Alternative proof for the existence of Green's function

We present a new method for the existence of a Green's function of 
nod-divergence form parabolic operator with Holder continuous coefficients. 
We also derive a Gaussian estimate. 
Main ideas involve only basic estimates and known results without a potential approach, 
which is used by E.E. Levi.

1. Introduction. We consider the following non-divergence form parabolic operator: where its coefficients satisfy symmetric and ellipticity condition. Namely, a ij = a ji , λ|ξ| 2 ≤ a ij (x, t)ξ i ξ j ≤ Λ|ξ| 2 (2) holds for some positive constants λ and Λ in a given domain Q := Ω × (0, T ), Ω is an open connected bounded set in R n for n ≥ 1. We study the Green's function (of the first kind) for the operator L in a smooth domain. Following [12], a function G defined in Q × Q, is called the Green's function if, for any fixed Y = (y, s) ∈ Q, LG(X, Y ) = 0, t > s, X = (x, t) ∈ Q, G(X, Y ) = 0 for X ∈ ∂ p Q, where ∂ p Q is the parabolic boundary of Q (see the definition below), δ y is the Dirac mass at y, and the last expression is understood in a distributional sense. For its fundamental feature, this has been a topic of many publications. See [13,9,12,15] and references therein.

SUNGWON CHO
Throughout this paper, we assume a ij to be a α-Hölder continuous with its norm of A α . Namely, for some fixed α ∈ (0, 1). In this case, the fundamental solution was constructed from the parametrix method of E. E. Levi [16,17], where volume potentials were use of essential. For more details and history, see [12,7,24,14].
In this paper, we present an alternative method for the existence of the Green's function along with its pointwise estimation. Our method does not involve any potential theory, but basic estimates including maximum principle and classical Schauder theory. Now we state our main result along with some remarks: Theorem 1.1. Let L be a non-divergence parabolic operator of the form (1) satisfying (2). Assume its coefficients a ij are α-Hölder continuous, and Ω be a bounded open connected set in R n with C 2,α boundary. Then the Green's function G of L in Q := Ω × (0, T ) exists and satisfies the following Gaussian estimate: for some positive constants C := C(n, α, λ, Λ, A α , diam(Ω), T ) and c := c(n, α, λ, Λ, A α , diam(Ω)), Remark 1. In the nondivergence elliptic case, the continuity of the Green's function and classical Harnack principle away from the pole (x = y) are proved in [23] and [1], respectively. With only continuous assumption on coefficients a ij , the Green function is not even bounded. See [4].

Remark 2.
For the measurable coefficients case, two-sided pointwise estimations of fundamental solution (the Green function when Ω = R n ) become available by Escauriaza [10], using a certain adjoint solution for the bound functions. In the proof, he used the concept of normalized adjoint solution, which was first introduced by Bauman [5] in elliptic, and Fabes, Garofalo, Salsa [11] in parabolic case, respectively.
Remark 3. More extensive results are available in divergence case. See Aronson [2], and [3,19,25,21,20,8,22]. Our main estimation (6) is not optimal if we consider G is vanishing near its parabolic boundary. See [6] for two sided estimations including its boundary in the divergence case.
We conclude this section with a sketch of the paper and some notations. In section 2, we solve the Poisson problem with weakly singular right hand side along with pointwise estimate of the solution. In section 3, using Theorem 2.1, we prove our main result, Theorem 1.1. Now, we enlist some standard notations: . For X = (x, t), C r (X) denote the standard parabolic cylinder of radius r centered at X.
The diameter of Ω, sup x,y∈Ω {|x−y|}, will be denoted by diam(Ω). We write u ∈ C 2,1 if sup |u| and D t u, D i u, D ij u, i, j = 1, . . . , n exist and are bounded in a given domain. Also u ∈ C 2,1,α if u ∈ C 2,1 and for i, j = 1, . . . , n. The expression N (· · · ) denotes the various constants N which will be determined by the quantities described in the parentheses.
2. The Poisson problem with weakly singular right hand side. Let Ω be a bounded domain with C 2,α -boundary, and ..,n is strictly positive and symmetric, we have a symmetric, positive, and invertible matrix S : for t > s, X = (x, t) ∈ Q, and identically zero for t ≤ s. Here, S −1 denotes the inverse matrix of S, and N 1 is a fixed positive constant. Consider the following Poisson problem with weakly singular right hand side: The following theorem gives us the existence and an estimate of the solution u: There exists a solution u of the problem (PS) in the class C 2,1,α (Q\ C δ (Y )) for any δ > 0, and satisfying where N 2 = N 2 (n, α, λ, Λ, A α , diam(Ω), T ), N 3 := N 3 (n, α, λ, Λ, A α , diam(Ω)).
Proof. Dividing N 1 , using u ≡ 0 for t < s, we may assume N 1 = 1, y = 0, s = 0. Furthermore, assume temporarily f ∈ C α (Q), then the existence of solution u ∈ C 2,1,α (Q) ∩ C(Q) of the Poisson problem (PS) will be guaranteed by [18,Proposition 4.25]. We will show that the estimate (8) holds. Define for t > 0, X = (x, t), and identically zero for t ≤ 0, where S −1 := (s −1 ij ) is the inverse matrix of S, and N 4 and N 5 are positive constants which will be fixed later. Also, let It is easy to check the following by direct computations: for any N 5 ∈ (1, n n−α ). Fix N 5 := for some N 4 := N 4 (n, α, Λ). Here, we used the fact that for some constant N depending on N 5 , Λ, α. For these fixed constants N 4 and N 5 , for some N 6 := N 6 (n, α, λ, Λ, A α ). We can choose small 0 such that for |X| < 0 , where 0 depends on N 5 , N 6 , n, α, Λ. From (9), we have 4N 5 Λt for t > 0, and identically zero for t ≤ 0. Here the positive constant N 7 will be fixed later. Note for small t 0 depending on n, α, Λ, 0 , N 5 , and v ≥ h for |x| ≥ 0 for any N 7 ≥ N 4 .
For the general case, choose f m ∈ C α (Q) for any positive integer m, such that f m = f in C(Q \ C 1 m (Y )) and f m also satisfies the given singular estimate (7). We showed that there exists u m such that and verifying (8) independent of m. We claim that there exists u ∈ C 2,1,α (Q\C δ (Y )) such that u m has a subsequence converging to u in C 2,1 (Q \ C δ (Y )) for any δ > 0. For this, note that u m solves the following: Thus u m C 2,1,α (Q\C δ (Y )) ≤ N f m C α (Q\C δ (Y )) + sup (Q\C δ (Y )) u m by [18,Proposition 4.25] again. By Arzela-Ascoli theorem, there exists u ∈ C 2,1 (Q \ C δ (Y )) such that u m has a subsequence converging to u and Lu = f in Q \ C δ (Y ). By the standard Schauder theory and the uniqueness of the solution, u ∈ C 2,1,α (Q \ C δ (Y )).
Proof. Fix > 0. Since u is uniformly continuous in Q, we can find δ > 0 such that u ≤ ≤ w + on ∂ p Q δ , where Q δ := Ω × (δ, T ). By the comparison principle, we have u ≤ w + in Q δ . Observing that was arbitrary and δ → 0 + as → 0 + , we have the desired result.
3. Existence and estimates of the Green's function. In this section, we will prove Theorem 1.1. Fix Y ∈ Q. Similar to the proof of Theorem 2.1, for the strictly positive, symmetric matrix A := A(Y ) = (a ij (Y )) i,j=1,...,n , we have a symmetric, positive, and invertible matrix S := (s ij ) i,j=1,...,n such that A = S 2 . Define for t > s, X = (x, t), Y = (y, s), and identically zero for t ≤ s, where S −1 is the inverse matrix of S. By [12], H is the fundamental solution of L Y , where a ij (Y )D ij u.
We will construct G in the form where LG 1 = 0 in Q, G 1 = −H on ∂ p Q,

and
LG 2 = f in Q, G 2 = 0 on ∂ p Q, for f := (L Y − L)H. The existence of G 1 is well known. For example, see [18]. For Using a Hölder continuity of a ij , the function f satisfies (7) with y = 0, s = 0, N 1 = N 1 (n, λ, Λ, A α ). Thus, the existence of G 2 is proved by Theorem 2.1. Furthermore, it is immediate to see that G satisfies the definition of Green's function, (3)-(5). Lastly, we will prove (6). For this, note that G 1 ≤ 0 by the maximum principle, and the estimate holds for H and |G 2 |.