$W^{1,p}$ estimates for Schrödinger equation in the region above a convex graph

The purpose of this paper is to establish $W^{1,p}$ solvability for Schrödinger operator in the region above a convex graph. Precisely, let

where $\phi:\mathbb{R}^{d-1}\to\mathbb{R}$ is a convex function with $\|\nabla\phi\|_{L^{\infty}(\mathbb{R}^{d-1})}\leq M$. For $f\in L^{p}(\Omega,\mathbb{R}^{d})$ and $g\in B^{-\frac{1}{p},p}(\partial\Omega)$, we consider the Schrödinger equation

Following the tradition and physical significance, $V$ is referred to be the electric potential. Throughout this paper, we assume that $0<V\in B_{\infty}$, i.e., $V\in L_{loc}^{\infty}(\mathbb{R}^{d})$, and there exists a constant $C>0$ such that, for all ball $B\subset\mathbb{R}^{d}$

Examples of $B_{\infty}$ weights are $|x|^{a}$ with $0\leq a<\infty$.

To state the main result of the paper, let $n$ denote the outward unit normal to $\partial\Omega$ and $B^{\alpha,p}(\partial\Omega)$ with $0<\alpha<1$ and $1<p<\infty$ denote the Besov spaces on $\partial\Omega$.

The $W^{1,p}$ estimates for inhomogeneous equation $\Delta u=F$ in bounded non-smooth domains have been studied extensively. Indeed, it has been known since the 1990s that the Dirichlet problem and the Neumann problem is $W^{1,p}$ solvable in bounded Lipschitz domains for $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ when $d\geq 3$ where $\varepsilon>0$ depends on $\Omega$. For a general second order elliptic equation with coefficients belonging to $VMO$, $W^{1,p}$ estimates are reduced to the weak reverse Hölder estimates for local solutions and are established for $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ when $d\geq 3$. And the ranges of $p$ are sharp (see [8, 16, 11, 23]; also see [2, 3, 4, 5, 6, 7] for references on related work on boundary value problems in bounded Lipschitz domains or Reifenberg flat domains). It is worth noting that for every $p>3$ and $d\geq 3$ there is a Lipschitz domain such that $\nabla u$ cannot belong to $L^{p}(\Omega)$ even if the right side $F$ is in $C^{\infty}$. If a slightly stronger smoothness condition is imposed, that $\Omega$ is a bounded convex domain, the $W^{1,p}$ solvability was essentially established in [12] for the sharp range $1<p<\infty$. Regarding the convexity of $\Omega$, with the analysis tools developed in [13] at disposal, the weak reverse Hölder inequality

which is the sufficient condition to the $W^{1,p}$ estimate is established.

For Schrödinger equations, Z. Shen [19] obtained the $W^{1,p}$ estimate for Dirichlet problems for $2<p<3+\varepsilon$ when $d\geq 3$, and $2<p<4+\varepsilon$ when $d=2$ if $\Omega$ is a bounded Lipschitz domain and for $2<p<\infty$ if $\Omega$ is a bounded $C^{1}$ domain under the assumption that the potential $V$ is positive and bounded. In the region above a Lipschitz graph, Z. Shen [20] established the $L^{p}$ solvability for the Neumann problem and the Dirichlet problem if $V\in B_{\infty}$. The $W^{1,p}$ solvability is formulated in forthcoming paper [14]. One may notice that the region above a Lipschitz graph maybe unbounded, and results in bounded domains do not work. And it worthwhile to flagged up that, in [20], by the decay of solutions at infinity and the limit $R\to\infty$, the results in $\Omega_{R}=\{(x^{\prime},t):|x^{\prime}|<R\text{ and }\phi(x^{\prime})<t<\phi(x^{\prime})+R\}$ can be extended to $\Omega$. We remark that in [21], Z. Shen established $\|\nabla u\|_{L^{p}(\mathbb{R}^{d})}+\|V^{\frac{1}{2}}u\|_{L^{p}(\mathbb{R}^{d})}\leq C\|f\|_{L^{p}(\mathbb{R}^{d})}$ where $1\leq p\leq 2q$ and $q\geq d$ for $-\Delta u+Vu={\rm div}f$ in $\mathbb{R}^{d}$ with $V\in B_{q}$. More related work about the Schrödinger operator refers to [1, 17, 18, 22].

Motivated by [12, 20], we extend the results to the Schrödinger operator $-\Delta+V$ in the region above a convex graph. Our proof to Theorem 1.1 follows the approach in [11]. Employing a real-variable perturbation argument and John-Nirenberg inequality, we give a sufficient condition for $W^{1,p}$ estimates for weak solutions of (1.1) with $g=0$. Roughly speaking, we prove that for $p>2$, if the reverse Hölder’s inequality

holds for the solutions of $-\Delta v+Vv=0$ in $\Omega$ and $\frac{\partial v}{\partial n}=0$ on $B(x,2r)\cap\Omega$, then the $W^{1,p}$ estimate is established. Following similar line of [13], we demonstrate the condition (1.5) by the improved Fefferman-Phong inequality, the estimates for the Fefferman-Phong-Shen maximal function $m(x,V)$ as well as the convexity of $\Omega$. For the case of $1<p<2$, duality arguments and the estimates for the Neumann functions

and

play significant roles.

The present paper can be split into three portions. In the first part, we collect some known results for the Fefferman-Phong-Shen maximal function, boundary $L^{\infty}$ estimate and estimate for the Neumann function. The second portion presents a sufficient condition of $W^{1,p}$ estimate for $p>2$. Last segment is devoted to prove $W^{1,p}$ estimate for $1<p<2$.

We end this section with some notations. We will use $\fint_{E}u$ to denote the average of $u$ over the set $E$; i.e.

Let $B(x,r)$ denote the sphere centered at $x$ with radius $r$. Denote $D(x,r)=B(x,r)\cap\Omega$ and $\Delta(x,r)=B(x,r)\cap\partial\Omega$. For $R>0$ large sufficiently, let

where $\phi:\mathbb{R}^{d-1}\to\mathbb{R}$ is the convex function.

We first note that $V$ is a $B_{\infty}$ weight, defined by Franchi [10, p.153]. Then the measure $Vdx$ is doubling, i.e., there exists $C>0$ such that for any ball $B$ in $\mathbb{R}^{d}$,

Let

for $x\in\mathbb{R}^{d},r>0$, then the Fefferman-Phong-Shen maximal function is defined as

Several conclusions from [20] and [21] will be quoted in this section. These lemmas and definitions are related to the concepts of Fefferman and Phong discussed in [9]. We list some of them below.

Next we shall introduce the Fefferman-Phong type inequality.

A refine version of Lemma 2.4 was obtained in [1].

We end this section with a boundary $L^{\infty}$ estimate and the estimate for the Neumann function.

Let $\Gamma(x,y)$ denote the fundamental solution of the Schrödinger operator $-\Delta+V$ in $\mathbb{R}^{d}$. Fix $x\in\Omega$, let $v^{x}(y)$ be the solution of $-\Delta u+Vu=0$ in $\Omega$ with Neumann data $\frac{\partial\Gamma(x,y)}{\partial\nu_{y}}$. Let $N(x,y)=\Gamma(x,y)-v^{x}(y).$ Then we have the following estimate.

The following theorem is a refined real variable argument which established in [24, Theorem 3.2] (see also [25, Theorem 4.2.3]) and can be seen as a duality argument of the Calderón-Zygmund decomposition. With this, the $W^{1,p}$ estimates follow from the locally weak reverse Hölder inequality.

With the real variable method at disposal, we give the sufficient condition.

To establish the reverse Hölder inequality, we need an auxiliary lemma as follows.