Optimal Regularity of a Degenerate Elliptic Equation

In this paper, we consider the optimal regularity of solutions to elliptic equation with a Grushin-like operator. By using the Feynman–Kac formula, we first get the expression of heat kernel, and then by using the properties of heat kernel, the optimal regularity of solutions will be obtained. The novelty of this paper is that the Grushin-like operator is a degenerate operator.


Introduction
The regularity of solutions to second order elliptic equation has been extensively studied by many authors, see the book [5]. But for degenerate elliptic equation, there is few work about the regularity results until now. In this paper, we focus on a special degenerate elliptic equation-like Grushin elliptic equation. The main reason why we can deal with it is that we can get the expression of heat kernel by using the probability method.
For Grushin operator, many authors studied it. Beckner [2] obtained the Sobolev estimates for the Grushin operator in low dimensions by using hyperbolic symmetry and conformal geometry. Riesz transforms and multipliers for the Grushin operator was considered by Jotsaroop et al. [11]. Tri [16] studied the generalized Grushin equation. L p -estimates for the wave equation associated to the Grushin operator was studied by Jotsaroop and Thangavelu [12]. The fundamental solution for a degenerate parabolic pseudo-differential operator covering the Grushin operator was obtained by Tsutsumi [17,18]. Furthermore, Tsutsumi [19] constructed a left parametrix for a pesudo-differential operator. We remark that Tsutsumi did not give the exact expression of heat kernel. In the book [1], they gave the expression of heat kernel (see Page 191), but he expression is hard to use because they used the inverse Fourier transform. Yu [24] established Liouville type theorem for nonlinear elliptic equation involving Grushin operators. Chang and Li [3] studied the heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator. Similar degenerate elliptic equation was studied by Robinson and Sikora [15] and the Hardy inequalities for Grushin operator was considered by [23]. The blowup problem of Grushin operator was considered by Lv et al. [10]. Recently, Wang et al. [21,22] studied the regularity for subelliptic systems.
About the regularity of degenerate elliptic operator, Di Fazio et al. [4] considered the regularity of a class of strongly degenerate quasilinear operators, also see [14]. Guo et al. [6,8] studied a degenerate elliptic equations and obtained the well-posedness of the degenerate elliptic equations, also see [7]. In this paper, in view of probability point, we give a new expression and then get the regularity of solution by using the properties of heat kernel. This paper is arranged as follows. In next section, some preliminaries are given and the main results will be proved in Sect. 3. Throughout this paper, we write C as a general positive constant and C i , i = 1, 2, ⋯ as a concrete positive constant.

Main Results
Consider the Grushin operator which is the generator of the diffusion process Here W i t , i = 1, 2 are standard Brownian motion. It is easy to see that the process (X t , Y t ) is a two-dimensional Gaussian stochastic process. In this short paper, we consider the following operator It is easy to get the heat kernel of the operator L is which yields that In [20], the authors considered a degenerate elliptic equation on a manifold. Thus the heat kernel K is different from that in [20]. In paper [9], Li studied the Grushin operator Δ x + |x| 2 Δ u on ℝ n x × ℝ n � u with n ≥ 3 and n � = 1 , which is different from our cases. We only consider the operator on ℝ × ℝ . Meanwhile, we remark that Tsutsumi [19] considered the existence of left parameters of the operator where a , are constants, m ∈ ℤ , D j = j=1 j j . It is remarked that the operator A defined on ℝ 3 . Moreover, there is no concrete representation of heat kernel.
We first want to solve the following elliptic equation We say u is a solution of (1) if u satisfies There exists a 0 > 0 such that for any Next, we consider the L p -regularity of v(x, y).

Proof of Main Results
Denote [⋅] by the semi-norm of C . Set

Lemma 3.1 Assume that f ∈ A and u is defined as in (3). Then
Moreover, if f (⋅, y) ∈ C b (ℝ) for any y ∈ ℝ , it holds that Proof Simply calculations show that and where we used the fact that We also remark that the meaning of ‖f ‖ L ∞ (ℝ,C b (ℝ)) is that we take infinity norm for the first variable and take Hölder norm for the second variable.

Recall the following interpolation inequality
Now, if 0 < < 1 , applying the above inequality with = 0, = 1 and = (1 − ) , we have where we used the definition of semi-norm. Next, we consider the derivative of the second variable.
By dividing the real line into two parts, we have x,x,y,ŷ∈ℝ,x≠x,y≠ŷ x,x,y,ŷ∈ℝ,x≠x,y≠ŷ Let us estimate I 11 -I 14 . By using the form of heat kernel, we get Similarly, we can obtain Next, we consider I 13 . where 0 < < 1 and > 0 depends on h satisfying Denote By using the above fractional mean value formula with > and the interpolation inequality in Hölder space Lastly, by using the properties of heat kernel K, it is easy to see that Using the above equality and similar to the operation of I 13 , we have Substituting I 11 −I 14 into I 1 , we get Similarly, we can prove that if f (⋅, y) ∈ C b (ℝ) for any y ∈ ℝ, Summing the above discussion, we obtain Noting that the above inequality holds for 0 < < 1 . The proof of this lemma is complete. ◻ x,x,y,ŷ∈ℝ,x≠x,y≠ŷ x,x,y,ŷ∈ℝ,x≠x,y≠ŷ .

Remark 3.1
It is well known that if f ∈ C (ℝ n ) , then the solution u of the following equation belongs to C 2+ (ℝ n ) , which is the Schauder theory. Noting that the heat kernel of above equation is Gauss heat kernel, that is It is easy to see that x ∼ √ t . But in our case, different axis has different scaling, that is, Thus when we take derivative for variable x, we can get t − 1 2 , and take double derivative for variable x, we will get t −1 . But if we take derivative for variable y, we shall get t −1 , which is different from the classical case. In Schauder theory, we can get C 2+ (ℝ n ) estimates, but in our case the C 1+ (ℝ n ) should be optimal.
Like the classical case, we can get the C 2+ -estimate for the x-axis if f (⋅, y) ∈ C b (ℝ) for any y ∈ ℝ , but we can not get the same estimate for y-axis. If we want to get the C 2+ -estimate for the y-axis, we need more regularity about the second variable. In other words, we have the following results. Corollary 3.1 Assume that f (⋅, y) ∈ C b (ℝ) for any y ∈ ℝ and ∇ y f (x, ⋅) ∈ C b (ℝ) for any x ∈ ℝ . Then where 0 < < ( ∧ (1 − )) . That is to say, ∇ 2 u ∈ C b (ℝ 2 ).

Proof of Theorem 2.1
We use Picard's iteration to solve (1). Let u 0 = 0 and define for n ∈ ℕ, It follows from Lemma 3.1 that and u t − Δu = f , u 0 = 0, Choosing 0 be large enough so that C − ‖b‖ C b (ℝ 2 ) < 1∕4 for all ≥ 0 , we get and for all n ≥ m, Substituting them into (5) and (6), we obtain and for all n ≥ m, Hence there is a u ∈ C 1+ b (ℝ 2 ) such that (2) holds and and u solves Eq. (1) by taking limits for (4). The second result can be obtained similarly. The proof is complete. ◻

Proof of Theorem 2.2
For simplicity, we only consider a special case, that is, . Assume that f 1 ∈ L p (ℝ) and f 2 ∈ L q (ℝ) . Denote By using the above inequality, Minkowski's inequality and the properties of the heat kernel, we have

Furthermore, we can get
Thus if we take p = q = r , then we have ‖∇ 2 x u‖ L r (ℝ 2 ) ≤ C . However, if we deal with the second variable, it is difficult to get the decay estimate. More precisely, we have for 3 2r < 1 2p + 1 q , Hence we must add more regularity on the second variable. Meanwhile, we recall that if h ∈ W s,p (ℝ n ) with 0 < s < 1 , then If we assume that f 2 ∈ W s,q (ℝ) and let we get where m, n satisfy (7) and Moreover, under the condition that f 2 ∈ W s,p (ℝ) , we can similarly get Hence it is easy to see that we can take suitable s ∈ (0, 1), p > 1 and r > 1 such that −s − 3 2r + 1 2p + 1 ≤ 0 . That is to say, we have under the condition that f 2 ∈ W s,p (ℝ) . The proof is complete. ◻

Remark 3.2
It is well known that if f ∈ L p (ℝ n ) , then the solution u of the following equation belongs to W 2,p (ℝ n ) , which is the L p -theory. Noting that the heat kernel of above equation is Gauss heat kernel, and similar to Remark 3.1, it is easy to find the difference from the classical Laplacian operator. Due to the singularity of the variable y, we must give two different assumptions. Comparing the classical L p -theory, in our case the regularity of Theorem 2.2 should be optimal.
Author Contributions All authors contributed to the writing of the present article and they read and approved the final manuscript.
Funding This work was supported by NSFC of China Grants 11901158, 11571176 and the Startup Foundation for Introducing Talent of NUIST.

Conflict of interest
The authors declare that they have no competing interests.
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