On the Boundary Point Principle for divergence-type equations
Darya E. Apushkinskaya Alexander I. Nazarov

Chaos and the Flow Capture Problem: Polluting is Easy, Cleaning is Hard
    Julio M. Ottino
    December 30, 2023
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We provide some versions of the Zaremba-Hopf-Oleinik boundary point lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundaries of domains. 




§ INTRODUCTION

The Boundary Point Principle, known also as the “normal derivative lemma”, is one of the important tools in qualitative analysis of partial differential equations. This principle states that a supersolution of a partial differential equation with a minimum value at a boundary point, must increase linearly away from its boundary minimum provided the boundary is smooth enough.

The history of this famous principle begins with a pioneering paper of S. Zaremba <cit.> where the above assertion was established for the Laplace equation in a three-dimensional domain Ω satisfying an interior touching ball condition. Notice that the major part of  all known results on the normal derivative lemma concerns  equations with nondivergence structure and strong solutions.  A key contribution to the investigation of this problem for elliptic equations was made simultaneously and independently by E. Hopf <cit.> and O.A. Oleinik <cit.> (by this reason, all the statements of such type are often called the Hopf-Oleinik lemma). The corresponding comprehensive historical review can be found in <cit.>. 

The case of the divergence-type elliptic equations  

    Lu:=-D_i(a^ij(x)D_ju)+b^i(x)D_iu=0
 




is less studied. It is well known that the Boundary Point Principle fails for uniformly elliptic equations in divergence form with bounded and even continuous  coefficients a^ij(x) (see, for instance, <cit.>, <cit.>, <cit.> and <cit.>). Thus, the normal derivative lemma requires more smoothness of the leading coefficients. 

The sharp requirements on the regularity of the boundary of a domain, providing the validity of the Boundary Point Principle for the Laplace equation, were independently and simultaneously formulated in the papers <cit.> and <cit.>.

The first result for weak solutions of 
(<ref>) was proved  by R. Finn and D. Gilbarg <cit.>. They considered  a two-dimensional bounded domain with 𝒞^1,α-regular boundary, the Hölder continuous leading coefficients and continuous lower order coefficients.  Recently,  in <cit.> (see also <cit.>) the normal derivative lemma was established in n-dimensional domains (n ⩾ 3) for equations with the lower-order coefficients from the Lebesgue space L^q, q>n, under the same assumptions on the leading coefficients and on the boundary  as in <cit.>.

The history of the Boundary Point Principle for parabolic equations
is much shorter then for elliptic ones and begins with the papers of L. Nirenberg <cit.> and A. Friedman <cit.>. For a partial bibliography in the nondivergence case we refer the reader to <cit.>. 

As for the divergence-type parabolic equations 

    ℳu := ∂_t u-D_i(a^ij(x;t)D_ju)+b^i(x;t)D_iu=0,

we do not know such results. However, the normal derivative lemma 
for (<ref>) can be extracted from the lower bound estimates of the Green function for the operator M. These estimates were obtained in <cit.>, <cit.> and <cit.> under various assumptions on the coefficients of M and on the boundary of a domain. In particular, <cit.> deals with cylindrical domains with 𝒞^1,α-regular lateral surface, Dini-continuous leading coefficients
and lower-order coefficient from the so-called parabolic Kato class (see Remark <ref> below).


The goal of our paper is to prove the Boundary Point Principle for  the general divergence-type elliptic and parabolic equations under strongly weakened assumptions close to the necessary ones.

  



 §.§ Notation and conventions

Throughout the paper we use the following notation:

x=(x',x_n)=(x_1,…,x_n-1,x_n) is a point in
ℝ^n;

(x;t)=(x',x_n;t)=(x_1,…,x_n;t) is a point in ℝ^n+1;

ℝ^n_+={x∈ℝ^n : x_n>0},     ℝ^n+1_+=
{(x;t)∈ℝ^n+1 : x_n>0};

|x|, |x'| are the Euclidean norms in corresponding spaces;

B_r(x^0) is the open ball in ℝ^n with center
x^0 and radius r;   B_r=B_r(0);

Q_r(x^0;t^0)=B_r(x^0) × (t^0-r^2; t^0);   Q_r=Q_r(0;0);



D_i denotes the operator of (weak) differentiation with respect
to  x_i; 

D=(D',D_n)=( D_1, …,D_n-1,D_n);   ∂_t=∂∂ t.

 

We adopt the convention that the indices i and j run
from 1 to n. We also adopt the convention regarding summation
with respect to repeated indices.

We use standard notation for the functional spaces. For a bounded domain E⊂ℝ^n+1 we understand 𝒞^1,0_x,t ( E) as the space of u∈𝒞( E) such that Du∈𝒞( E).




We say that a function σ : [0,1]→ℝ_+ belongs to the class 𝒟 if 



  * σ is increasing, and σ (0)=0;

  * σ (τ)/τ is summable and decreasing.

 


 
It should be noted that our assumption about the decay of σ(τ)/τ is not restrictive (see <cit.> for details). 
Moreover, we claim that without loss of generality σ can be assumed continuously differentiable  on (0;1]. Indeed, for any  function σ∈𝒟, one can define

    σ̂ (r):=2∫_r/2^r σ (τ)/τ dτ,      r ∈ (0;1].

It is easy to see that σ̂∈𝒞^1(0;1]. Due to monotonicity properties of σ and σ (τ)τ, we have

    σ̂' (r)   =2/r(σ (r)-σ (r/2)) ⩾ 0,
    (σ̂(r)/r)'   =1/r[ 2 σ (r)/r -σ (r/2)/(r/2)-2/r∫_r/2^r σ (τ)/τ dτ] ⩽ 0,

and  for all r∈ (0;1]

    σ (r) ⩽σ̂ (r) ⩽ 2σ (r/2).

The second inequality in (<ref>) provides σ̂∈𝒟. Finally, the first inequality in (<ref>) allows us to use σ̂ instead of σ in all estimates, and the claim follows.


For σ∈𝒟 we define the function 𝒥_σ as

    𝒥_σ(s):=∫_0^s 
    σ (τ)/τ dτ .



Let E be a bounded domain in ℝ^n. We say that a function ζ :  E→ℝ  belongs to the class 𝒞^0, 𝒟( E),  if



  * |ζ(x)-ζ (y)|⩽σ (|x-y|) for all x,y ∈ E, and σ belongs to the class 𝒟.






Similarly, suppose that E is a bounded domain in ℝ^n+1. A function
ζ :  E→ℝ is said to  belong to the class 𝒞_p^0, 𝒟( E),  if



  * |ζ(x;t)-ζ (y;s)|⩽σ (√(|t-s|+|x-y|^2)) for all (x;t), (y;s) ∈ E, 
and σ belongs to the class 𝒟.









We use the letters C and N (with or without indices) to denote various constants. To indicate that, say, C depends on some parameters, we list them in parentheses: C(…).






§ ELLIPTIC CASE

Let Ω be a bounded domain in ℝ^n with boundary ∂Ω, and let d(x) denote the distance between x and ∂Ω.

We suppose that ∂Ω satisfies the interior 𝒞^1,𝒟-paraboloid condition. The latter means that  in a local coordinate system ∂Ω is given by the equation
x_n=F(x'), where F is a  𝒞^1-function such that F(0)=0 and the inequality

    F(x') ⩽ |x'|·σ (|x'|).

holds true in some neighborhood of the origin. Here σ is a 𝒞^1-function belonging to the class 𝒟 (see Remark <ref>).

Let an operator L be defined by the formula (<ref>).
Assume that the coefficients of ℒ satisfy the following conditions:  

    ν I_n≤ (a^ij(x))≤ν^-1 I_n,
    
    a^ij∈𝒞^0,𝒟(Ω)    for all   i,j=1,…,n,

 and

    ω(r):=sup_x ∈Ω∫_B_r(x)∩Ω|𝐛(y)|/|x-y|^n-1·d(y)/d(y)+|x-y| dy
    → 0     as   r → 0.

Here   ν is a positive constant, I_n is identity (n× n)-matrix, while 𝐛(y)=( b^1(y), …, b^n(y)). 


 
Notice that condition (<ref>) says that the function |𝐛(y)||x-y|^n-1·d(y)d(y)+|x-y| is integrable uniformly with respect to x. Moreover, in any strict interior subdomain of Ω condition (<ref>) means that 𝐛 is an element of the Kato class K_n,1. (For the definition of the scale of the Kato classes K_n,α with α <n the reader is referred to the paper <cit.>). However, in the whole domain Ω our condition (<ref>) is weaker then 𝐛∈ K_n,1.




The main result of this Section is stated as follows.
 
Let Ω be a bounded domain in ℝ^n with the boundary ∂Ω satisfying the interior 𝒞^1,𝒟-paraboloid condition,  let ℒ be defined by (<ref>), and let assumptions (<ref>)-(<ref>) be fulfilled. 

In addition, assume that a nonconstant function u∈𝒞^1(Ω) satisfies, in the weak sense, the inequality

    ℒu⩾ 0     in  Ω.

Then, if u attends its minimum at a point x^0∈∂Ω, we have

    ∂ u/∂𝐧(x^0) <0.

Here ∂/∂𝐧 is the  derivative with respect to  the exterior normal on ∂Ω.



Notice that  without restriction, we may assume that x^0=0 and ∂Ω is locally a paraboloid x_n=|x'|·σ (|x'|) with  a smooth function σ∈𝒟. Further,
all the assumptions on a^ij and 𝐛 are invariant under the 𝒞^1,𝒟-regular change of variables.  So, we may consider ∂Ω  locally  as a flat boundary x_n=0 and assume, without loss of generality, that   B_R ∩ℝ^n_+ ⊂Ω for some R>0.


Consider for 0<ρ<R/2 the point x^ρ=(0,…,0,ρ) and the annulus

    A_ρ:={x: ρ/2 < |x-x^ρ|<ρ}⊂Ω.

Let x^* be an arbitrary point in A_ρ. Following <cit.> (see also <cit.>)  we define the auxiliary functions z and ψ_x^* as solutions of the problems

    {ℒ_0z   =0   in A_ρ,
    
    z   =1   on ∂ B_ρ /2(x^ρ),
    
    z   =0   on ∂ B_ρ(x^ρ),
    .
          {ℒ_0^x^*ψ_x^*   =0   in A_ρ,
    ψ_x^*   =1   on ∂ B_ρ /2(x^ρ),
    ψ_x^*   =0   on ∂ B_ρ(x^ρ),
    .

where the operators ℒ_0 and ℒ_0^x^* are determined by the formulas 

    ℒ_0z:=-D_i(a^ij(x)D_jz)     and    ℒ_0^x^*ψ_x^*:=-D_i(a^ij(x^*)D_jψ_x^*),
 respectively. It is well known that ψ_x^*∈𝒞^∞(A_ρ), and the existence of (unique) weak solution z follows from the general elliptic theory.

 
There exists C_1=C_1(n,ν, σ)>0 such that the inequality 

    |Dz(x^*)-Dψ_x^* (x^*)| ⩽ C_1 𝒥_σ(2ρ)/ρ

holds true for all ρ⩽ R/2. 



Setting w^(1)=z-ψ_x^* we observe that w^(1) vanishes  on ∂ A_ρ. Hence, w^(1) can be represented in A_ρ as

    w^(1)(x)=∫_A_ρ G^x^*_ρ(x,y)ℒ_0^x^*w^(1)(y) dy(⋆)=
    ∫_A_ρ G^x^*_ρ(x,y) (ℒ_0^x^*z(y)-ℒ_0 z(y)) dy,

where G^x^*_ρ stands for the Green function of the operator ℒ_0^x^* in A_ρ. The equality (⋆) follows from the relation ℒ_0^x^*ψ_x^*=ℒ_0 z=0, see (<ref>).

Applying integration by parts we get another version of the representation formula:

    w^(1)(x)=∫_A_ρD_y_iG_ρ^x^*(x,y)(a^ij(x^*)-a^ij(y))D_jz(y) dy.

Differentiating both sides of equality (<ref>) with respect to x_k we get

    D_kw^(1)(x^*)=∫_A_ρD_x_kD_y_iG_ρ^x^*(x^*,y)(a^ij(x^*)-a^ij(y))D_jz(y) dy,
    
    k=1,…,n.


According to Lemma 3.2 <cit.>, z∈𝒞^1(A_ρ), and the following estimate holds for y∈A_ρ: 

    |Dz(y)|⩽N_1/ρ,

where N_1 depends only on  n, ν, and σ. Moreover, due to Theorem 3.3 <cit.> we have also the estimate for the Green function G_ρ^x^*(x,y):

    |D_xD_yG_ρ^x^*(x,y)|⩽N_2/|x-y|^n     x,y ∈ A_ρ,

where N_2 is completely determined by  n, ν, and σ.

Finally, combination of (<ref>)-(<ref>) with condition (<ref>) implies 

    |Dw^(1)(x^*)|⩽N_1N_2/ρ∫_B_2ρ(x^*)σ (|x^*-y|)/|x^*-y|^n dy,

and (<ref>) follows.


Further, we introduce the barrier function v defined as the weak solution of the Dirichlet problem

    {ℒv   =0   in   A_ρ,
    
    v   =1   on  ∂ B_ρ/2(x^ρ),
    
    v   =0   on  ∂ B_ρ(x^ρ).
    .


 
There exists ρ_0>0 such that for all ρ⩽ρ_0 the problem (<ref>) admits a unique solution v∈𝒞^1(A_ρ). Moreover, the inequality

    |Dv(x)-Dz(x)|⩽ C_2 ω (2ρ)/ρ

holds true for any x∈ A_ρ. Here C_2=C_2(n, ν, σ)>0, ρ_0 is completely defined by  n, ν, σ, and ω, while z∈𝒞^1(A_ρ) is defined in (<ref>).





Consider in A_ρ the auxiliary function
w^(2)=v-z.  We observe that w^(2) vanishes  on ∂ A_ρ, and 

    ℒ_0w^(2)=ℒ_0v=ℒv-b^iD_iv=-b^i (D_iw^(2)+D_iz)   in   A_ρ.

Hence, w^(2) can be represented in A_ρ via corresponding Green function G_0,ρ(x,y) as

    w^(2)(x)=-∫_A_ρ G_0,ρ(x,y)b^i(y)  (D_iw^(2)(y)+D_iz(y)) dy.


Differentiation with respect to x_k  gives


    D_kw^(2)(x)=-∫_A_ρD_x_kG_0,ρ(x,y)  b^i(y)(D_iw^(2)(y)+D_iz(y)) dy.

Therefore, we get the relation

    (𝕀 +𝕋_1 )Dw^(2) =-𝕋_1 Dz,

where 𝕀 stands for the identity operator, while 𝕋_1 denotes the matrix operator whose (k,i) entries are  integral operators with kernels D_x_kG_0,ρ(x,y)b^i(y). 


The statement of Theorem follows from the next assertion.

 
The operator 𝕋_1 is bounded in C(A_ρ), and

    𝕋_1_ C→ C⩽ C_3 ω(2ρ),

where C_3 depends only on n, ν, and σ.


 
Theorem 3.3 <cit.> provides the estimate

    |D_xG_0,ρ(x,y)| ⩽ N_3 min{|x-y|^1-n; {y, ∂ A_ρ} |x-y|^-n}

for any x,y ∈ A_ρ. Here N_3 is the constant depending only on n, ν, and σ. 

Since {y, ∂ A_ρ}⩽ d(y) for any y∈ A_ρ, the combination of estimate (<ref>) with condition (<ref>) gives

    ∫_B_r(x)∩ A_ρ|D_xG_0,ρ(x,y)| |𝐛(y)| dy⩽ 2N_3ω(r),      x∈A_ρ,    r⩽ 2ρ.



For arbitrary vector function f∈ C(A_ρ) we have

    |𝕋_1 f(x)|⩽ f_ C(A_ρ)·∫_A_ρ|D_xG_0,ρ(x,y)| |𝐛(y)| dy⩽ 2N_3 ω(2ρ) · f_ C(A_ρ),   x∈A_ρ.


It remains to show that 𝕋_1 f∈ C(A_ρ). For x,x̃∈A_ρ and any small δ>0 we have

    (𝕋_1 f)(x)-(𝕋_1 f) (x̃) =J_1+J_2
       := (∫_A_ρ∩ B_δ(x̃)+∫_A_ρ∖ B_δ(x̃))(D_xG_0,ρ(x,y)-D_xG_0,ρ(x̃,y))  [𝐛(y)· f(y)] dy.

If |x-x̃|⩽δ/2 then (<ref>) gives

    |J_1|   ⩽ f_ C(A_ρ)·∫_B_δ(x̃)∩ A_ρ(|D_xG_0,ρ(x,y)|+|D_xG_0,ρ(x̃,y)|) |𝐛(y)| dy
       ⩽ 2 N_3 ω(3δ/2) · f_ C(A_ρ).

Thus, given ε we can choose δ such that |J_1|⩽ε.

On the other hand, D_xG_0,ρ(x,y) is continuous for x y. Thus, it is equicontinuous on the compact set 

    {(x,y): x∈B_δ/2(x̃)∩A_ρ,  y∈A_ρ∖ B_δ(x̃) }.

Therefore, for chosen δ we obtain, as |x-x̃|→ 0,

    |J_2|⩽ f_ C(A_ρ)·∫_A_ρ |𝐛(y)| dy ·max_y∈A_ρ∖ B_δ(x̃) |D_xG_0,ρ(x,y)-D_xG_0,ρ(x̃,y)| → 0,

and the Lemma follows.


We continue the proof of Theorem <ref>. Choose the value of ρ_0 so small that ω (2ρ_0) ⩽(2C_3)^-1, where C_3 is the constant from Lemma <ref>. Then by the Banach theorem the operator 𝕀 +𝕋_1 in
(<ref>) is invertible. This gives the
existence and uniqueness of w^(2)∈ C^1(A_ρ), and thus, the
unique solvability of the problem (<ref>). Moreover, Lemma <ref> and inequality (<ref>) provide (<ref>). The proof is complete.


To prove Theorem <ref> we need the following maximum principle.

 
Let ℒ be defined by (<ref>), and let assumptions (<ref>)-(<ref>) be satisfied in a domain E. Suppose that a function w∈ C^1( E) satisfies ℒw ⩾ 0 in E.
If w attains its minimum in an interior point of E then w=const.


 In the paper <cit.> the Harnack inequality was established for the divergence-type operators  with the Hölder continuous coefficients a^ij and b^i belonging to the Kato class K_n,1. However, it is mentioned in <cit.> that the assumption of the Hölder continuity of leading coefficients is needed only for the pointwise gradient estimate of the Green function for the corresponding parabolic operator M_0 without lower order coefficients, see (<ref>) below. Since by Theorem 2.6 <cit.> this estimate holds for operators with Dini coefficients, and 𝐛∈ K_n,1 in any strict interior subdomain of Ω (see Remark <ref>),
the strong maximum principle holds for the operator ℒ in Ω.




It is well known that the Boundary Point Principle holds true  for the operator with constant coefficients. 
Using this statement for the operator ℒ_0^x^* (see (<ref>)) with x^*=0 in the annulus A_1 and rescaling A_1 into A_ρ  we get the estimate




    D_nψ_0(0)⩾N_4(n, ν)/ρ>0.


Furthermore, the inequalities (<ref>) and (<ref>) imply for sufficiently small ρ

    D_nv(0)    ⩾ D_nψ_0(0)-|Dz(0)-Dψ_0(0)|-|Dv(0)-Dz(0)|
       ⩾N_4/ρ-C_1 𝒥_σ(2ρ)/ρ-C_2 ω (2ρ)/ρ⩾N_4/2ρ.

We fix such a ρ. Since u is nonconstant, Lemma <ref> ensures u-u(0)>0 on ∂ B_ρ/2(x^ρ). Therefore,
we have for sufficiently small ε

    L(u-u(0)-ε v)⩾ 0  in  A_ρ;     u-u(0)-ε v⩾ 0   on  ∂ A_ρ.

By Lemma <ref> the estimate u-u(0) ⩾ε v holds true in A_ρ, with equality at the origin. This gives

    ∂ u/∂𝐧(0)=-D_nu(0)⩽ -ε D_nv(0),

which completes the proof.


 
Notice that the statement of Theorem <ref> is also valid for weak supersolutions of equation (<ref>). Namely, let Ω and the coefficients of ℒ be the same as in Theorem <ref>, and let a nonconstant function u ∈ W^1_2(Ω) with |𝐛· Du|∈ L^1(Ω) satisfy in Ω the inequality ℒu ⩾ 0 in the weak sense. Then, if u attends its minimum at a point x^0∈∂Ω, we have

    lim inf_ε→ 0u(x^0-ε n(x^0))-u(x^0)/ε >0.



 
The assumptions on the lower-order coefficients b^i (i=1,…, n) can be also weakened. In fact, one can take as coefficients b^i the signed measures, satisfying condition (<ref>). Indeed, all our arguments require a convergence of the corresponding integrals only.






§ PARABOLIC CASE

Let Q be a bounded domain in ℝ^n+1 with topological boundary ∂ Q. We define the parabolic boundary ∂' Q as the set of all points (x^0;t^0) ∈∂ Q such that for any ε >0, 
we have Q_ε (x^0;t^0)∖Q∅.

By d_p(x;t) we denote the parabolic distance between (x;t) and ∂' Q  which is defined as follows:

    d_p(x;t):=sup{ρ >0 : Q_ρ (x; t) ∩∂' Q = ∅}.

Next, we define the lateral surface ∂” Q as the set of all points (x^0;t^0) ∈∂' Q such that Q_ε (x^0;t^0)∩ Q∅ for any ε >0. 


We suppose that Q satisfies the parabolic interior 𝒞^1, 𝒟-paraboloid condition. It means that in a local coordinate system ∂” Q is given by the equation x_n=F(x';t), where F is a  𝒞^1-function such that F(0;0)=0 and the inequality 

    F(x';t) ⩽√(|x'|^2 -t)·σ (√(|x'|^2 -t) )     for   t⩽ 0

holds in some neighborhood of the origin. Here σ is a 𝒞^1-function belonging to the class 𝒟 (see Remark <ref>).



Let an operator M be defined by the formula (<ref>).
Suppose that the coefficients of ℳ satisfy the following conditions:  

    ν I_n≤ (a^ij(x;t))≤ν^-1 I_n,
    
    a^ij∈𝒞_p^0,𝒟(Q)    for all   i,j=1,…,n,

and 

    ω^-_p(r) → 0    and  ω^+_p(r) → 0      as   r → 0,

where

    ω^-_p(r):=sup_(x;t) ∈ Q∫_Q_r(x;t)∩ Q   |𝐛(y;s)|/(t-s)^(n+1)/2·exp(-γ |x-y|^2/t-s)×
    ×    d_p(y;s)/d_p(y;s)+√(|x-y|^2+t-s) dy ds;


    ω^+_p(r):=sup_(x;t) ∈ Q∫_Q_r(x;t+r^2)∩ Q   |𝐛(y;s)|/(s-t)^(n+1)/2·exp(-γ |x-y|^2/s-t)×
    ×    d_p(y;s)/d_p(y;s)+√(|x-y|^2+s-t) dy ds.

Here ν and ℐ_n are the same as in Section 2, 𝐛(y;s)=(b^1(y;s), …, b^n(y;s)), and γ is a positive constant to be determined later, depending only on n, ν and on the moduli of continuity of the coefficients a^ij. 


 
Similarly to the elliptic case, in any strict interior subdomain of Q∖∂'Q condition (<ref>) means that 𝐛 is an element of the parabolic Kato class K_n, see <cit.>. 
Indeed, in this case (<ref>) can be rewritten as follows:

    sup_(x;t) ∈ Q∫_(t-r^2,t+r^2)× B_r(x)|𝐛(y;s)|/|s-t|^(n+1)/2·exp(-γ |x-y|^2/|s-t|) dy ds→0

as r→0.

This condition differs from Definition 3.1 <cit.> only in that the integration in <cit.> is over (t-r^2,t+r^2)×ℝ^n. However, using the covering of ℝ^n∖ B_r(x) by the balls of radius r/3 one can check that corresponding suprema converge to zero simultaneously.

In the whole domain Q our condition (<ref>) is weaker then 𝐛∈ K_n.



To formulate the parabolic counterpart of Theorem <ref> we need the following notion.

For a point (x;t)∈Q we define its dependence set as the set of all points (y;s)∈Q admitting a vector-valued map 𝔉 : [0,1] ↦ℝ^n+1 such that the last coordinate function 𝔉_n+1 is strictly increasing and

    𝔉(0)=(y;s);   𝔉(1)=(x;t);   𝔉 ((0,1))⊂ Q.

If Q is a right cylinder with generatrix parallel to the t-axis, then for any (x;t)∈Q the dependence set is Q∩{s <t}. 



 
Let Q be a bounded domain in ℝ^n+1, let ∂” Q satisfy the interior parabolic 𝒞^1, 𝒟-paraboloid condition, let ℳ be defined by (<ref>), and let assumptions (<ref>)-(<ref>) be satisfied.

In addition, assume that a function u ∈𝒞^1,0_x,t (Q) satisfies, in the weak sense, the inequality

    ℳu ⩾ 0     in   Q.

Then, if u attends its minimum at a point (x^0;t^0) ∈∂” Q, and u is nonconstant on the dependence set of (x^0;t^0), we have

    ∂ u/∂𝐧 (x^0;t^0) <0.

Here ∂/∂𝐧 denotes the derivative with respect to the spatial exterior normal on ∂” Q∩{t=t^0}.



Notice that we do not care of the behavior of u after t^0. Thus, without loss of generality we suppose Q=Q ∩{ t<t_0}. Moreover, similarly to the elliptic case, we may assume that (x^0; t^0)=(0;0), and ∂”Q is locally a paraboloid 

    x_n=𝒫(x';t):=√(|x'|^2-t)·σ (√(|x'|^2-t)),
 
where σ∈𝒟 is smooth.


Next, we flatten the boundary of the paraboloid by the coordinate transform

    x̃'=x';   t̃=t;   x̃_n=x_n-𝒫(x';t).


 
Assumptions (<ref>) and (<ref>) on a^ij and 𝐛 remain valid under transform (<ref>).



 It is easy to see that 
 |D'𝒫| ∈𝒞_p^0, 𝒟(Q_R ∩ℝ^n+1_+)
for some R>0. (Here, we consider D'𝒫 as a function of (x;t)-variables, which is independent on x_n). Therefore, the “new” coefficients ã^ij satisfy (<ref>) in Q_R ∩ℝ^n+1_+. 

It is also evident that the transformed “old” coefficients b^i satisfy (<ref>). However, the coordinate change (<ref>) generates an additional term b̃_n which admits the estimate

    |b̃_n (x̃;t)| ⩽ C |∂_t 𝒫(x'; t)|= C ( σ (√(|x'|^2-t))/√(|x'|^2-t) +
    σ' (√(|x'|^2-t)) ).

Estimating the integral  entering in the definition of ω^±_p, it suffices to assume that x'=0 and t=0. This gives 

    ω^-_p(r)⩽ C ∫_Q_rexp(-γ |y|^2/-s)/(-s)^(n+1)/2·( σ (√(|y'|^2-s))/√(|y'|^2-s) +
    σ' (√(|y'|^2-s)) )  dy ds.

After integration over y_n we make change of variables 

    ϱ=|y'|/√(-s);     τ=√(|y'|^2-s),

and arrive at 

    ω^-_p(r)   ⩽ C ∫_0^r√(2)∫_0^∞exp(-γϱ^2)ϱ^n-2/√(ϱ^2+1)(σ (τ)/τ +σ' (τ) )
    dϱ dτ
       ⩽ C(n,γ) ( 𝒥_σ (r√(2)) + σ (r √(2))),

and the lemma follows.

 
 Thus, we may consider ∂” Q  locally  as a flat boundary x_n=0 and assume, without loss of generality, that Q_R ∩ℝ^n+1_+ ⊂ Q. 
 
 Next, we take  for 0<ρ⩽ R/2 the cylinder A_ρ=Q_ρ(x^ρ;0) (as in the elliptic case,  x^ρ=(0,…,0,ρ)). Define the auxiliary function z as the solution of the initial-boundary value problem

    {ℳ_0z:=∂_t z-D_i(a^ij(x;t)D_jz)   =      0       in        A_ρ,
    z   =      0       on       ∂” A_ρ,
    
    z(x;-ρ^2)   =      φ(x-x^ρρ)       for       x∈ B_ρ(x^ρ),
    .

where φ is a smooth cut-off function such that 

    φ(x)=1   for  
    |x|<1/2;
        φ(x)=0   for  
    |x|>3/4.

The existence of (unique) weak solution z follows from the general parabolic theory.


The function z belongs to 𝒞^1,0_x,t( A_ρ) for sufficiently small ρ. Moreover, there exists a positive constant ρ_0⩽ R/2 depending only on n, ν and σ, such that the inequality

    |Dz(x;t)| ⩽C_4(n,ν)/ρ,     (x;t)∈ A_ρ,

holds true for all ρ⩽ρ_0. 


 We partially follow the line of proof of Lemma <ref>. Let (x^*;t^*) be an arbitrary point in 𝒜_ρ. We introduce the auxiliary function ψ_x^*, t^* as the solution of the problem

    {ℳ_0^x^*, t^*ψ_x^*, t^*   =      0 
          in        A_ρ,
    ψ_x^*, t^*   =      0       on       ∂” A_ρ,
    ψ_x^*, t^*(x;-ρ^2)   =      φ(x-x^ρρ)       for       x∈ B_ρ(x^ρ),
    .

where ℳ_0^x^*, t^*:=∂_t 
-D_ia^ij(x^*;t^*)D_j is operator with constant coefficients frozen at the point (x^*;t^*). It is well known that ψ_x^*,t^*∈𝒞^∞( A_ρ), and 

    |Dψ_x^*,t^*(y;s)|⩽N_5(n,ν)/ρ,     (y;s)∈𝒜_ρ.




Setting w^(3)=z-ψ_x^*,t^* we observe that w^(3) vanishes  on ∂' 𝒜_ρ. Hence, w^(3) can be represented in the cylinder 𝒜_ρ as

    w^(3)(x;t)=∫_𝒜_ρ∩{s⩽ t}Γ^x^*, t^*_ρ(x,y;t,s)ℳ_0^x^*, t^*w^(3)(y;s) dyds,

where Γ^x^*, t^*_ρ stands for the Green function of the operator ℳ_0^x^*, t^* in 𝒜_ρ. 

Similarly to (<ref>), we integrate by parts and obtain

    w^(3)(x;t)=∫_𝒜_ρ∩{s⩽ t}
    D_y_iΓ_ρ^x^*, t^*(x,y;t,s)(a^ij(x^*;t^*)-a^ij(y;s))D_jz(y;s) dyds.

Differentiating both sides with respect to x_k, k=1,…,n, we get the system of equations

    D_kz(x;t)-∫_ A_ρ∩{s⩽ t}D_x_kD_y_iΓ^x^*, t^*_ρ(x,y;t,s)×
    ×(a^ij(x^*;t^*)-a^ij(y;s))D_jz(y;s) dyds
    =D_kψ_x^*, t^*(x;t).


Now we put (x^*;t^*)=(x;t) and get the relation

    (𝕀 -𝕋_2 )Dz = Ψ,

where

    Ψ=Dψ_x^*, t^*(x;t)|_(x^*;t^*)=(x;t)

while 𝕋_2 denotes the matrix integral operator whose kernel is matrix T_2 with entries

    T_2^kj(x,y;t,s)=D_x_kD_y_iΓ^x^*, t^*_ρ(x,y;t,s)|_(x^*;t^*)=(x,t)·(a^ij(x;t)-a^ij(y;s))χ_{s≤ t}.

It is easy to see that Ψ∈ C( A_ρ). Therefore, the statement of Theorem follows from the next assertion.

 
The operator 𝕋_2 is bounded in C( A_ρ), and

    𝕋_2_ C→ C⩽ C_5 𝒥_σ (2√(2)ρ),

where C_5 depends only on n and ν.


 
The following estimate for the Green function Γ^x^*, t^*_ρ(x,y;t,s) is well known:

    |D_xD_yΓ^x^*, t^*_ρ(x,y;t,s)|⩽N_6/(t-s)^(n+2)/2exp(-N_7 |x-y|^2/t-s),

where N_6 and N_7 are completely determined by  n and ν.

Combination of (<ref>) with condition (<ref>) gives for r⩽ 2ρ and (x;t)∈ A_ρ

    ∫_Q_r(x;t)∩ A_ρ|T_2(x,y;t,s)| dyds
    ⩽∫_t-r^2^t∫_B_r(x)N_6 σ (√(t-s+|x-y|^2))/(t-s)^ (n+2)/2exp(-N_7 |x-y|^2/t-s) dyds.

Change of variables ϱ=|x-y|/√(t-s), 
τ=√(t-s+|x-y|^2) gives

    ∫_Q_r(x;t)∩ A_ρ|T_2(x,y;t,s)| dyds
    ⩽   ∫_0^r√(2)∫_0^∞N_8
    exp(-N_7ϱ^2)ϱ^n-1σ (τ)/τ  
    dϱ dτ⩽ C_5 𝒥_σ (r√(2))

(N_8 and C_5 depend only on n and ν).

For a vector function f∈ C( A_ρ) and for all (x;t)∈ A_ρ we have

    |𝕋_2 f(x;t)|⩽ f_ C( A_ρ)·∫_ A_ρ|T_2(x,y;t,s)| dyds⩽ C_5 𝒥_σ (2√(2)ρ) · f_ C( A_ρ).


It remains to show that 𝕋_2 f∈ C( A_ρ). For (x;t),(x̃;t̃)∈ A_ρ and any small δ>0 we have

    (𝕋_2 f)(x;t)-(𝕋_2 f) (x̃;t̃) =J_1+J_2
       := (∫_ A_ρ∩ Q_δ(x̃;t̃)+∫_ A_ρ∖ Q_δ(x̃;t̃))(T_2(x,y;t,s)-T_2(x̃,y;t̃,s))  f(y;s) dyds.

Similarly to the proof of Theorem <ref>, if 
√(|t-s|+|x-y|^2)⩽δ/2 then (<ref>) gives

    |J_1|⩽ 2 C_5 𝒥_σ (3√(2)ρ/2) · f_ C( A_ρ).

Thus, given ε we can choose δ such that |J_1|⩽ε. 

Next, D_xD_yΓ^x^*, t^*_ρ(x,y;t,s) is continuous w.r.t. (x;t) and w.r.t. (x^*;t^*) for (x;t) (y;s). Therefore, T_2(x,y;t,s) is continuous w.r.t. (x;t) for (x;t) (y;s). Similarly to the proof of Theorem <ref>, for chosen δ we obtain, as (x;t)→(x̃;t̃),

    |J_2|⩽ f_ C( A_ρ)·meas( A_ρ)·max_(y;s)∈ A_ρ∖ Q_δ(x̃;t̃) |T_2(x,y;t,s)-T_2(x̃,y;t̃,s)| → 0,

and the Lemma follows.


We continue the proof of Theorem <ref>. Choose the value of ρ_0 so small that 𝒥_σ (2√(2) ρ_0) ⩽(2C_5)^-1, where C_5 is the constant from Lemma <ref>. Then by the Banach theorem the operator 𝕀 -𝕋_2 in
(<ref>) is invertible. This gives z∈𝒞^1,0_x,t ( A_ρ). Moreover, Lemma <ref> and inequality (<ref>) provide (<ref>). 
The proof is complete.


For ρ⩽ρ_0 we introduce the Green function Γ_0,ρ(x,y;t,s) of the operator ℳ_0 in the cylinder 𝒜_ρ. By Theorem 2.6 <cit.>, D_xΓ_0,ρ(x,y;t,s) is continuous for (x;t) (y;s), and the estimate 

    |D_xΓ_0,ρ(x,y;t,s)|    ⩽ N_8 min{1/(t-s)^(n+1)/2; {y, ∂ B_ρ(x^ρ)}/(t-s)^(n+2)/2}×
       ×exp(-N_9 |x-y|^2/t-s)

holds for any (x;t), (y;s) ∈ A_ρ, s<t. Here N_8 and N_9 are the constants depending only on n, ν, and σ. 


Further, we introduce the barrier function v defined as the weak solution of the initial-boundary value problem

    {ℳv   =      0       in        A_ρ,
    v   =      0       on       ∂” A_ρ,
    v(x;-ρ^2)   =      φ(x-x^ρρ)       for       x∈ B_ρ(x^ρ),
    .

where φ is the same as in (<ref>).

 
Let b satisfy the first relation in (<ref>) with γ=N_9(n,ν,σ) (here N_9 is the constant in (<ref>)). Then there exists a positive ρ_0⩽ρ_0 such that for all ρ⩽ρ_0 the problem (<ref>) admits a unique solution v∈𝒞^1,0_x,t( A_ρ). Moreover, the inequality

    |Dv(x;t)-Dz(x;t)|⩽ C_6 ω^-_p (2ρ)/ρ

holds true for any (x;t)∈ A_ρ. Here C_6=C_6(n, ν, σ)>0, ρ_0 is completely defined by  n, ν, σ, and ω, while z∈𝒞^1,0_x,t( A_ρ) is defined in (<ref>).




We follow the line of proof of Theorem <ref>.
Consider in A_ρ the auxiliary function
w^(4)=v-z.  We observe that w^(4) vanishes  on ∂'  A_ρ, and 

    ℳ_0w^(4)=-b^i (D_iw^(4)+D_iz)   in   A_ρ.

Similarly to the proof of Theorem <ref>, D_kw^(4) can be represented in A_ρ as

    D_kw^(4)(x;t)=    -∫_ A_ρ∩{s⩽ t} D_x_kΓ_0,ρ(x,y;t,s)×
       × b^i(y;s)  (D_iw^(4)(y;s)+D_iz(y;s)) dyds.

Therefore, we get the relation

    (𝕀 +𝕋_3 )Dw^(4) =-𝕋_3 Dz,

where 𝕋_3 denotes the matrix operator whose (k,i) entries are integral operators with kernels D_x_kΓ_0,ρ(x,y;t,s)b^i(y;s)χ_{s≤ t}. 


The statement of Theorem follows from the next assertion.

 
The operator 𝕋_3 is bounded in C( A_ρ), and

    𝕋_3_ C→ C⩽ C_7 ω^-_p(2ρ),

where C_7 depends only on n, ν, and σ.


 Recall that ρ≤ R/2 and Q_R ∩ℝ^n+1_+⊂ Q. Thus {y, ∂ B_ρ(x^ρ)}⩽ d_p(y;s) for any (y;s)∈ A_ρ, and the combination of estimate (<ref>) with the first relation in (<ref>) gives for r⩽ 2ρ

    ∫_Q_r(x;t)∩ A_ρ
    |D_xΓ_0,ρ(x,y;t,s)| |𝐛(y;s)| dyds
    ⩽ N_10(n)N_8 ω^-_p(r),    
    x∈ A_ρ,

(here N_8 is the constant in (<ref>)).

The rest of the proof repeats literally the proof of Lemma <ref>.


We continue the proof of Theorem <ref>. Choose the value of ρ_0 so small that ω (2ρ_0) ⩽(2C_7)^-1, where C_7 is the constant from Lemma <ref>. Then by the Banach theorem the operator 𝕀 +𝕋_3 in
(<ref>) is invertible. This gives the existence and uniqueness of w^(4)∈𝒞^1,0_x,t( A_ρ), and thus, the unique solvability of the problem (<ref>). Moreover, Lemma <ref> and inequality (<ref>) provide (<ref>). The proof is complete.



To prove Theorem <ref> we need the following maximum principle.

 
Let ℳ be defined by (<ref>), and let assumptions (<ref>)-(<ref>) be satisfied in a domain E⊂ℝ^n+1. Let a function w∈𝒞^1,0_x,t( E) satisfy ℳw ⩾ 0 in E. If w attains its minimum in a point (x^0;t^0)∈ E∖∂' E then w=const on the closure of the dependence set of (x^0;t^0).


 The Harnack inequality for parabolic divergence-type operators was established in <cit.> under the assumptions that the leading coefficients a^ij are Hölder continuous and 𝐛 satisfy (<ref>) with arbitrary γ>0 (and integration over (t-r^2,t+r^2)×ℝ^n that is inessential, see Remark <ref>). 

As it was mentioned in the proof of Lemma <ref>, the first assumption can be replaced by the Dini continuity. Further, in fact only (<ref>) with a certain γ occuring in the estimate of DΓ_0,ρ is used in <cit.>. The latter coincides with the assumption 𝐛∈ K_n.

Since our assumption (<ref>) implies 𝐛∈ K_n in any strict interior subdomain of Q∖∂' Q (see Remark <ref>), the strong maximum principle holds for the operator ℳ.



This Lemma is the only point where we need the second relation in (<ref>). If we could prove at least weak maximum principle for the operator M using only the quantity ω^-_p, we did not need ω^+_p at all. Unfortunately, we cannot do it, and the question whether the second relation in (<ref>) is necessary for the Boundary Point Principle remains open.




It is well known that the Boundary Point Principle holds true  for the operator with constant coefficients. 
Using this statement for the operator ℳ_0^x^*, t^* with x^*=0, t^*=0 in the cylinder A_1 and rescaling A_1 into A_ρ  we get the estimate

    D_nψ_0,0(0;0)⩾N_11(n, ν)/ρ.


Next, the relation (<ref>), Lemma <ref>, and inequality (<ref>) imply for sufficiently small ρ

    D_nz(0;0) ⩾ D_nψ_0,0(0;0)-
    𝕋_2Dz_ C( A_ρ)⩾N_11/ρ-C_4C_5𝒥_σ (2√(2)ρ)/ρ⩾N_11/2ρ.

The relation (<ref>) gives for sufficiently small ρ

    D_nv(0;0) ⩾ D_nz(0;0)- |Dv(0;0)-Dz(0;0)|⩾N_11/2ρ-C_6 ω^-_p (2ρ)/ρ⩾N_11/4ρ.

We fix such a ρ. Since u is nonconstant on the dependence set of (0;0), Lemma <ref> ensures 

    u(x;-ρ^2)-u(0;0)>0     for   x∈ B_3ρ/4(x^ρ).
 
Therefore, we have for sufficiently small ε

    M(u-u(0;0)-εv)⩾ 0  in   A_ρ;     u-u(0;0)-εv⩾ 0   on  ∂'  A_ρ.

By Lemma <ref> the estimate u-u(0;0) ⩾εv holds true in A_ρ, with equality at the origin. This gives

    ∂ u/∂𝐧(0;0)=-D_nu(0;0)⩽ -ε D_nv(0;0),

which completes the proof.


 
As in elliptic case, the statement of Theorem <ref> is also valid for weak supersolutions of equation (<ref>). Namely, let Q and the coefficients of ℳ be the same as in Theorem <ref>. Suppose that for a  function u∈ L^2(Q) with Du ∈ L^2(Q) the  assumptions

    sup_tu(·;t)_L^2<∞,   and   |𝐛· Du|∈ L^1(Q)

are fulfilled. Finally, let u satisfy
the inequality ℳu ⩾ 0 in the weak sense. 
 
 Then, if u attends its minimum at a point (x^0;t^0) ∈∂” Q, and u is nonconstant on the dependence set of (x^0;t^0), we have

    lim inf_ε→ 0u(x^0-ε n(x^0);t^0)-u(x^0;t^0)/ε >0.



 
Similarly to Remark <ref>, one can take as coefficients b^i the signed measures, satisfying condition (<ref>).







§ SOME SUFFICIENT CONDITIONS FOR THE VALIDITY OF (<REF>) AND (<REF>)


In this section we list several simple sufficient conditions on lower-order coefficients providing the validity of assumptions (<ref>) and (<ref>) for elliptic and parabolic operators, respectively. These conditions are close to ones imposed in <cit.>, where equations in non-divergence form were studied.

Throughout this section we will denote various constants depending on n only by the letter N without indices, and the constants depending on n and γ only by the letter N without indices. 
To simplify the notation, we assume that b is extended by zero outside of Ω (of Q).

In the elliptic case we consider two types of restrictions: distributed drift

    𝐛∈ L^n(Ω);     sup_x∈Ω𝐛_n, B_ρ(x)⩽ C σ (ρ),

and near-boundary drift

    |𝐛(y)| ⩽ C σ(d(y))/d(y)

(recall that σ∈𝒟).

 
Any of the restrictions (<ref>) and (<ref>) implies the validity of condition (<ref>). 


 
Let (<ref>) hold. We set B^k=B_r/2^k and estimate

    ω(r)   ⩽sup_x∈Ω ∑_k=0^∞  ∫_B^k∖ B^k+1| b(x-y)|/|y|^n-1 dy
       ⩽sup_x∈Ω ∑_k=0^∞ (∫_B^k∖ B^k+1| b(x-y)|^n dy)^1/n·(∫_B^k∖ B^k+1dy/|y|^n)^n-1/n
       ⩽ NC∑_k=0^∞σ(r/2^k)⩽ NC 𝒥_σ(2r),

and (<ref>) follows.

Now 
let (<ref>) hold. We use the decay of σ (τ)/τ to estimate

    σ(d(y))/d(y)+|x-y|⩽σ(d(y)+|x-y|)/d(y)+|x-y|⩽σ(|x-y|)/|x-y|.

Therefore,

    ω(r)⩽ C∫_B_r(x)σ(|x-y|)/|x-y|^n⩽ NC𝒥_σ(r),

and (<ref>) again follows.


In the parabolic case we consider the following  analogue of (<ref>): 

    𝐛∈ L^n+1(Q);     sup_(x;t)∈ Q𝐛_n+1, Q_ρ(x;t)⩽ C σ (ρ)ρ^1/(n+1),

as well as the analog of (<ref>):

    |𝐛(y;s)| ⩽ C σ(d_p(y;s))/d_p(y;s).



 
Any of the restrictions (<ref>) and  (<ref>) implies the validity of condition (<ref>) with arbitrary γ>0.


 
We estimate the quantity ω^-_p(r); the case of ω^+_p(r) is considered along the same lines.

Let (<ref>) hold. We set Q^k=Q_r/2^k and obtain

    ω^-_p(r)   ⩽sup_(x;t)∈ Q ∑_k=0^∞ ∫_Q^k∖ Q^k+1|𝐛(x-y;t+s)|/(-s)^(n+1)/2·exp(-γ |y|^2/-s) dy ds
    
       ⩽sup_(x;t)∈ Q ∑_k=0^∞ (∫_Q^k∖ Q^k+1
    | b(x-y;t+s)|^n+1 dyds)^1/(n+1)·Φ_k^n/(n+1)
       ⩽ C∑_k=0^∞σ(r/2^k)
    ·(r/2^k)^1/(n+1)·Φ_k^n/(n+1),

where

    Φ_k=∫_Q^k∖ Q^k+1exp(-γ n+1/n |y|^2/-s)·dyds/(-s)^(n+1)^2/2n.


Change of variables ϱ=|y|/√(-s), 
τ=√(|y|^2-s) gives

    Φ_k≤∫_r/2^k+1^r√(2)/2^k∫_0^∞exp(-γ n+1/n ϱ^2)·2ϱ^n-1(ϱ^2+1)^1/2n/τ^1+1/n dϱ dτ⩽N/(r/2^k)^1/n.

Thus,

    ω^-_p(r)⩽NC∑_k=0^∞σ(r/2^k)
    ⩽NC 𝒥_σ(2r),

and the first relation in (<ref>) follows.

Now 
let (<ref>) hold. As in Lemma <ref>, we use the decay of σ (τ)/τ to estimate

    σ(d_p(y;s))/d_p(y;s)+√(|x-y|^2+t-s)⩽σ(√(|x-y|^2+t-s))/√(|x-y|^2+t-s).

Therefore,

    ω^-_p(r)⩽ C∫_Q_r(x;t)exp(-γ |x-y|^2/t-s)·σ(√(|x-y|^2+t-s))/√(|x-y|^2+t-s) dyds/(t-s)^(n+1)/2.

Change of variables ϱ=|x-y|/√(t-s), 
τ=√(t-s+|x-y|^2) gives

    ω^-_p(r)⩽ C∫_0^r√(2)∫_0^∞exp(-γϱ^2)·ϱ^n-1/√(ϱ^2+1) σ (τ)/τ  dϱ dτ⩽NC𝒥_σ(r√(2)),

and the first relation in (<ref>) again follows.






















