Second-Order Regularity Estimates for Singular Schrödinger Equations on Convex Domains

Let Ω ⊂ R 𝑛 be a nonsmooth convex domain and let 𝑓 be a distribution in the atomic Hardy space 𝐻 𝑝𝑎𝑡 (Ω) ; we study the Schr¨odinger equations –div (𝐴∇𝑢) + 𝑉𝑢 = 𝑓 in Ω with the singular potential 𝑉 and the nonsmooth coefficient matrix 𝐴 . We will show the existence of the Green function and establish the 𝐿 𝑝 integrability of the second-order derivative of the solution to the Schr¨odinger equation on Ω with the Dirichlet boundary condition for 𝑛/(𝑛 + 1) < 𝑝 ≤ 2 . Some fundamental pointwise estimates for the Green function are also given.


Introduction and Main Results
The regularity theory is fundamental to the partial differential equation in nonsmooth domain. Usually, the estimate of the second-order derivative of the weak solution required the smoothness of the coefficients and the smoothness of the domain. Early in 1951, Ladyzhenskaya [1] found a solution to the problem of describing the domain of the closure in 2 (Ω) of an elliptic operator L with the Dirichlet boundary condition. The solvability of the problems is based on a priori estimate, ‖ ‖ 2,2 (Ω) ≤ (Ω) (‖L ‖ 2 (Ω) + ‖ ‖ 2 (Ω) ) ; (1) here L is a second-order elliptic operator with smooth coefficients, Ω is a bounded domain in R with smooth boundary, and is a function in 2,2 (Ω) that vanishes on the boundary or satisfies a nondegenerate homogeneous boundary condition of the first order. The significance of this result for the theory of differential operators, including the boundary value problem and the spectral theory, can hardly be overestimated. It's certainly valuable and challenging to deduce the regularity estimate (1) for elliptic operators with rough coefficients in nonsmooth domain. In 1964, Kadlec [2] took use of the geometric properties of the convex domain to show that if Ω is a bounded convex domain in R , > 2, and ∈ 2 (Ω), then there is a unique solution ∈ 1,2 0 (Ω) solving the Laplace equation −Δ = in Ω, and further ‖∇ 2 ‖ 2 (Ω) ≤ ‖ ‖ 2 (Ω) . In 1993, Adolfsson [3] extended Kadlec's results to get the integrability of ∇ 2 whenever ∈ (Ω) for 1 < ≤ 2.
In the present paper, let Ω be a bounded or unbounded convex domain in R , > 2, we consider the following singular Schrödinger operator: in Ω, where is a nonnegative singular potential belonging to the class B for some ≥ /2, and ( ) = ( ( )) is a real symmetric matrix. We call that the potential satisfies the reverse Hölder class B for 1 < < ∞, if ≥ 0 belongs to loc (R ) and there exists a positive constant such that for all balls ⊂ R . One sees that B ⊆ B for 1 < ≤ ≤ ∞.
Here we remark that for general convex domain Ω and the nonsmooth coefficient matrix ( ), the associated operator ∇ 2 L −1 is not always a Calderón-Zygmund operator, and the methods used in [4,5,7,8] cannot be applied to these cases.
The purpose of the paper is to give an elemental proof of the (Ω) boundedness and the (Ω) boundedness for the operator ∇ 2 L −1 on the convex domain Ω ⊂ R without assumption (A3). Equivalently, we will study the existence and the regularity of the weak solution = L −1 to the following Dirichlet problem in the convex domain Ω, that is for ∈ (Ω) with /( + 1) < ≤ 1, the atomic Hardy spaces, or ∈ (Ω) with 1 < ≤ 2, where is the trace operator on the boundary Ω of the domain Ω.
For our purpose, let be the integer and let 0 < < ∞, we denote by the Sobolev spaces, and denote by , 0 (Ω) the closure of ∞ 0 (Ω) in , (Ω). We call ( ) a -atom, if ( ) is a bounded measurable function defined in Ω and the following conditions (i), (ii), and (iii) hold: The atomic Hardy space in domain Ω, (Ω), is then defined as the collection of all = ∑ in the sense of distributions, where { } is a sequence of -atoms and { } is a sequence of real numbers with ∑ | | < ∞. The norm of is defined by One might see [9] for space over open subsets in R . It's worthy to point out that if Ω is a Lipschitz domain or a convex domain then we can see from the works in [10,11] that (Ω) = (Ω), where (Ω) is the following local Hardy space in domain, We also notice that the dual space of (Ω) with /( + 1) < < 1 is the space of Hölder continuous functions, ( ) (Ω), with the exponent ( ) = (1 − )/ . Thus the paring between an element of (Ω) and the function in ( ) (Ω) is well defined. One could refer to [12] for related boundary value problems. For ∈ (Ω), /( +1) < < 1, we say is a solution to the Dirichlet problem (7), if ∈ 1,2 0 (Ω) ∩ 1, (Ω) satisfies for any test function ( ) ∈ ( ) (Ω) ∩ 1,2 0 (Ω). Applying the Lax-Milgram theorem, we will prove that for the Lipschitz domain Ω and the function ∈ 2 (Ω), there is a unique solution ∈ 1,2 0 (Ω) to the Dirichlet problem (7); see Theorem 10 below. We will then show the existence of the Green function related to the operator L and the domain Ω and give the point-wise estimates for the Green function which is fundamental to us. Moreover, we will give the 2 boundedness for the second-order derivative of the solution, see Theorems 20 and 21 below. Our main aim is to further establish the second-order regularity estimates for the equation L = in Ω with ∈ (Ω).

Theorem 2.
Let Ω be the region above a convex Lipschitz graph, and let ∈ B . If ∈ (Ω) for /( + 1) < ≤ 1 and with the constant independent of .
By the interpolation argument between the 2 -estimate (12) and Theorems 1 and 2, we have the following -regularity estimates for 1 < ≤ 2.
Remark 5. One can see from Theorem 10 in Section 2 and the arguments for the proof of Theorems 1 and 2 that the condition ∈ B couldn't be reduced for the second-order derivative estimates of the solution, but the condition ∈ B /2 is enough for the existence of 1,2 0 (Ω)-solution to the Dirichlet problem (7).
The paper is organized in the following way. In Section 2, after recalling some properties for the class B , we will show the solvability and uniqueness of the 1,2 0 (Ω) solution to the Dirichlet problem (7); see Theorem 10 below. We will also give some useful point-wise estimates for the Green function ( , ) and its gradient ∇ ( , ) related to the singular Schrödinger operator L in the convex domain Ω; see Lemmas 12-16 below for details. In Section 3, we will deduce some important estimates for the solution to L = in Ω, especially, the local 2 -estimates for the second-order derivative of the solution ; see Theorems 20 and 21 below. In Section 4, we will give the proofs of Theorems 1 and 2.

The 1,2 0 (Ω) Solutions and the Green Function
In this section, we will show the existence of the 1,2 0 (Ω) solution to the singular Schrödinger equation L = in Lipschitz domain Ω for ∈ 2 (Ω) and give some estimates about the Green function related to the operator L in Ω. To this end, we need to use an auxiliary function and some properties for the singular potential . Let ∈ B /2 , we can define the auxiliary function ( , ) by Recall that ∈ B /2 implies that ( ) is a doubling measure, and ∈ B for some > /2. Thus, by the Hölder inequality, for any 0 < ≤ < ∞, with some > 0. Therefore, the auxiliary function ( , ) is well defined and 0 < ( , ) < ∞. For example, if ( ) is a polynomial of degree and ( ) = | ( )|, then Lemma 6 (see [4]). Let ∈ B for ≥ /2, then there exist constants > 0, > 0 and 0 > 0 such that for any , in R , The estimates of (i), (ii), and (iii) in Lemma 6 were proved in [4], while the estimate (iv) can be derived from the estimates (ii) and (iii).
Lemma 8 (see [8]). Let Γ( , ) denote the fundamental solution to equation L = 0 in R . Then for any integer > 0, one has that (i) if ∈ B /2 , then there exists a constant such that for any , in R ; (ii) if ∈ B , also we assume ( ) satisfies ‖ ( )‖ ( ) ≤ 1 with constant 1 > 0 and ∈ (0, 1], then there exists a constant such that for any , in R .
The following lemma is useful for proving the 2 solvability to the Dirichet problem, which extends the Fefferman-Phong inequality and has been showed in [14] for the case Ω = R . Here we thank the referee for pointing out that Lemma 9 below can be generalized to more general domains by applying the embedding estimates in [15] among others.
Proof. Along the same lines as that in [14], we can claim the following Poincaré inequality: where = ( 0 , ) ∩ Ω for 0 ∈ Ω. The inequality (24) for case Ω = R was founded in [14]; here we adapt the argument and give the simple lines of the proof for completeness. In fact, for , ∈ , one notes that is a convex domain and so one can write that Let = ∇ and, for , ∈ , we define It's clear that + (( − )/| − |) ∈ for 0 < < | − |, and so Also, by the Fubini theorem, the doubling property of measure , and the inequality (18), one can deduce that (see page 527 in [14]) Combining the inequalities (27) and (28), we get by interpolation. By summation and the Minkowski inequality, we have Since Ω is a Lipschitz domain and 0 ∈ Ω, there exists ( 0 , ) ⊂ for some 0 ∈ and > 0, depending only on the Lipschitz character of Ω. Thus, by the doubling property of , This, together with (30), implies (24). Let 0 = 1/ ( , 0 ), by (31) and the definition of ( , 0 ), one sees that Now applying the Poincaré inequality (24), we obtain that Abstract and Applied Analysis 5 We integrate both sides of (33) and (34), respectively, with respect to 0 over Ω. By the Fubini theorem and Lemma 6 we will obtain the inequalities (23). The lemma is proved.
Next we let H(Ω) be the class of all functions ∈ 1,2 0 (Ω) such that Then H(Ω) is a Hilbert space and 1 0 (Ω) is dense in H(Ω). Let then we can see from the elliptic condition of the matrix ( ) and Lemma 9 that for all , V ∈ H(Ω); and by Lemma 9, there is a positive constant independent of such that Thus ( , V) is a bounded, coercive bilinear form on the Hilbert space H(Ω).
Next in this section, we suppose that Ω is a bounded convex domain or the region above a Lipschitz graph. Noting that one may take a cone of arbitrary height and fixed opening angle at any boundary point of the Lipschitz graph, by similar argument as that of Theorems 1.8 and 1.9 in [16], we can deduce the following Hölder estimate for the Green function ( , ).
In order to get the derivative estimates for the Green function ( , ), we need to show the following lemma.
Proof. Begin by assuming that Ω is a convex domain with a 2 boundary. Let the vector field = ∇ and the ball ⊃⊃ supp , then ⋅ = 0 for any tangent vector on the boundary Ω and ≡ 0 near , so the Kadlec formula on page 134 in [17] implies where is the normal vector and tr B is the trace of the second fundamental quadratic form on the boundary, that is, the mean curvature of the boundary. For a convex domain we have tr B ≤ 0, and consequently (63) After a direct computation and using the inequality 2 ≤ (1/ ) 2 + 2 , we have that with the absolute constant = ( ) > 0 independent of . Thus by the nonsingular change of variables, we get, for any with the absolute constant = ( , ) > 0 independent of 0 and . Applying the inequality (65) and using the standard perturbation procedure, we have the absolute constant = ( , ) > 0 such that Now we decompose R into a sequence of cubes { } such that R = ∪ , ∩ = 0 for any ̸ = , and diam(2 ) = < (3 ) −1 for all . Let the cut-off functions ∈ ∞ 0 (2 ) be the partition of the unity; namely, we can write that = ∑ ∞ =1 . One can see from the inequality (66) that From this and the inequality (67), we can deduce the inequality (61). A routine limiting argument, see [17] for example, yields the inequality (61) for all convex bounded domains or the unbounded region above a convex Lipschitz graph. The lemma is proved.
We also need the following lemma about the local estimates of the derivatives.
Proof. Noting that ∈ 1,2 0 (Ω) satisfies which, together with the ellipticity of the matrix , follows the inequality (69). Moreover, by this and the Cauchy inequality and the Poincaré inequality, we have that Taking = (2 ) −2 in the inequality above, we deduce the inequality (70).
From Lemmas 17-19, we have the following local estimates of the second-order derivatives of the solution for the Dirichlet problem (7).