The main goal of this paper is finding the minimal regular nonlinearities to study a global estimate in weighted Lorentz spaces for the gradient of weak solution to a nonlinear elliptic obstacle problem over a bounded nonsmooth domain. Let Ω be a bounded nonsmooth domain of \(\mathbb{R}^{d\ge2}\) to be specified later. For a given obstacle \(\Psi\in W^{1,2}(\Omega)\) with \(\Psi\le0 \mbox{ a.e. on } \partial\Omega\), we denote the set of admissible functions by
$$\begin{aligned} \mathcal{A} = \bigl\{ \phi\in W^{1,2}_{0}(\Omega): \phi\ge \Psi \text{ a.e. in } \Omega \bigr\} . \end{aligned}$$
For \(u \in\mathcal{A}\), we focus on considering the following variational inequalities:
$$ \int_{\Omega} \mathbf{a}(Du, x) \cdot D ( \phi- u ) \,dx \ge \int _{\Omega} \mathbf{f} \cdot D ( \phi- u ) \,dx\quad \text{for all } \phi\in \mathcal{A}, $$
(1.1)
where the inhomogeneous term f is a given vector-valued function in \(L^{2}(\Omega; \mathbb{R}^{d})\), and the nonlinearities \(\mathbf{a}(\xi, x): \mathbb{R}^{d}\times\Omega\rightarrow\mathbb {R}^{d}\) are, as usual, so-called Carathéodory functions satisfying the following conditions:
$$\begin{aligned} \textstyle\begin{cases} \mathbf{a}(\xi, x) \text{ measurable in } x\in\Omega \text{ for all } \xi\in\mathbb{R}^{d}, \\ \mathbf{a}(\xi, x) \text{ differentiable in } \xi\in\mathbb{R}^{d} \text{ for almost all } x\in\Omega. \end{cases}\displaystyle \end{aligned}$$
We call such a function \(u \in\mathcal{A}\) weak solution to the variational inequalities (1.1). To ensure the solvability in \(L^{2}(\Omega)\) to (1.1), it is quite necessary to impose additional assumptions on the given datum.
(H1) (ellipticity and growth) There exist two constants \(0<\lambda\le\Lambda<\infty\) such that
$$\begin{aligned} \textstyle\begin{cases} \langle D_{\xi}\mathbf{a}(\xi,x) \eta\cdot\eta \rangle\ge \lambda \vert \eta \vert ^{2}, \\ \vert \mathbf{a}(\xi,x) \vert + \vert \xi \vert \vert D_{\xi}\mathbf{a}(\xi,x) \vert \le\Lambda \vert \xi \vert \end{cases}\displaystyle \end{aligned}$$
(1.2)
for a.e. \(x \in\Omega\) and \(\xi,\eta\in\mathbb{R}^{d}\).
It is clear that relations (1.2) immediately yield the following monotonicity conditions:
$$\begin{aligned} \mathbf{a}(0,x)=0 \quad\text{and}\quad \bigl\langle \mathbf{a}(\xi,x)- \mathbf {a}(\eta,x), \xi-\eta \bigr\rangle \ge\lambda \vert \xi-\eta \vert ^{2}. \end{aligned}$$
(1.3)
With the nonlinearities satisfying (1.2), by way of classical estimate we make sure that there exists a unique weak solution \(u\in\mathcal{A}\) to the variational inequality (1.1) with the usual \(L^{2}\) estimate
$$\begin{aligned} \Vert Du \Vert _{L^{2}(\Omega)} \le C \bigl( \Vert \mathbf{f} \Vert _{L^{2}(\Omega)} + \Vert D\Psi \Vert _{L^{2}(\Omega)} \bigr), \end{aligned}$$
(1.4)
where the constant C is independent of \(u, \mathbf{f}, \Psi\), and Ω; see Lemma 2.1 in [6]. In this paper, we are interested in the Calderón–Zygmund-type theory in the scale of weighted Lorentz spaces regrading the variational inequality (1.1) by imposing some minimal regular assumptions on the given datum. More precisely, we are interested in finding small partially BMO requirements on the nonlinearities and Reifenberg flat geometric structure of the domain to ensure the Calderón–Zygmund estimate for the gradient of weak solution in the weighted Lorentz spaces \(L^{(p,q)}_{\omega}(\Omega)\), which essentially shows that
$$\begin{aligned} \Vert D u \Vert _{L^{(p,q)}_{\omega}(\Omega)}\le C \bigl( \Vert \mathbf{f} \Vert _{L ^{(p,q)}_{\omega}(\Omega)} + \Vert D\Psi \Vert _{L ^{(p,q)}_{\omega}(\Omega )} \bigr) \end{aligned}$$
(1.5)
for \(\omega\in A_{p/2}\), \(2< p<\infty\) and \(0< q\le\infty\), where the constant C is independent of \(u, \mathbf{f}\), and Ψ.
A key ingredient under consideration concerning the nonlinearities \(\mathbf{a}(\xi,x)\), apart from \(C^{1}\) in ξ is that we also require them to be small BMO of codimension one with respect to the spatial variable x, which means that, in a neighborhood of each point in Ω, there is a local coordinate system such that \(\mathbf{a}(\xi,x)\) is only measurable in one direction and has small bounded mean oscillation in the remaining \((n-1)\) orthogonal directions. In fact, this was first introduced by Kim and Krylov [21], and later employed by Dong and Kim [11–13] and Byun and Wang [8] in the study of weighted \(L^{p}\) theory for divergence and nondivergence of linear elliptic and parabolic equations/systems. It has actually proved to be a sort of minimal regular requirement imposed on the leading coefficients of the elliptic operator to ensure a satisfactory Calderón–Zygmund theory for all \(p > 1\). Here, we would also like to point out that Byun and Palagachev [8] derived a global weighted \(W^{1,p}\)-estimate for \(2< p<\infty\) to Dirichlet problems of linear elliptic equations in Reifenberg flat domains, provided that the coefficients are vanishing of codimension one (also called small partially BMO) based on a different geometric approach instead of the pointwise estimates of sharp functions from Dong, Kim and Krylov’s papers. Furthermore, we also remarked that Byun et al. have employed their argument to derive \(L^{p}\) estimates to Dirichlet problems of quasilinear principal coefficients \(a_{ij} (x,u)\) (see [9]) and nonlinearities \(\mathbf{a}(x,Du)\) (see [7]) with small partially BMO “coefficients” in x-variables. Very recently, Erhardt [14] obtained a local Calderón–Zygmund estimate for localizable solutions of parabolic obstacle problems with nonstandard growth, and Liang and Zheng [22]showed the \(W^{1,\gamma(\cdot )}\)-regularity for nonlinear nonuniformly elliptic equations with small BMO coefficients.
As a refined version of Lebesgue spaces, Lorentz spaces are a two-parameter scale of spaces [3, 24]. The regularity in Lorentz spaces concerning partial differential equations was originated from Talenti’s work [28] based on symmetrization. Since then, there is a lot of papers to study the Lorentz regularity of various problems of PDEs; see some recent references in [2, 4, 5, 23], and we also refer the reader to Xiao [32], who characterized a nonnegative Radon measure μ on \(\mathbb{R}^{d}\) to produce a continuous map \(I_{\alpha}\) from the Lorentz space \(L^{(p,1)}\) to the Lebesgue space \(L^{p}_{\mu}\). We would like to mention that Baroni [4, 5] showed the Lorentz estimates for evolutionary p-Laplacian systems and obstacle parabolic p-Laplacian, respectively, by using the large-M-inequality principle introduced by Acerbi and Mingione [1]. Meanwhile, Mengesha and Phuc [23] and Zhang and Zhou [33] attained gradient estimates in weighted Lorentz spaces for quasilinear elliptic p-Laplacian and \(p(x)\)-Laplacian equations based on a rather different geometrical approach used in [8], respectively. Tian and Zheng [29, 30] very recently derived a globally weighted Lorentz estimate and a variable Lorentz estimate to linear elliptic problems over Reifenberg flat domains under the assumptions of partially BMO coefficients, respectively. In addition, Zhang and Zheng [34] studied weighted Lorentz estimates of the Hessian of strong solution for nondivergent linear elliptic equations with partially BMO coefficients. We notice that in these papers concerning nonlinear problems mentioned, an important regular assumption on the “nonlinearity coefficients” is VMO or small BMO in all x beyond the settings of linear PDEs.
Motivated by recent progress [7, 9] in particular involved in partially regular coefficients to nonlinear problems, in the present paper, we essentially want to study the Lorentz estimates (1.5) to the variational inequalities (1.1) and relevant nonlinear elliptic equations with controlled growth under the minimal assumption with partially regular nonlinearities \(\mathbf{a}(\xi,x)\). More precisely, we assume that there is no regular requirement on the nonlinearities \(\mathbf{a}(\xi ,x)\) with respect to the variable \(x_{1}\), which implies that the nonlinearities \(\mathbf{a}(\xi,x)\) might have jumps along the \(x_{1}\) variable, whereas the nonlinearities \(\mathbf{a}(\xi,x)\) are controlled in terms of small BMO, such as small multipliers of the Heaviside step function, along the remaining variables. Of course, our consideration is a natural outgrowth of Byun and Kim’s paper [7] concerning the Calderón–Zygmund estimate for nonlinear elliptic problems with measurable nonlinearities. Here we would like to mention that this is a kind of minimal regular requirement on the “coefficients” even for the settings of linear equations in accordance with the famous counterexample by Ural’tseva [31], who constructed an example of an equation in \(\mathbb{R}^{d}\ (d\ge3)\) with the coefficients depending only on the first two coordinates, so that we get that there is no unique solvability in Sobolev spaces \(W^{1,p}\) for any \(p>1\). Its particular interest under consideration is due to a subtle link with application to medium composite materials [20]. Also, these are closely related to some important problems arising in modeling of deformations in composite materials, in the mechanics of membranes and films of simple nonhomogeneous materials that form linear laminated medium [26].
Before stating main results, let us recall some basic concepts and facts. In the context, let us denote a type point by \(x = (x_{1},\ldots, x_{d}) = (x_{1}, x')\in\mathbb{R}^{d}\) with \(x'=(x_{2},\ldots, x_{d}) \). Set
$$B_{r}(x)= \bigl\{ y\in\mathbb{R}^{d}: \vert x-y \vert < r \bigr\} ,\qquad B'_{r} \bigl(x' \bigr)= \bigl\{ y'\in\mathbb {R}^{d-1}: \bigl\vert x'-y' \bigr\vert < r \bigr\} , $$
and a typical cylinder
$$Q_{r}(x)= (x_{1}-r, x_{1}+r)\times B'_{r} \bigl(x' \bigr). $$
For convenience, we sometimes write \(B_{r}=B_{r}(0)\) and \(B'_{r}=B'_{r}(0')\). We denote the average of f on \(Q_{r}\) with \(r>0\) by
$$\begin{aligned} \fint_{Q_{r}} f(x) \,dx = \frac{1}{ \vert Q_{r} \vert } \int_{Q_{r}} f(x) \,dx, \end{aligned}$$
where \(\vert Q_{r} \vert \) is the d-dimensional Lebesgue measure of \(Q_{r}\), and we also denote the \((d-1)\)-dimensional average with respect to \(x'\) by
$$\begin{aligned} \bar{f}_{B'_{r}}(x_{1}) = \fint_{B'_{r}} f \bigl(x_{1},x' \bigr) \,dx' = \frac {1}{ \vert B'_{r} \vert } \int_{B'_{r}} f \bigl(x_{1},x' \bigr) \,dx' \end{aligned}$$
with \(\vert B'_{r} \vert \) as the \((d-1)\)-dimensional Lebesgue measure of \(B'_{r}\). We are now in a position to impose an additional partially regular assumption on the nonlinearities \(\mathbf{a}(\xi, x)\) just like in [7]. For this, we recall the function \(\beta (\mathbf {a}, Q_{ r} )(x)\) on \(Q_{r}\) with \(r>0\) defined by
$$\begin{aligned} \beta (\mathbf{a}, Q_{r} ) (x) = \sup _{\xi\in\mathbb {R}^{d} \backslash\{0\}}\frac{ \vert \mathbf{a}(\xi,x)- \bar{\mathbf {a}}_{B'_{r}}(\xi, x_{1}) \vert }{ \vert \xi \vert }. \end{aligned}$$
Assumption 1.1
We say that (\(\mathbf{a}(\xi,x), \Omega\)) is (\(\delta, R_{0}\))-vanishing of codimension one if for every point \(x_{0}\in \Omega\), there exists a constant \(R_{0}>0\) such that, for any \(0< r \le R_{0}\) with
$$\begin{aligned} \operatorname{dist}(x_{0}, \partial\Omega) = \min_{z\in\partial\Omega} \operatorname{dist}(x_{0}, z) > \sqrt{2} r, \end{aligned}$$
there exists a coordinate system depending only on \(x_{0}\) and r, whose variables are still denoted by x, such that, in the new coordinate system with \(x_{0}\) as the origin,
$$\begin{aligned} \fint_{Q_{r}} \bigl\vert \beta (\mathbf{a}, Q_{r} ) (x) \bigr\vert ^{2} \,dx \le\delta^{2}; \end{aligned}$$
whereas, for \(x_{0}\in\overline{\Omega}\) with
$$\begin{aligned} \operatorname{dist}(x_{0}, \partial\Omega) = \min_{z\in\partial\Omega} \operatorname{dist}(x_{0}, z) = \operatorname{dist} (x_{0}, z_{0}) \le \sqrt{2} r, \end{aligned}$$
where \(z_{0}\in\partial\Omega\), there exists a coordinate system depending on \(x_{0}\) and \(0< r< R_{0}\) such that, in the new coordinate system with \(z_{0}\) as the origin,
$$\begin{aligned} Q_{3r}\cap\{x_{1}\ge 3\delta r\} \subset Q_{3r}\cap\Omega\subset Q_{3r}\cap\{x_{1}\ge-3 \delta r\} \end{aligned}$$
(1.6)
and
$$\begin{aligned} \fint_{Q_{3r}} \bigl\vert \beta(\mathbf{a}, Q_{3r}) (x) \bigr\vert ^{2} \,dx \le \delta^{2}, \end{aligned}$$
(1.7)
where \(a(x,\xi)\) is zero extended from \(Q_{3r}\cap\Omega\) to \(Q_{3r}\), and the parameter \(\delta>0\) will be specified later.
Here we point out that the boundary geometric structure (1.6) implies that Ω is a \((\delta, R)\)-Reifenberg flat domain. It is also obvious that this is an A-type domain with the relations
$$ \underset{0< r\leq R }{\sup}\underset{y\in\Omega}{\sup} \frac { \vert B_{r}(y) \vert }{ \vert B_{r}(y)\cap\Omega \vert } \leq \biggl(\frac{2}{1-\delta} \biggr)^{n} \leq \biggl(\frac{16}{7} \biggr)^{n} $$
(1.8)
for \(0<\delta<\frac{1}{8}\) by a scaling transformation [10].
Considering that our estimates are concerned with the weighted Lorentz spaces, it is necessary to recall some basic definitions involved in weight functions and Lorentz spaces.
Definition 1.2
For \(1< s<\infty\), a nonnegative function \(\omega(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})\) is called a weight in Muckenhoupt class \(A_{s}\), denoted by \(\omega\in A_{s}\), if
$$\begin{aligned}{} [\omega]_{s} = \sup_{B} \biggl( \fint_{B}\omega(x)\,dx \biggr) \biggl( \fint _{B}\omega^{\frac{-1}{s-1}}(x)\,dx \biggr)^{s-1}< \infty, \end{aligned}$$
(1.9)
where the supremum is taken over all balls \(B \subset\mathbb{R}^{d}\), and the constant \([\omega]_{s}\) is referred to be the \(A_{s}\) constant of ω.
For a given measurable set \(E\subset\mathbb{R}^{d}\) and weight ω, we set
$$ \omega(E)= \int_{E}\omega(x)\,dx. $$
Definition 1.3
Let E be an open subset in \(\mathbb{R}^{d}\), and let ω be a weight function. The weighted Lorentz space \(L^{(p,q)}_{\omega}(E)\) with \(p\in[1, +\infty)\) and \(q\in(0, + \infty)\) is the set of measurable functions \(g: E \rightarrow\mathbb{R}^{d}\) such that
$$\begin{aligned} \Vert g \Vert _{L^{(p,q)}_{\omega}(E)}:= \biggl(p \int_{0}^{\infty} \bigl(\gamma^{p}\omega \bigl( \bigl\{ x\in E: \bigl\vert g(x) \bigr\vert >\gamma \bigr\} \bigr) \bigr)^{\frac {q}{p}}\frac{d\gamma}{\gamma} \biggr)^{\frac{1}{q}} < + \infty. \end{aligned}$$
For \(q=\infty\), the space \(L^{(p,\infty)}_{\omega}(E)\) is the classical Marcinkiewicz space with quasinorm
$$\begin{aligned} \Vert g \Vert _{L^{(p,\infty)}_{\omega}(E)}:= \sup_{\gamma>0} \bigl(\gamma ^{p} \omega \bigl( \bigl\{ x\in E: \bigl\vert g(x) \bigr\vert > \gamma \bigr\} \bigr) \bigr)^{\frac{1}{p}} < + \infty. \end{aligned}$$
It is rather clear that if \(p=q\), then the Lorentz space \(L^{(p,p)}_{\omega}(E)\) is nothing but the usual weighted Lebesgue space \(L^{p}_{\omega}(E)\), which is equivalently defined by
$$\begin{aligned} \Vert g \Vert _{L^{p}_{\omega}(E)}= \biggl( \int_{E} \bigl\vert g(x) \bigr\vert ^{p} \omega(x) \,dx \biggr)^{\frac{1}{p}} < + \infty; \end{aligned}$$
more specifically, if \(\omega(x)=1\), then \(\omega(E)= \int_{E}\,dx= \vert E \vert \), which implies
$$\begin{aligned} L^{(p,q)}_{\omega}(E) = L^{(p,q)}(E) \quad\text{and}\quad L^{p}_{\omega}(E) = L^{p}(E). \end{aligned}$$
(1.10)
We are now ready to summarize our main result.
Theorem 1.4
Let
\(\omega\in A_{p/2}\)
be a weight function with
\(2< p<\infty\). For
\(0< q\le\infty\)
and
\(R_{0}>0\), there exists a positive constant
\(\delta= \delta(d,p,q,\lambda,\Lambda,[\omega]_{p/2})\)
such that (\(\mathbf{a}(\xi,x), \Omega\)) satisfies (\(\delta, R_{0}\))-vanishing of codimension one (Assumption 1.1). If
\(\vert \mathbf{f} \vert \)
and
\(\vert D\Psi \vert \in L ^{(p,q)}_{\omega}(\Omega)\), then, for a weak solution
\(u \in\mathcal{A}\)
of variational inequalities (1.1) satisfying (H1), we have
\(\vert Du \vert \in L^{(p,q)}_{\omega}(\Omega)\)
with the estimate
$$\begin{aligned} \Vert D u \Vert _{L^{(p,q)}_{\omega}(\Omega)}\le C \bigl( \Vert \mathbf{f} \Vert _{L ^{(p,q)}_{\omega}(\Omega)} + \Vert D\Psi \Vert _{L ^{(p,q)}_{\omega}(\Omega )} \bigr), \end{aligned}$$
(1.11)
where the constant
C
is independent of
\(u, \mathbf{f}\), and Ψ.
As a consequence of Theorem 1.4, by taking a special weight we also get the following Lorentz–Morrey estimate for the gradient of weak solution to variational inequalities (1.1). Let us recall the so-called Lorentz–Morrey spaces
\(L^{p,q;\theta}(E)\) for \(1< p<\infty\), \(0< q\le\infty\), and \(0<\theta\le d\). We say that \(g(x)\in L^{(p,q)}(E)\) belongs to \(\mathcal{L}^{p,q;\theta}(E)\) if for \(\delta=\operatorname{diam}(E)\), we have
$$\begin{aligned} \Vert g \Vert _{\mathcal{L}^{p,q;\theta}(E)}:= \sup_{z\in E, 0< r< \delta} r^{\frac{\theta-d}{p}} \Vert g \Vert _{L^{(p,q)}(B_{r}(z)\cap E)}< + \infty. \end{aligned}$$
Clearly, \(\mathcal{L}^{p,q;\theta}(E) \subset\mathcal {L}^{(p,q)}(E)\) for all \(\theta\in(0,d]\). For \(p=q\), the space \(\mathcal{L}^{p,q;\theta}(E)\) is the usual Morrey space \(\mathcal {L}^{p;\theta}(E)\); see [15, 16].
Corollary 1.5
For
\(2< p<\infty, 0< q\le\infty\), \(0<\theta\le d\), and
\(R_{0}>0\), there exists a positive constant
\(\delta= \delta(d,p,q,\theta ,\lambda,\Lambda)\)
such that (\(\mathbf{a}(\xi,x), \Omega\)) satisfies (\(\delta, R_{0}\))-vanishing of codimension one (Assumption 1.1). If
\(\vert \mathbf{f} \vert \)
and
\(\vert D\Psi \vert \in \mathcal{L}^{p,q;\theta}(\Omega)\), then, for weak solution
\(u\in \mathcal{A}\)
of variational inequalities (1.1) satisfying (H1), we have
\(\vert Du \vert \in\mathcal{L}^{p,q;\theta}(\Omega)\)
with the estimate
$$\begin{aligned} \Vert Du \Vert _{\mathcal{L}^{p,q;\theta}(\Omega)}\le C \bigl( \Vert \mathbf{f} \Vert _{\mathcal{L}^{p,q;\theta}(\Omega)} + \Vert D\Psi \Vert _{\mathcal {L}^{p,q;\theta}(\Omega)} \bigr), \end{aligned}$$
(1.12)
where the constant
C
is independent of
\(u, \mathbf{f}\), and Ψ.
Finally, as an application of main Theorem 1.4, we further present a global Lorentz estimate to the following Dirichlet problem for nonlinear elliptic equations with controlled growth under very weak assumptions on given datum. Let us consider the Dirichlet problem
$$\begin{aligned} \textstyle\begin{cases} - \operatorname{div} ( \mathbf{a}(Du, x) ) = B(Du, x) & \text{in } \Omega, \\ u=0 & \text{on } \partial\Omega, \end{cases}\displaystyle \end{aligned}$$
(1.13)
where the nonlinearities \(\mathbf{a}(\xi, x)\): \(\mathbb{R}^{d} \times\Omega \rightarrow\mathbb{R}^{d}\) satisfy the hypothesis (1.2), and \(B(\xi, x)\): \(\mathbb{R}^{d} \times\Omega \rightarrow\mathbb{R}\) satisfies the following controlled growth condition:
(H2) (controlled growth) There exist a constant \(\mu >0\) and a nonnegative function
$$\begin{aligned} \psi\in L^{(p, q)}(\Omega) \end{aligned}$$
(1.14)
with \(p \ge\frac{2d}{d+2}\) and \(0 < q \le\infty\) such that
$$\begin{aligned} \bigl\vert B(\xi, x) \bigr\vert \le\mu \bigl( \psi(x) + \vert \xi \vert ^{\frac{d+2}{d}} \bigr) \end{aligned}$$
(1.15)
for a.e. \(x\in\Omega\) and all \(\xi\in \mathbb{R}^{d}\), where inequality (1.15) is usually said to be the controlled growth.
This problem is inspired by the following achievements on this topic. It is well known that nonlinear PDEs with controlled growth were always very important research subjects coming from variational problems [17, 18]. Regarding the setting with discontinuous coefficients, Zheng and Feng [35] showed an optimal Hölder regularity of weak solutions to quasilinear elliptic systems under controlled growth with VMO coefficients. Later, Dong and Kim [12] obtained an \(L^{p}\) estimate for quasilinear elliptic equations under controlled growth with coefficients satisfying VMO in spatial variables. Also, Byun and Palagachev derived a refined Morrey regularity for the gradient of weak solution to a quasilinear elliptic equation with lower-order term of Riccati type under the assumption of partially BMO nonlinearity in x (small BMO in the remainders except an independent variable, say \(x_{1}\)), and very recently they also dealt with the Sobolev–Morrey estimate for general quasilinear equations of p-Laplacian type with BMO nonlinearities in all x under controlled growth. Here, our aim under consideration is to attain a global Lorentz gradient estimate to problem (1.13) over Reifenberg flat domain under a very weak regular Assumption 1.1, based on an elaborate bootstrap argument, which implies that
$$\begin{aligned} \psi \in L^{(p, q)}(\Omega) \quad\Longrightarrow\quad Du\in L^{(p^{*}, q)}(\Omega) \end{aligned}$$
with
$$ p^{*}= \textstyle\begin{cases} \frac{dp}{d-p}\ge2 & \text{if } p < d,\\ \text{any number} > p & \text{if } p \ge d. \end{cases} $$
(1.16)
More precisely, we have the following:
Theorem 1.6
Let
\(u\in W^{1,2}_{0}(\Omega)\)
be a weak solution to Dirichlet problem (1.13) with nonlinear structural conditions (H1) and (H2) and
\(R_{0}>0\). There exists a positive constant
\(\delta= \delta(d, p, q, \lambda, \Lambda, \mu)\)
such that if (\(\mathbf{a}(\xi, x), \Omega\)) satisfies (\(\delta, R_{0}\))-vanishing of codimension one as in Assumption 1.1, then we have that the gradient
Du
belongs to an appropriate Lorentz space:
$$\begin{aligned} Du \in L^{(p^{*}, q)}(\Omega) \end{aligned}$$
for any
\(p \ge\frac{2d}{d+2}\)
and
\(0< q\le\infty\).
Here, we would like to point out that there is no usual restriction with \(p>\frac{d}{2}\) to Dirichlet problem (1.13) since we do not employ the boundedness of weak solution of (1.13). In fact, this makes the weak solution of (1.13) possibly unbounded since it is invalid for the De Giorgi–Moser–Nash iterating argument in \(p\le\frac{d}{2}\). Finally, we complete the proof by enhancing the index of gradient integrability of weak solution to the linearized problem in accordance with a successive application of the bootstrapping argument. Also, we would remark that in the particular case \(p=q\), Theorem 1.6 is just a classical Calderón–Zygmund property of (1.13) also in the framework of Lebesgue scales as in [12].
The remainder of this paper is organized as follows. We denote by \(C(d, \mu, \Lambda, \ldots)\) and \(N_{i}(d, \mu, \Lambda,\partial\Omega , \ldots)\) for \(i=1,2, \ldots \) universal constants depending only on prescribed quantities and possibly varying from line to line. We recall some usual auxiliary results in the next section. In Sect. 3, we show local interior and boundary estimates of the gradient to weak solution of the reference problem to variational inequality (1.1). We prove main Theorem 1.4 and Corollary 1.5 in Sect. 4. Finally, we give a proof of Theorem 1.6 regarding the Dirichlet problem (1.13) with controlled growth in Sect. 5.