A^-_l( E_r) ∩ B'_1/2) ≥3/4ℋ^n-1(B'_1/2).



If E_r does not lie below x_n=(1-η_0)a in B_1/2' then, after using the same argument “upside-down", we find that 

    ℋ^n-1(A^+_l( E_r) ∩ A^-_l( E_r) ∩ B'_1/2) ≥1/2ℋ^n-1(B'_1/2),

where A^+_l( E_r) denotes the union of cubes Q'_l with the property that 

    Q'_l × [1/2 a,  a]   ∩ E_r ≠∅.

On the other hand, by projecting E_r along e_n, we find that for all cubes Q'_l,

    P(E_r, Q'_l × [-a, a]) ≥ℋ^n-1(Q'_l),

with equality only if E_r coincides with a hyperplane x_n = const. in the cylinder Q_l' × [-a,a].  
This inequality can be improved as

    P(E_r, Q'_l × [-a, a]) ≥ (1+ c_2)ℋ^n-1(Q'_l),      Q'_l ∈ A^+_l( E) ∩ A^-_l( E).

This follows by compactness and the fact that (<ref>), (<ref>) 
together with the density estimate (<ref>) imply that 
both E_r and its complement E^c_r have uniform density estimates 
in the two half cylinders Q_l' × [-a,0] and Q_l' × [0,a], hence E_r cannot be close (say in L^1 sense) to a half space in the e_n direction.

  
Therefore, using (<ref>) combined with (<ref>),(<ref>) we obtain 

    P(E, B'_1/2×  [-a, a]) ≥(1+ c_2/4) ℋ^n-1(B'_1/2).

Now we compare E and F:=E ∖ (B'_r/2× [-ra, ∞)) in B_r, and by quasi-minimality,

    P(E,  B'_r/2× [-r/2×r/2]) ≤ (1+ r^α) (1+ Ca)ℋ^n-1(B'_r/2) + C r^α + n-1.

This gives

    P(E_r, B'_1/2×  [-a,a]) ≤ (1+ C a + C r^α)ℋ^n-1(B'_1/2),

and we contradict (<ref>) if a is small enough.




Next we provide the proof of the improvement of flatness Lemma <ref>.



Proof of Lemma <ref>. We argue by compactness. Assume by contradiction that there exist a sequence r_k → 0 and of quasi-minimizers E^k such that E^k_r_k satisfies the assumption of the lemma but not the conclusion.

Denote by 
    E^k_*:= {(x', r_k^-_0x_n) : x ∈ E^k_r_k}  ⊂ B_1' × [-1,1].
 
Then, in view of Lemma <ref> (up to extracting a subsequence), E^k_* converges in the Hausdorff distance sense to the graph E^*:={x_n=w(x')} of a Hölder continuous function w. We wish to prove that w is harmonic. This leads to a contradiction since the graph of a harmonic function satisfies the improvement of flatness with respect to linear functions.

To show that w is harmonic, let us assume by contradiction that a quadratic polynomial Q(x') with 
    Δ Q > δ >0
 
touches w strictly from below, say for simplicity at 0, in a neighborhood B'_δ. Then, by the uniform convergence of E^k_* to E_* we conclude that for k large (rescaling back)

    E^k ⊂{x_n > Q_k:= r_k^1+_0(Q(r^-1x')-c_k)},    c_k → 0   as k →∞.


Notice that the polynomial Q_k can be written as 

    Q_k= γ_k (Q̃ +o(1)),     γ_k:= r_k^_0 -1,

and Q̃ satisfies the hypotheses of Remark <ref> with ρ_k:= δ r_k. We obtain  

    c(δ) γ_k |A_k| ≤ C ρ_k^α+n-1,

where A_k is defined as in the proof of Lemma <ref>.
On the other hand, in view of the Hölder continuity of w and the uniform convergence of the E^k_* to E_*, we have 

    |A_k| ≥ c(δ) ρ_k^n-1(γ_k ρ_k^2).
 Therefore, we reach a contradiction if 

    c(δ) γ_k^2 ρ_k^n+1≥ C ρ_k^α+n-1.

This is the case for k large, as long as 

    _0 < α/2.







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