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

Abstract

Let be a nonsmooth convex domain and let be a distribution in the atomic Hardy space ; we study the Schrödinger equations in with the singular potential and the nonsmooth coefficient matrix . We will show the existence of the Green function and establish the integrability of the second-order derivative of the solution to the Schrödinger equation on with the Dirichlet boundary condition for . Some fundamental pointwise estimates for the Green function are also given.

1. 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 of an elliptic operator with the Dirichlet boundary condition. The solvability of the problems is based on a priori estimate, here is a second-order elliptic operator with smooth coefficients, is a bounded domain in with smooth boundary, and is a function in 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 is a bounded convex domain in , , and , then there is a unique solution solving the Laplace equation in , and further . In 1993, Adolfsson [3] extended Kadlec's results to get the integrability of whenever for .

In the present paper, let be a bounded or unbounded convex domain in , , we consider the following singular Schrödinger operator: in , where is a nonnegative singular potential belonging to the class for some , and is a real symmetric matrix. We call that the potential satisfies the reverse Hölder class for , if belongs to and there exists a positive constant such that for all balls . One sees that for .

Recently, the regularity for the Schrödinger operator in , rather than in domain , has been studied in [4, 5]. Shen [4] proved that if belongs to the reverse Hölder class , then is a Calderón-Zygmund operator in , which means that if is the solution to in , then for . We also remark that Ladyzhenskaya, see Theorem III.9.1 in [6], had found the estimate if is a bounded convex domain and for some .

To discuss the singular Schrödinger equation , we need to introduce the following assumptions (A1) and (A2) for the matrix :(A1)there exists a constant such that (A2)there exists a positive constant such that

The last assumption above in (A2) means that we can rewrite the operator as . In the case , by the Calderón-Zygmund singular integral theory, Avellaneda and Lin [7] showed the -boundedness of the operator for under assumptions (A1), (A2), and(A3)there is such that Kurata and Sugano [8] also obtained the weighted -boundedness of the operator for under the assumptions (A1), (A2), and (A3).

Here we remark that for general convex domain and the nonsmooth coefficient matrix , the associated operator 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 boundedness and the boundedness for the operator on the convex domain without assumption (A3). Equivalently, we will study the existence and the regularity of the weak solution to the following Dirichlet problem in the convex domain , that is for with , the atomic Hardy spaces, or with , where is the trace operator on the boundary of the domain .

For our purpose, let be the integer and let , we denote by the Sobolev spaces, and denote by the closure of in .

We call a -atom, if is a bounded measurable function defined in and the following conditions (i), (ii), and (iii) hold:(i)there is a cube satisfying supp ;(ii);(iii) for any multi-index with .

The atomic Hardy space in domain , , 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 . It’s worthy to point out that if is a Lipschitz domain or a convex domain then we can see from the works in [10, 11] that , where is the following local Hardy space in domain,

We also notice that the dual space of with is the space of Hölder continuous functions, , with the exponent . Thus the paring between an element of and the function in is well defined. One could refer to [12] for related boundary value problems.

For , , we say is a solution to the Dirichlet problem (7), if satisfies for any test function .

Applying the Lax-Milgram theorem, we will prove that for the Lipschitz domain and the function , there is a unique solution to the Dirichlet problem (7); see Theorem 10 below. We will then show the existence of the Green function related to the operator and the domain and give the point-wise estimates for the Green function which is fundamental to us. Moreover, we will give the 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 in with .

Theorem 1. Suppose that is a bounded convex domain, and satisfies assumptions (A1) and (A2). If for and solves the Dirichlet problem (7), then one has with the constant independent of .

Theorem 2. Let be the region above a convex Lipschitz graph, and let . If for and satisfies in , then one has with the constant independent of .

By the interpolation argument between the -estimate (12) and Theorems 1 and 2, we have the following -regularity estimates for .

Corollary 3. Suppose that is a bounded convex domain, and satisfies assumptions (A1) and (A2). If for and solves the Dirichlet problem (7), then one has with the constant independent of .

Corollary 4. Let be the region above a convex Lipschitz graph, and let . If for and satisfies in , then one has with the constant independent of .

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 couldn’t be reduced for the second-order derivative estimates of the solution, but the condition is enough for the existence of -solution to the Dirichlet problem (7).

The paper is organized in the following way. In Section 2, after recalling some properties for the class , we will show the solvability and uniqueness of the 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 in the convex domain ; see Lemmas 12–16 below for details. In Section 3, we will deduce some important estimates for the solution to in , especially, the local -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.

2. The Solutions and the Green Function

In this section, we will show the existence of the solution to the singular Schrödinger equation in Lipschitz domain for and give some estimates about the Green function related to the operator in . To this end, we need to use an auxiliary function and some properties for the singular potential . Let , we can define the auxiliary function by Recall that implies that is a doubling measure, and for some . Thus, by the Hölder inequality, for any , with some . Therefore, the auxiliary function is well defined and . For example, if is a polynomial of degree and , then

Lemma 6 (see [4]). Let for , then there exist constants , and such that for any in ,(i), if ;(ii);(iii);(iv).

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 7 (see [13]). Let , , , and is sufficiently large, then there are positive constants , , and such that for any and .

Lemma 8 (see [8]). Let denote the fundamental solution to equation in . Then for any integer , one has that(i)if  , then there exists a constant such that for any in ;(ii)if , also we assume satisfies with constant and , then there exists a constant such that for any in .

The following lemma is useful for proving the solvability to the Dirichet problem, which extends the Fefferman-Phong inequality and has been showed in [14] for the case . 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.

Lemma 9. Let be a convex domain in and let for . Then, for , with the absolute constant .

Proof. Along the same lines as that in [14], we can claim the following Poincaré inequality: where for . The inequality (24) for case 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 , 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 is a Lipschitz domain and , there exists for some and , depending only on the Lipschitz character of . Thus, by the doubling property of , This, together with (30), implies (24).
Let , by (31) and the definition of , one sees that Now applying the Poincaré inequality (24), we obtain that We integrate both sides of (33) and (34), respectively, with respect to over . By the Fubini theorem and Lemma 6 we will obtain the inequalities (23). The lemma is proved.

Next we let be the class of all functions such that Then is a Hilbert space and is dense in . Let then we can see from the elliptic condition of the matrix and Lemma 9 that for all ; and by Lemma 9, there is a positive constant independent of such that Thus is a bounded, coercive bilinear form on the Hilbert space .

On the other hand, for and , let for . Then by the Hölder inequality and the Poinceré inequality one gets which means , a bounded linear functional on .

Thus using the Lax-Milgram theorem we obtain the following solvability of the Dirichlet problem (7).

Theorem 10. Suppose that is a bounded convex domain, and for . Let and let , then there is a unique weak solution solving the singular Schrödinger equation in , and further with the absolute constant .

Proof. The estimate (40) follows from Lemma 9, the inequalities (38) and (39).

Remark 11. Checking the argument above, we note that if then the results of Theorem 10 are still true for any unbounded convex domain .

In this paper, we always let for some , thus by Theorem 10 and Remark 11, we have the Green function defined on for any convex domain such that, for each and any , , and in the distribution sense. Noting , we know by maximal principle that for any , with the constant independent of . Moreover, we can show the following decay estimates as Lemma 2.7 in [13] or Lemma 1.21 in [14]; here we omit the details of the proof.

Lemma 12. Let be any integer, then where is the constant independent of .

Next in this section, we suppose that 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 .

Lemma 13. Let be any integer, then there are constants and , , such that for all ,

Using the similar arguments as that of Theorems  3.3(ii) and  3.4(ii) in [16], we also have the following estimate.

Lemma 14. Let be any integer, then there is such that, for all , where .

In order to get the derivative estimates for the Green function , we need to show the following lemma.

Lemma 15. Suppose . If satisfies the equation in , then there exists a positive constant independent of and such that

Proof. Let be the cut-off function, then we have and so which implies the following Caccioppoli inequality: with the absolute constant independent of and with .
Observing that and letting the cut-off function such that on and , then we have Hence, from this, and using Lemma 8 and the Caccioppli inequality (48), we obtain the desired estimate of the lemma.

Lemma 16. Suppose , and that is any integer. Then where is independent of .

Proof. For , we let and denote by . If , thus . One notes that satisfies in for the fixed . By Lemmas 12 and 15, one can see that
On the other hand, if , we observe that satisfies in for the fixed . Thus by Lemma 15 we have We can choose a point such that and a point satisfying . Then a direct computation implies that From this and by using Lemma 14, we get that where we have used Lemma 6 in the last inequality. The proof of the lemma is complete.

3. Local Second-Order Regularity for the Dirichlet Problem

In this section, we will give some useful a priori estimates for the second-order derivative of the solution to the Dirichlet problem (7), we will show in Theorems 20 and 21 some estimates of for the solution and any smooth function . This local second-order regularity will play an important role in the regularity argument for the Dirichlet problem.

Lemma 17. Suppose is an open domain, and , then the solution of the Dirichlet problem (7) satisfies the following uniform estimate: with the constant .
Moreover, for any smooth function , the estimate holds with the constant independent of .

Proof. Applying the Green representation formula, the Hölder inequality and Lemma 7, we have that Hence, which is the desired inequality (57).
Noting and , and so by assumption (A2) for the matrix , we can obtain the estimate (58) from the inequality (57). The lemma is proved.

The following a priori estimate is crucial to us.

Lemma 18. Suppose that is a bounded convex domain or an unbounded region above a convex Lipschitz graph, and . Let be a solution of the Dirichlet problem (7). Then for any , and one has with the constant .

Proof. Begin by assuming that is a convex domain with a boundary. Let the vector field and the ball , then for any tangent vector on the boundary and near , so the Kadlec formula on page 134 in [17] implies where is the normal vector and 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 , and consequently After a direct computation and using the inequality , we have that with the absolute constant independent of . Thus by the nonsingular change of variables, we get, for any , with the absolute constant independent of and .
Applying the inequality (65) and using the standard perturbation procedure, we have the absolute constant such that Now we decompose into a sequence of cubes such that , for any , and for all . Let the cut-off functions be the partition of the unity; namely, we can write that . One can see from the inequality (66) that with the constant independent of and . Noting that the sequence of cubes has the finite intersect property, thus there are constants and such that, for every , 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.

Lemma 19. Suppose that is a bounded convex domain or an unbounded region above a convex Lipschitz graph, and . Let be a solution of the Dirichlet problem (7). Then, for any , one has with the constant independent of , and .

Proof. Noting that 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 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).

Theorem 20. Suppose that is a bounded convex domain or an unbounded region above a convex Lipschitz graph, and . Let be a solution of Dirichlet problem (7). Then for any , and one has with the constant independent of , and .
Particularly, if is a bounded convex domain, one has with the constant .

Proof. Applying Lemmas 17–19, we obtain the inequality (73) from (58), (61), and (69) and get the inequality (74) from (58), (61), and (70). Letting for in the inequality (74), we obtain the inequality (75).

Theorem 21. Suppose that is an unbounded region above a convex Lipschitz graph, and . Let be a solution to in . Then , and for any , and one has with the constant independent of . Moreover,

Proof. Repeating the proof of Theorem 20, and using the inequality (64) in place of the inequality (61), we can then deduce estimate (76). Next, by choosing on and , , and by taking the limit as tends to infinity, we find that and the global estimate (77) holds.

4. The Proof of the Main Theorems

Proof of Theorem 1. By the atomic decomposition theory, it suffices to show the uniform estimates for the solution of the equation in with the -atom and .
Without loss of generality, we assume that , and , where denotes the cube with center and side length . Using the inequality (75) of Theorem 20 and the size condition of the atom , we see that, for , with the constant independent of .
Recall the inequality (74) of Theorem 20 and let the cut-off function satisfy in and with an absolute constant independent of and , , then we see from the Hölder inequality and the support of the atom that, for , Observing and using the estimates for the Green function, Lemma 16 and the size condition of the -atom , we can estimate the last integral in the inequality (79) above. Then we have that
The inequalities (78) and (81) follow that with the constant independent of , which implies the theorem.

Proof of Theorem 2. It’s enough to consider the case , a -atom with its support cube . Then the Hölder inequality and the -estimate (77) in Theorem 21 give that
On the other hand, if we use the inequality (76) and take the cut-off function satisfying in and with the constant independent of and , then we have
Now using the Hölder inequality and the estimates of the , by the similar arguments as that in the proof of Theorem 1, we obtain that with the constant independent of the -atom , which implies the theorem.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was partly supported by National Natural Science Foundation of China under Grants no. 11171306 and no. 11071065 and funded by Zhejiang provincial Natural Science Foundation of China under Grant no. LY12A01024.