1 Introduction
The aim of this paper is to consider the following quasilinear elliptic systems with VMO discontinuous coefficients under natural growth: with Ω a bounded domain of \(\mathbb{R}^{n} \) for \(n\ge2\). Here, we assume that \(B(x,u,Du)=(B_{i}(x,u,Du))\) takes a value in \(\mathbb{R}^{N} \) with \(N\ge1\) and \(A(x,u)=(A_{ij}^{\alpha\beta }(x,u))\) takes a value in \(\mathbb{R}^{n^{2}N^{2}}\). In the context, we adopt Einstein’s convention by summing over repeated indices with \(\alpha,\beta=1,2,\ldots,n\) and \(i,j=1,2,\ldots,N\), then one can briefly rewrite (1.1) by Consequently, we rewrite a vectorial-valued function \(u\in W_{\mathit{loc}}^{1,2}(\Omega,\mathbb{R}^{N})\) to be a weak solution of the systems (1.1) in the sense of distribution as follows. As we know, it is very necessary that some structural and regular assumptions are imposed on the tensorial-valued operator \(A(x,u)\) and the inhomogeneity \(B(x,u,Du)\). Here, we suppose that the uniformly elliptic operator \(A(x,u)\) satisfies a VMO-condition in a.e. \(x\in \Omega\) uniformly with respect to \(u\in\mathbb{R}^{N}\) and is continuous in u uniformly with respect to \(x\in\Omega\), and \(B(x,u,D u)\) satisfies the natural growth. More precisely, let us first recall some related notations.
$$ -\sum_{\alpha,\beta=1}^{n}\sum _{j=1}^{N}D_{\alpha}\bigl(A_{ij}^{\alpha\beta }(x,u)D_{\beta}u^{j}\bigr)=B_{i}(x,u,Du),\quad\text{a.e. } x\in\Omega, i=1,\ldots,N, $$
(1.1)
$$ -\operatorname {div}\bigl(A(x,u)Du\bigr)=B(x,u,Du),\quad \text{a.e. } x\in\Omega. $$
(1.2)
$$ \int_{\Omega}A(x,u)Du\cdot D\varphi\, dx= \int_{\Omega}B(x,u,Du)\varphi \,dx,\quad\forall\varphi\in C_{0}^{\infty}\bigl(\Omega,\mathbb{R}^{N}\bigr). $$
(1.3)
Definition 1.1
A locally integrable function
u
is said to belong to
\(\operatorname {BMO}(\Omega)\) (bounded mean oscillation in Ω), if
\(u \in L^{1}_{\mathit{loc}}(\Omega)\)
and for any
\(0< s<\infty\), we have
where
\(\Omega(x, \rho)=\Omega\cap B(x, \rho)\)
with any open ball
\(B(x, \rho)\)
in
\({\mathbb {R}}^{n}\)
centered at
x
of radius
ρ, and
.
$$M_{s}(u, \Omega)=\sup_{x\in \Omega, 0< \rho< s}\bigl\vert \Omega(x, \rho)\bigr\vert ^{-1} \int_{\Omega(x, \rho)}\bigl\vert u(y)-u_{x, \rho}\bigr\vert \,dy< + \infty, $$
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Definition 1.2
A function
\(u\in L^{1}_{\mathit{loc}}(\Omega)\)
is said to be in
\(\operatorname {VMO}(\Omega)\) (vanishing mean oscillation in Ω), if
$$M_{0}(u)=\lim_{s\rightarrow0}M_{s}(u, \Omega)=0. $$
Definition 1.3
Let
\(p\ge1\)
and
\(\lambda\ge0 \). The Morrey space is defined by
with its norm
where
\(R_{0}=\operatorname {diam}(\Omega)\).
$$ L^{p,\lambda}{(\Omega)}= \biggl\{ u\in L^{p}(\Omega):\mathop{\sup _{x\in\Omega}}_{ 0< \rho\leq{R_{0}} } \biggl(\rho^{-\lambda} \int_{\Omega(x,\rho)}\vert u\vert ^{p}\,dy \biggr)^{\frac{1}{p}}< +\infty \biggr\} $$
$$\Vert u\Vert _{L^{p,\lambda}{(\Omega)}}=\mathop{\sup_{x\in\Omega}}_{ 0< \rho\leq{R_{0}} } \biggl(\rho^{-\lambda} \int_{\Omega(x,\rho)}\vert u\vert ^{p}\,dy \biggr)^{\frac{1}{p}}, $$
We are now in a position to state the assumptions which are imposed on \(A(x,u)\) and \(B(x,u,Du)\) as follows:
-
H1 (Uniform ellipticity) There exist two constants \(0<\lambda\le\Lambda\) such that$$ \lambda \vert \xi \vert ^{2}\le A^{\alpha\beta}_{ij}(x,u) \xi_{\alpha}^{i}\xi _{\beta}^{j}\le\Lambda \vert \xi \vert ^{2} , \quad\forall x\in\Omega, u\in{\mathbb {R}}^{N}, \xi\in{\mathbb {R}}^{nN}. $$(1.4)
-
H2 (Minimal regularity on \(A(x,u)\)) Suppose that \(A(\cdot ,u)\) is VMO in x with uniformly respect to \(u\in\mathbb{R}^{N}\) and is continuous in u with uniformly respect to \(x\in\Omega\). That is, \(\lim_{s\rightarrow{0}}M_{s}(A(\cdot,u_{0}))=0\), and there exist a constant C and a continuous concave function1 \(\omega:\mathbb {R}^{+}\rightarrow\mathbb {R}^{+}\) with \(\omega(0)=0\), \(0\leq\omega\leq 1\), such that$$ \bigl\vert A^{\alpha\beta}_{ij}(x,u)-A^{\alpha\beta}_{ij}(x,v) \bigr\vert \leq C\omega\bigl(\vert u-v\vert ^{2}\bigr), \quad \forall u, v \in {\mathbb {R}}^{N}, x\in\Omega. $$(1.5)
-
H3 (Natural growth) The lower order term \(B(x,u,Du)\) satisfies the following natural growth: for \(u\in W^{1,2}\cap L^{\infty}(\Omega,\mathbb{R}^{N})\) with \(M=\Vert u\Vert _{L^{\infty}(\Omega)}\), we havewith$$ \bigl\vert B_{i}(x, u, D u)\bigr\vert \le\mu(M) \bigl( \vert D u\vert ^{2}+f^{i}(x) \bigr),\quad i=1, 2, \ldots, N, $$(1.6)$$2\mu(M)M< \lambda,\quad \text{and} \quad f^{i}\in L^{q}(\Omega) \quad \text{with } q>\frac{n}{2}. $$
Before stating the main conclusion, let us first briefly review some recent studies involving related problems. Note that the discontinuous coefficient is not so crucial for Hölder continuity of the weak solutions to the scalar partial differential equations, which is due to the famous De Giorgi-Moser-Nash iterative technique; see [1]. However, in the case of \(N>1\) one cannot in general expect that the weak solutions will be classical \(C^{\alpha}\) solutions with some \(0<\alpha\le1\), which was first shown by a counterexample from De Giorgi’s work [2], also see Giaquinta’s monograph [3]. For the systems (1.1), Giaquinta and Modica [3, 4] first studied partial regularity of their weak solutions in the Morrey space and in the Campanato space [3, 5] when each entry of the leading coefficients \(A(x,u)\) is assumed to be continuous.
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However, it is an important observation that many stochastic processes with discontinuous coefficients reappeared in connected with the diffusion approximation [6], which reminds us of the significance to treat particular cases of discontinuity. Sarason [7] in 1975 introduced the function spaces of the vanishing mean oscillations (briefly called VMO), which not only contains discontinuous functions but also owns a good property similar to the class of continuous functions that is not shared by general bounded measurable functions. In recent years, Calderón-Zygmund’s theory of linear and nonlinear PDEs with VMO coefficients was immensely developed, which naturally originated from the singular integral operators and the estimates of commutators with a VMO function [8, 9]. In the meantime, Morrey’s regularity of the weak solutions of elliptic and parabolic PDEs with the discontinuous leading coefficients was also investigated in a similar approach by Di Fazio and Ragusa [10] and Fan et al. [11]. Among them there are some main different arguments to deal with elliptic and parabolic PDEs with VMO leading coefficients, for example a few celebrated approaches of Chiarenza et al. [8], Syun and Wang [12] and Krylov et al. [13, 14].
For the elliptic systems (1.1), Zheng [15] and Zheng and Feng [16] derived an optimal partial regularity in Morrey spaces by using a reverse Hölder inequality and perturbation argument when \(A(x,u)\) is VMO in \(x\in\Omega\) and continuous in \(u\in\mathbb {R}^{N}\) with controllable growth and natural growth, respectively. On the other hand, Chen and Tan [17] got an interior partial regularity for the nonlinear elliptic systems with controllable growth by the A-harmonic approximation, but their principal coefficients \(A(x,u,Du)\) are essentially Hölder continuous in \((x,u)\). Inspired by those achievements, in this paper we will provide an alternative approach to get Hölder continuity with an optimal Hölder exponent for the quasilinear elliptic systems (1.1) with VMO coefficients under the natural growth (cf. [15]). Here, we simply employ a modification of the A-harmonic approximation argument based on Duzaar and Grotowski’s technique, which can avoid the use of the reverse Hölder inequality and perturbation approach. As pointed out by Duzaar and Grotowski [18], the reformulation of the A-harmonic approximation could simplify the proofs of the regularity for PDEs, we here apply the A-harmonic approximation approach to prove the regularity of our previous problem from [15]. More precisely, we have
Theorem 1.4
Suppose that
\(u\in W_{\mathit{loc}}^{1,2}\cap L^{\infty}(\Omega,\mathbb{R}^{N})\)
is a locally weak solution of the systems (1.1), and
\(A(x,u)\), \(B(x,u,Du)\)
satisfy the basic assumptions (H1)-(H3). Then there exists an open subset
\(\Omega_{0}\subset\Omega\)
with
\(\dim_{H}(\Omega\setminus\Omega_{0})\le n-2\)
such that
\(u\in C^{0,\alpha}_{\mathit{loc}}(\Omega_{0},\mathbb{R}^{N}) \)
with
\(\alpha=1-\frac{n}{2q} \), where
and
\(\dim_{H}\)
is the Hausdorff dimension. Moreover, we have
\(Du\in L^{2,\gamma}(\Omega_{0},\mathbb{R}^{nN})\)
with
\(\gamma=n-\frac{n}{q}\).
We close the introduction with briefly describing the strategy of the proof. At first, we establish the so-called second Caccippoli’s inequality, then we give an estimate to a certain energy functional which measures the oscillations of the solution u except a small excess quantity by the modified A-harmonic approximation. On the basis of the iteration lemma we get the boundedness of the functional, which leads to the desired regularity in points while the excess quantity is small. We believe that this new approach should also be particularly helpful to understand some elliptic and parabolic systems including degenerate settings.
2 Preliminaries
Throughout the paper we adopt the usual convention of denoting by C a general constant, which may vary from expression to expression in the same chain of inequalities. Let us first establish the so-called second Caccioppoli’s inequality. As we know, Caccioppoli’s inequality is always a standard preliminary tool to obtain the partial regularity to elliptic and parabolic PDEs; see [3].
Lemma 2.1
Let
\(u\in W_{\mathit{loc}}^{1,2}\cap L^{\infty}(\Omega,\mathbb{R}^{N})\)
be any weak solution of the systems (1.1) under the assumption (H1)-(H3) satisfying
\(\Vert u\Vert _{L^{\infty}(\Omega)}= M< \infty\). If
\(2\mu(M)M<\lambda\), then for any
\(B_{\rho}(x_{0})\subset \Omega\), we have
where
is an integral average over
\(B_{\rho}(x_{0})\).
$$ \int_{B_{\frac{\rho}{2}}(x_{0})}\vert Du\vert ^{2}\,dx\leq \frac{C_{1}}{\rho^{2}} \int _{B_{\rho}(x_{0})}\vert u-u_{x_{0},\rho} \vert ^{2}\,dx +C_{2}\rho^{n+2-\frac{2n}{q}}\Vert f\Vert ^{2}_{L^{q}}, $$
(2.1)
Proof
For any \(x_{0}\in\Omega\) and \(0<\rho<\operatorname {dist}(x_{0},\partial\Omega)\), denoting \(B_{\rho}:=B_{\rho}(x_{0})\), we take \(\eta\in C^{\infty}_{0}(B_{\rho}(x_{0}))\) as a cut-off function with \(0\le\eta\le1\), \(\vert D\eta \vert \le \frac {4}{\rho}\) and \(\eta\equiv1\) on \(B_{\frac{\rho}{2}}(x_{0})\). As usual, we choose the test function \(\varphi=\eta^{2}(u-u_{x_{0},\rho})\). Note that \(D\varphi=2\eta D\eta(u-u_{x_{0},\rho})+\eta^{2}Du\), and we substitute it in (1.3) finding which implies By the ellipticity (H1) and the natural growth (H3) we get where the last term of the right-hand side is obtained by the Hölder inequality. Thanks to the Hölder inequality, the Young inequality with \(\varepsilon>0\), and the boundedness of u it follows that that is,
$$\begin{aligned}& \int_{B_{\rho}}A(x,u)Du\cdot\bigl[2\eta D\eta(u-u_{x_{0},\rho})+ \eta ^{2}Du\bigr]\,dx \\& \quad = \int_{B_{\rho}}B(x,u,Du)\eta^{2}(u-u_{x_{0},\rho})\,dx, \end{aligned}$$
$$\begin{aligned}& \int_{B_{\rho}}\eta^{2} A(x,u)Du\cdot Du \,dx \\& \quad =-2 \int_{B_{\rho }}A(x,u)Du\cdot\bigl(\eta(u-u_{x_{0},\rho})D\eta \bigr)+ \int_{B_{\rho }}B(x,u,Du)\eta^{2}(u-u_{x_{0},\rho})\,dx. \end{aligned}$$
$$\begin{aligned}& \lambda \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx \\& \quad \le2 \Lambda \int_{B_{\rho }}\vert \eta Du\vert \cdot\bigl\vert (u-u_{x_{0},\rho})D\eta\bigr\vert \,dx+ \mu(M) \int_{B_{\rho}}\eta^{2} \bigl(\vert Du\vert ^{2}+\bigl\vert f(x)\bigr\vert \bigr) \vert u-u_{x_{0},\rho} \vert \,dx \\& \quad \le \mu(M) \int_{B_{\rho}} \vert u-u_{x_{0},\rho} \vert \cdot \vert \eta Du \vert ^{2} \,dx+\mu (M) \int_{B_{\rho}} \vert u-u_{x_{0},\rho} \vert \cdot\bigl\vert f(x)\bigr\vert \,dx \\& \quad\quad {}+2\Lambda \biggl( \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx \biggr)^{\frac {1}{2}} \biggl( \int_{B_{\rho}} \vert D\eta \vert ^{2}\vert u-u_{x_{0},\rho} \vert ^{2} \,dx \biggr)^{\frac{1}{2}}, \end{aligned}$$
(2.2)
$$\begin{aligned} \lambda \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx \le& 2 \mu(M)M \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx \\ &{}+\mu(M) \biggl( \int _{B_{\rho}}\biggl\vert \frac{u-u_{x_{0},\rho}}{\rho}\biggr\vert ^{2} \,dx \biggr)^{\frac{1}{2}} \biggl( \int_{B_{\rho}}\bigl\vert \rho f(x)\bigr\vert ^{2} \,dx \biggr)^{\frac {1}{2}} \\ &{}+2\Lambda\varepsilon \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx+C( \varepsilon )\cdot\frac{1}{\rho^{2}} \int_{B_{\rho}} \vert u-u_{x_{0},\rho} \vert ^{2} \,dx , \end{aligned}$$
$$\begin{aligned} \lambda \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx \le& \bigl(2\mu(M)M+2\Lambda\varepsilon\bigr) \int_{B_{\rho}} \vert \eta Du\vert ^{2} \,dx+ \frac{C}{\rho^{2}} \int_{B_{\rho}} \vert u-u_{x_{0},\rho} \vert ^{2} \,dx \\ &{}+C\rho^{2} \int _{B_{\rho}}\bigl\vert f(x)\bigr\vert ^{2} \,dx . \end{aligned}$$
If we choose ε small enough such that \(2\mu(M)M+2\Lambda \varepsilon<\lambda\), it follows from the Hölder inequality that which yields the desired result. □
$$\int_{B_{\frac{\rho}{2}}}\vert Du\vert ^{2} \,dx\le \frac{C_{1}}{\rho^{2}} \int _{B_{\rho}(x_{0})}\vert u-u_{x_{0},\rho} \vert ^{2}\,dx +C_{2}\rho^{n+2-\frac{2n}{q}}\Vert f\Vert ^{2}_{L^{q}}, $$
We will make use of a modification of the so-called A-harmonic approximation lemma [19, 20]. Let us first recall the so-called locally A-harmonic.
Definition 2.2
(A-harmonic)
Let
\(A\in \operatorname {Bil}({B_{R}(x_{0})}\times\mathbb{R}^{N},\mathbb{R}^{n^{2}\times N^{2}})\)
be a bilinear form with constant coefficients, which satisfies the assumptions of (1.4). We call a map
\(h\in W^{1,2}({B_{R}(x_{0})},\mathbb{R}^{N})\)
A-harmonic in
\({B_{R}(x_{0})}\)
if it satisfies
$$\int_{{B_{R}(x_{0})}}A(Dh,D\varphi)\,dx=0,\quad \forall \varphi\in C^{1}_{0}\bigl({B_{R}(x_{0})}, \mathbb{R}^{N}\bigr). $$
We should emphasize that \(A\in \operatorname {Bil}({B_{R}(x_{0})}\times\mathbb {R}^{N},\mathbb{R}^{n^{2}\times N^{2}})\) is a bilinear form with constant tensorial coefficients, therefore it is well known that for any A-harmonic h we have the following inequality, which is similar to harmonic functions (cf. [1, 3]).
Lemma 2.3
Let
\(h(x)\in W^{1,2}({B_{R}(x_{0})},\mathbb{R}^{N})\)
be a weak solution of the following system:
where
\(A_{ij}^{\alpha\beta}\)
is a constant tensor. Then there exists a constant
\(C=C(n,\lambda,\Lambda)\)
such that, for any
\(0<\rho<R\le \operatorname {dist}(x_{0}, \partial\Omega)\)
with
\(x_{0}\in\Omega\), we have
$$D_{\alpha}\bigl(A_{ij}^{\alpha\beta}D_{\beta}h_{j}\bigr)=0,\quad i=1,\ldots, N, $$
$$ \int_{B_{\rho}(x_{0})}\bigl\vert Dh-(Dh)_{x_{0},\rho}\bigr\vert ^{2}\,dx\leq C \biggl(\frac{\rho }{R} \biggr)^{n+2} \int_{B_{R}(x_{0})}\bigl\vert Dh-(Dh)_{x_{0},R}\bigr\vert ^{2}\,dx. $$
(2.3)
Next, we recall the notation of the A-harmonic approximation introduced by Duzaar and Grotowski [18, 21]. In the sequel, suppose that there exist two constants \(0<\lambda\le\Lambda<\infty\) such that the bilinear form \(A\in \operatorname {Bil}({B_{R}(x_{0})}\times\mathbb{R}^{N},\mathbb {R}^{n^{2}\times N^{2}})\) satisfies
In the main proof, as a main approach we will employ the following modification of the A-harmonic approximation [19, 20].
$$ A^{\alpha\beta}_{ij}(x,u)\xi_{\alpha}^{i} \xi_{\beta}^{j}\geq\lambda \vert \xi \vert ^{2},\quad \text{for all }\xi\in\mathbb{R}^{nN}, $$
(2.4)
$$ A^{\alpha\beta}_{ij}(x,u)\xi_{\alpha}^{i} \bar{\xi}_{\beta}^{j}\leq \Lambda \vert \xi \vert \vert \bar{ \xi} \vert ,\quad \text{for all }\xi,\bar{\xi} \in\mathbb{R}^{nN}. $$
(2.5)
Lemma 2.4
Let
\(0<\lambda\le\Lambda <\infty\)
and
\(n\ge2\). Then, for any given
\(\varepsilon>0\)
there exists a positive constant
\(k=k(n,N,\lambda,\Lambda,\varepsilon)>0\)
with the following property: for any
\(A\in \operatorname {Bil}(B_{R}(x_{0})\times\mathbb{R}^{N},\mathbb{R}^{n^{2}\times N^{2}})\)
satisfying (2.4), (2.5) and any
\(u\in W^{1,2}(B_{R}(x_{0}),\mathbb{R}^{N})\), there exists an A-harmonic function
\(h\in W^{1,2}(B_{R}(x_{0}),\mathbb{R}^{N})\)
such that
moreover, there exists
\(\varphi\in C_{0}^{\infty}(B_{R}(x_{0}),\mathbb {R}^{N})\)
with
such that
$$ \int_{B_{R}(x_{0})}\vert Dh\vert ^{2}\,dx\leq \int_{B_{R}(x_{0})}\vert Du\vert ^{2}\,dx; $$
(2.6)
$$ \Vert D\varphi \Vert _{L^{\infty}(B_{R}(x_{0}),\mathbb{R}^{N})}\leq\frac{1}{R}, $$
(2.7)
$$ \int_{B_{R}(x_{0})}\vert u-h\vert ^{2}\,dx\leq\varepsilon R^{2} \int _{B_{R}(x_{0})}\vert Du\vert ^{2}\,dx+k(\varepsilon) \biggl[ R^{4-n} \biggl( \int_{B_{R}(x_{0})}ADu\cdot D\varphi \,dx \biggr)^{2} \biggr]. $$
(2.8)
To give a Hölder estimate to the solution by using Morrey’s lemma we employ the following iterating lemma; see [3].
Lemma 2.5
Let
\(\Phi(\rho)\)
be a non-negative and non-decreasing function on
\((0,R)\). Suppose that
with non-negative constants
A, B, α
and
β, and
\(\alpha >\beta\). Then there exist two constants
\(\epsilon_{0}=\epsilon_{0}(A,\alpha,\beta)\)
and
\(C=C(A,\alpha,\beta)\)
such that for any
\(0<\epsilon<\epsilon_{0}\)
we have
for any
\(0<\rho\leq R\leq R_{0}=\operatorname {dist}(x_{0},\partial\Omega)\).
$$\Phi(\rho)\le A \biggl[ \biggl(\frac {\rho}{R} \biggr)^{\alpha}+ \epsilon \biggr]\Phi(R)+BR^{\beta},\quad \forall 0< \rho< R< R_{0}= \operatorname {dist}(x_{0},\partial\Omega), $$
$$\Phi(\rho)\le C \biggl[ \biggl(\frac{\rho}{R} \biggr)^{\beta}\Phi (R)+B\rho^{\beta} \biggr], $$
In the process of proving our main results, we need to describe the decay of integral on the ball with respect to the radius σ. Let us recall the concept of Campanato space; see [22].
Definition 2.6
Let
\(p\ge1\)
and
\(\lambda\ge0 \). The Campanato space is defined as
with its norm
where
\(u_{x,\rho}\)
is the average of
\(u(x) \)
over
\(B_{\rho}(x)\).
$$ {\mathcal {L}}^{p,\lambda}{(\Omega)}= \biggl\{ u\in L^{p}( \Omega),[u]^{p}_{p,\lambda;\Omega}= \mathop{\sup_{x\in\Omega}}_{ 0< \rho\leq{R_{0}} } \rho^{-\lambda} \int_{\Omega(x,\rho)}\vert u-u_{x,\rho} \vert ^{p}\,dy< + \infty \biggr\} $$
$$\Vert {u}\Vert _{{\mathcal {L}}^{p,\lambda}{(\Omega)}} =\Vert {u}\Vert _{L^{p}{(\Omega )}}+[u]_{p,\lambda;\Omega}, $$
Lemma 2.7
(Theorem 3.1 in [22])
For
\(n<\lambda\leq n+2\), we have
with
\(\alpha=\frac{\lambda-n}{2}\).
$${\mathcal {L}}^{ 2,\lambda}_{\mathit{loc}}(\Omega)\cong C^{0,\alpha}_{\mathit{loc}}( \Omega) $$
Finally, we need the following estimate lemma of the Hausdorff dimensional measure while estimating the singular set to the weak solution.
Lemma 2.8
Let Ω be an open subset of
\({\mathbb {R}}^{n}\)
and
\(u\in L^{1}_{\mathit{loc}}(\Omega, {\mathbb {R}}^{N})\). Then for
\(0\le s< n\)
and setting
we have the following estimate:
see page 77 in Chapter 2 of [23].
$$ E_{s}:= \biggl\{ x\in\Omega:\lim\inf_{\rho\rightarrow 0} \rho^{-s} \int_{B_{\rho}(x) }\vert u\vert \,dy>0 \biggr\} , $$
(2.9)
$$H^{s}(E_{s})=0, $$
3 Proof of main result
In the section, we prove our main result on the basis of the modification of the A-harmonic approximation argument.
Proof of Theorem 1.4
Given \(x_{0}\in\Omega\) and \(R:=\operatorname {dist}(x_{0},\partial\Omega)\), for fixed \(\rho: 0<\rho<R\) let \(\bar{A}=(A(\cdot,u_{x_{0},\rho}))_{x_{0},\frac {\rho}{2}}\) be defined by Note that u is a weak solution of the systems (1.1) with test function \(\varphi\in C_{0}^{\infty}(B_{\frac{\rho}{2}},\mathbb{R}^{N})\), Now we take the test function φ with \(\sup_{B_{\frac{\rho }{2}}}\vert D\varphi \vert \le\frac{2}{\rho}\) and \(\sup_{B_{\frac{\rho }{2}}}\vert \varphi \vert \le2\). Based on natural growth (H3) and Hölder inequality, then we obtain In the following, we estimate the items I, II, and III, respectively. For the estimate of I, thanks to the VMO regular assumption (H2) on \(A(x,u)\) and the Caccioppoli inequality of Lemma 2.1, if we take \(m=u_{\rho}\), then one finds
For simplicity, we denote \(B_{\rho}(x_{0})=B_{\rho}\), \(u_{x_{0},\rho}=u_{\rho}\) in the following. For any \(\varphi\in C_{0}^{\infty}(B_{\frac{\rho }{2}},\mathbb{R}^{N})\), we may write
$$\begin{aligned} \int_{B_{\frac{\rho}{2}}}\bar{A}Du\cdot D\varphi \,dx =& \int _{B_{\frac{\rho}{2}}}\bigl(\bar{A}-A(\cdot,u_{\rho})\bigr)Du\cdot D\varphi \,dx + \int_{B_{\frac{\rho}{2}}}\bigl(A(\cdot,u_{\rho})-A(x,u)\bigr)Du\cdot D \varphi \,dx \\ &{}+ \int_{B_{\frac{\rho}{2}}}A(x,u)Du\cdot D\varphi \,dx. \end{aligned}$$
(3.1)
$$\begin{aligned}& \biggl\vert \int_{B_{\frac{\rho}{2}}}\bar{A}Du\cdot D\varphi \,dx\biggr\vert \\& \quad \le \int_{B_{\frac{\rho}{2}}}\bigl\vert \bar{A}-A(\cdot,u_{\rho})\bigr\vert \cdot \vert Du\vert \cdot \vert D\varphi \vert \,dx \\& \quad\quad{} + \int_{B_{\frac{\rho}{2}}}\bigl\vert A(\cdot,u_{\rho})-A(x,u)\bigr\vert \cdot \vert Du\vert \cdot \vert D\varphi \vert \,dx + \int_{B_{\frac{\rho}{2}}}\bigl\vert B(x,u,Du)\bigr\vert \cdot \vert \varphi \vert \,dx. \end{aligned}$$
$$\begin{aligned}& \biggl\vert \int_{B_{\frac{\rho}{2}}}\bar{A}Du\cdot D\varphi \,dx\biggr\vert \\& \quad \le \frac{2}{\rho} \int_{B_{\frac{\rho}{2}}} \bigl\vert \bar{A}-A(\cdot ,u_{\rho})\bigr\vert \cdot \vert Du\vert \,dx \\& \quad\quad{} +\frac{2}{\rho} \int_{B_{\frac{\rho}{2}}}\bigl\vert A(\cdot,u_{\rho })-A(x,u)\bigr\vert \cdot \vert Du\vert \,dx +2\mu(M) \int_{B_{\frac{\rho}{2}}}\vert Du\vert ^{2}+\vert f\vert \,dx \\& \quad \le\frac{2}{\rho} \biggl( \int_{B_{\frac{\rho}{2}}}\bigl\vert \bar {A}-A(\cdot,u_{\rho})\bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int_{B_{\frac {\rho}{2}}}\vert Du\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\& \quad\quad{} +\frac{2}{\rho} \biggl( \int_{B_{\frac{\rho}{2}}}\bigl\vert A(\cdot,u_{\rho })-A(x,u)\bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int_{B_{\frac{\rho}{2}}}\vert Du\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\& \quad\quad{} +2\mu(M) \int_{B_{\frac{\rho}{2}}}\vert Du\vert ^{2}\,dx+2\mu(M)\rho ^{n(1-\frac{1}{q})}\Vert f\Vert _{L^{q}} \\& \quad := I+\mathit{II}+\mathit{III}. \end{aligned}$$
(3.2)
$$\begin{aligned} I =& \frac{2}{\rho} \biggl( \int_{B_{\frac{\rho}{2}}}\bigl\vert \bar {A}-A(\cdot,u_{\rho})\bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int_{B_{\frac {\rho}{2}}}\vert Du\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\ \le&\frac{C}{\rho} \bigl(\rho^{n}M_{\rho}\bigl(A( \cdot,u_{\rho})\bigr) \bigr)^{\frac{1}{2}} \biggl(\frac{C_{1}}{\rho^{2}} \int_{B_{\rho}} \vert u-u_{\rho} \vert ^{2}\,dx +C_{2}\rho^{n+2-\frac{2n}{q}}\Vert f\Vert ^{2}_{L^{q}} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.3)
To estimate II, by using the continuity hypothesis (H2) in u and the Caccioppoli inequality in Lemma 2.1, where we use the concavity of the function \(\omega(\cdot)\) in the last inequality.
(3.4)
For the estimate of III, by the Caccioppoli inequality as above we deduce Now we are in a position to substitute (3.3), (3.4), and (3.5) into (3.2), then it follows that Note that \(2-\frac{n}{q}>0\) due to the assumptions \(q>\frac{n}{2}\), then we can take \(\rho>0\) such that \(\rho^{2-\frac{n}{q}}\Vert f\Vert _{L^{q}}\le1\), and we rewrite (3.6): where .
$$\begin{aligned} \mathit{III} =& 2\mu(M) \int_{B_{\frac{\rho}{2}}}\vert Du\vert ^{2}\,dx+2\mu(M)\Vert f \Vert _{L^{q}}\bigl(\alpha_{n}\rho^{n} \bigr)^{1-\frac{1}{q}} \\ \le& \frac{C}{\rho^{2}} \int_{B_{\rho}} \vert u-u_{\rho} \vert ^{2}\,dx +C_{2}\rho^{n+2-\frac{2n}{q}}\Vert f\Vert ^{2}_{L^{q}}+C_{2} \rho^{n(1-\frac {1}{q})}\Vert f\Vert _{L^{q}}. \end{aligned}$$
(3.5)
(3.6)
(3.7)
According to Lemma 2.4 on the modification of the A-harmonic approximation, Caccioppoli’s inequality (2.1) and the estimate (3.7) above, for a given \(\varepsilon >0\) there exists an A-harmonic \(h\in W^{1,2}(B_{\frac{\rho }{2}},\mathbb{R}^{N})\) such that In order to estimate the integral \(\int_{B_{\sigma }(x_{0})}\vert u-u_{x_{0},\sigma} \vert ^{2} \,dx\) for any \(0<\sigma<\frac{\rho}{2}\), by the A-harmonic property in Lemma 2.3 we have To estimate the last item in the right-hand side above, in terms of the Poincaré inequality, the inequality (2.6), and Caccioppoli’s inequality it follows that where we use the inequality \(\rho^{2-\frac{n}{q}}\Vert f\Vert _{L^{q}}\le1\) and in the last step.
(3.8)
$$\begin{aligned} \int_{B_{\sigma}} \vert u-u_{\sigma} \vert ^{2} \,dx \le& \int_{B_{\sigma }}\vert u-h_{\sigma} \vert ^{2} \,dx \le 2 \int_{B_{\sigma}} \vert u-h\vert ^{2} \,dx+2 \int_{B_{\sigma}} \vert h-h_{\sigma }\vert ^{2} \,dx \\ \le&2 \int_{B_{\frac{\rho}{2}}}\vert u-h\vert ^{2} \,dx+C \biggl( \frac{\sigma }{\rho} \biggr)^{n+2} \int_{B_{\frac{\rho}{2}}}\vert h-h_{\frac{\rho }{2}}\vert ^{2} \,dx. \end{aligned}$$
(3.9)
(3.10)
Now let us put the estimates (3.8) and (3.10) into the inequality (3.9), Note that if we assume on any \(x\in\Omega_{0}\) with some \(\Omega _{0}\subset\Omega\) as \(\rho\rightarrow0\), we have \(\varepsilon+M_{\rho}(A(\cdot,u_{\rho}))+\omega(J)+J<\epsilon_{0}\) with \(\epsilon_{0}\) as the same in Lemma 2.5 with a small \(\rho>0\), which is due to \(M_{\rho}(A(\cdot,u_{\rho}))\rightarrow0\) by the VMO property of \(A(x,u)\) in \(x\in\Omega\) and the continuous modulus of \(\omega(J)\). Therefore, by the iteration Lemma 2.5 we have which implies In virtue of Lemma 2.7 on the equivalence between Campanato’s spaces and Hölder continuous spaces, \(u\in C^{0,\alpha }_{\mathit{loc}}(\Omega_{0},\mathbb{R}^{N})\) with \(\alpha=1-\frac{n}{2q}\) due to \(n< n+2-\frac{n}{q}< n+2\).
$$\begin{aligned}& \int_{B_{\sigma}(x_{0})}\vert u-u_{x_{0},\sigma} \vert ^{2}\,dx \\& \quad \le C \biggl( \biggl(\frac{\sigma}{\rho} \biggr)^{n+2}+\varepsilon +M_{\rho}\bigl(A(\cdot,u_{\rho})\bigr)+\omega(J)+J \biggr) \\& \quad\quad{}\times \biggl( \int_{B_{\rho}(x_{0})}\vert u-u_{\rho} \vert ^{2}+ \rho^{n+2-\frac{n}{q}}\Vert f\Vert _{L^{q}} \biggr). \end{aligned}$$
$$ \int_{B_{\sigma}(x_{0})}\vert u-u_{x_{0},\sigma} \vert ^{2}\,dx \le C\sigma^{n+2-\frac{n}{q}}, $$
(3.11)
$$u\in\mathcal{L}^{2,n+2-\frac{n}{q}}\bigl(\Omega_{0}, \mathbb{R}^{N} \bigr). $$
Further, by the Caccippoli inequality from Lemma 2.1, we conclude that which implies that \(Du\in L^{2,\gamma}\) with \(\gamma=n-\frac{n}{q}\).
$$\begin{aligned}& \sup_{x\in\Omega_{0},\sigma\in(0,\frac{\rho}{2})}\sigma^{\frac {n}{q}-n} \int_{B_{\frac{\sigma}{2}}(x_{0})}\vert Du\vert ^{2}\,dx \\& \quad \leq\sup_{x\in\Omega_{0},\sigma\in(0,\frac{\rho}{2})}\sigma ^{\frac{n}{q}-n} \biggl[ \frac{C_{1}}{\sigma^{2}} \int_{B_{\sigma }(x_{0})}\vert u-u_{x_{0},\sigma} \vert ^{2}\,dx +C_{2}\sigma^{n+2-\frac{2n}{q}}\Vert f\Vert ^{2}_{L^{q}} \biggr] \\& \quad \le \sup_{x\in\Omega_{0},\sigma\in(0,\frac{\rho}{2})}\sigma ^{\frac{n}{q}-n} \biggl[ \frac{C}{\sigma^{2}}\cdot\sigma^{n+2-\frac {n}{q}}+C_{2}\sigma^{n+2-\frac{2n}{q}} \Vert f\Vert ^{2}_{L^{q}} \biggr] \\& \quad \le C\bigl(1+\rho^{2-\frac{n}{q}}\bigr)< +\infty, \end{aligned}$$
Finally, we are in a position to estimate the Hausdorff dimension of the singular set \(\Omega\setminus\Omega_{0}\). Recall that the singular set is written by On the basis of Lemma 2.8, it yields which implies \({\mathcal {H}}^{n-2}(\Omega\setminus\Omega_{0})=0\). This completes the proof of Theorem 1.4. □
Notice that by the Poincaré’s inequality one deduces
that is,
$$ \Omega\setminus\Omega_{0}\subset\Omega\setminus\Sigma_{0}= \Bigl\{ x\in\Omega:\liminf_{\rho\rightarrow 0}E(\rho)>0 \Bigr\} . $$
(3.12)
$${\mathcal {H}}^{n-2}(\Omega\setminus\Sigma_{0})=0, $$
Acknowledgements
This work is supported by the National Science Foundation of China grant 11371050. The authors would like to thank the referees for their valuable comments and suggestions.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.