Boundary regularity result for quasilinear elliptic systems

Abstract

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under controllable growth condition, and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

1 Introduction

In this article we are concerned with boundary regularity for weak solutions of quasilinear elliptic systems of the following type:

$$-D_{\alpha }\left(A_{ij}^{\alpha \beta }\left(x,u\right)D_{\beta }{u}^j\right)=B^i\left(x,u,Du\right),$$

(1.1)

where Ω is a bounded domain in Rⁿ, n ≥ 2, u and Bⁱtake values in Rⁿ, N > 1. Here each $A_{ij}^{\alpha \beta }$ maps Ω × R^Ninto R, and each Bⁱmaps Ω × R^N× R^nNinto R. In this article, we shall be concerned with weak solutions. In order to define weak solutions to (1.1), one needs to impose certain structural and regularity conditions on $A_{ij}^{\alpha \beta }$ and inhomogeneity Bⁱas follows:

(H1) There exists L > 0 such that

$$A_{ij}^{\alpha \beta }\left(x,\xi \right)\left(v,ṽ\right)\le L\left|v\right|\left|ṽ\right|\text{for all }\left(x,\xi \right)\in \bar{\Omega }\times R^N,v,ṽ\in R^{nN}.$$

(H2)$A_{ij}^{\alpha \beta }\left(x,\xi \right)$ is uniformly strongly elliptic, that is, for some λ > 0 we have

$$A_{ij}^{\alpha \beta }\left(x,\xi \right)\left(v,v\right)\ge \lambda |v{|}^2\text{for all}\left(x,\xi \right)\in \bar{\Omega }\times R^N,v\in R^{nN}.$$

(H3) There exists a monotone nondecreasing concave function ω(t, s): [0, ∞) → [0, ∞) with ω(t, 0) = 0, continuous at 0, such that

$$|A_{ij}^{\alpha \beta }\left(x,u\right)-A_{ij}^{\alpha \beta }\left(y,v\right)|\le \omega \left(|x-y{|}^2+|u-v{|}^2\right),$$

for all x, y ∈ Ω, u, v ∈ R^N .

(H4) Bⁱfulfill the following controllable growth condition:

$$\left|B^i\left(x,\xi ,v\right)\right|\le C\left(|v{|}^{2\left(1-\frac{1}{r}\right)}+|\xi {|}^{r-1}+1\right),$$

where $r=\frac{2n}{n-2}$ if n > 2, or any exponent if n = 2; for all $x\in \bar{\Omega },\xi \in R^N$ and ν ∈ R^nN.

(H5) There exists s with s > n and a function g ∈ H^1,s(Ω, Rⁿ), such that there holds:

$$u{|}_{\partial \Omega }=g{|}_{\partial \Omega }.$$

Note that we trivially have $g\in H^{1,2}\left(\Omega ,R^N\right)$. Further, by Sobolev embedding theorem we have g ∈ C^0,κ(Ω, Rⁿ) for any $\kappa \in \left[0,1-\frac{n}{s}\right].\text{If}g{|}_{\partial \Omega }\equiv 0,$we will take g ≡ 0 on Ω.

We will be particularly concerned with a model case in which the domain is the upper half unit ball B⁺. We reformulate the boundary condition for this case:

(H5)′ There exists s with s > n and a function g ∈ H^1,s(B⁺, Rⁿ) such that there holds:

$$u{|}_D=g{|}_D.$$

For x₀ ∈ R^{n− 1}× {0} we write D _ρ (x₀) = {x ∈ Rⁿ: x _n = 0, |x − x₀| < ρ}, and set D _ρ = D _ρ (0), D = D₁.

Definition 1.1. By a weak solution of (1.1) we mean a vector valued function u ∈ W^1,2(Ω, Rⁿ) such that

$$\underset{\Omega }{\int }{A}_{ij}^{\alpha \beta }\left(x,u\right)\left(D_{\beta }{u}^j,D_{\alpha }{\phi }^i\right)dx=\underset{\Omega }{\int }{B}^i\left(x,u,Du\right)\cdot {\phi }^{i}dx,$$

(1.2)

holds for all test-functions $\phi \in C_0^{\infty }\left(\Omega ,R^N\right)$ and, by approximation, for all $\phi \in W_0^{1,2}\left(\Omega ,R^N\right).$ Where we have introduced the notation

$$A_{ij}^{\alpha \beta }\left(x,\xi \right)\left(v,ṽ\right)=\left(A_{ij}^{\alpha \beta }\left(x,\xi \right)v\right)\cdot ṽ.$$

(1.3)

In the current situation the Sobolev embedding theorem yields the existence of a constant C _s depending only on s, n and N such that there holds:

$$\underset{B_{\rho }^{+}\left(x_0\right)}{\text{sup}}|g-g_{x_0,\rho }^{\prime }|\le C_s{\rho }^{1-\frac{n}{s}}{∥g∥}_{H^{1,s}\left(B_{\rho }^{+}\left(x_0\right),R^N\right)},$$

(1.4)

for x₀ ∈ D, ρ ≤ 1−|x₀|. Obviously, the inequality remains true if we replace $\left|\right|g|{|}_{H^{1,s}\left(B_{\rho }^{+}\left(x_0\right),R^N\right)}$ by $\left|\right|g|{|}_{H^{1,s}\left(B^{+},R^N\right)}$, which we will henceforth abbreviate simply as $\left|\right|g|{|}_{H^{1,s}}$.

We also note here that the Poincare inequality in this setting yields:

$$\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|g-g_{x_0,\rho }^{\prime }{|}^{2}dx\le C_p{\rho }^2\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|Dg{|}^{2}dx,$$

(1.5)

for a constant C _p which depending only on n.

Finally, we fix an exponent σ ∈ (0, 1) as follows: if g ≡ 0, σ can be chosen arbitrary (but henceforth fixed); otherwise we take σ fixed in $\left(0,1-\frac{n}{s}\right]$.

Under such assumptions on the structural conditions, full regularity of (1.1) at the boundary cannot, in general, be expected [1], even if the boundary data is smooth. Then, our goal is to establish a partial boundary regularity for weak solutions of systems (1.1).

There are some previous partial regularity results for quasilinear systems. For example, Arkhipova has studied regularity up to the boundary for nonlinear and quasilinear systems [2–4]. And there are more results of particular forms. For systems in diagonal form, boundary regularity was first established by Wiegner [5], and the proof was generalized and extended by Hildebrandt-Widman [6]. The further discussions in this case can be seen in [7, 8]. In the case of minima of functionals of the form ∫_ΩA(x, u)|Du|²dx, Jost-Meier [8] established full regularity in a neighborhood of the boundary. Further more, the dimension of singular set of solutions to non-differentiable elliptic systems are reduced by Kristensen and Mingione [9, 10]

The result which is most closely related to that given here was shown that in [11] Grotowski obtained the boundary partial regularity results for more general systems of the form: $-D_{\alpha }{A}_i^{\alpha }\left(\cdot ,u,Du\right)=f^i\left(\cdot ,u,Du\right)$, i = 1,..., N, under the quadratic growth and natural condition. And in [12], he got the analogous boundary regularity of (1.1) under natural growth condition. However the results in the current article do not follow from those in [11]; in the current situation we need only impose weaker structure conditions, and at the same time can obtain stronger conclusions.

The comparison is made possible by the technique of A-harmonic approximation. This technique is the natural extension of the technique of harmonic approximation. The harmonic approximation technique has its origins in Simon's [13] proof of the regularity theorem of Allard [14]. The technique of A-harmonic approximation then refers to the direct analog of the above situation. The interior version of this technique has previously been applied by Duzaar and Grotowski [15] to obtain a new, elementary proof for interior partial regularity for systems of the form (1.3). Then it has been extended and developed in [16, 17].

In this article, we shown optimal boundary regularity of (1.1) under controllable growth condition. As the argument for combining the boundary and interior estimates is relatively standard, we omit it and get the following results:

Theorem 1.1. Let Ω be a bounded domain in Rⁿ, with boundary of class C¹. Let u be a weak solution of (1.1) satisfying the boundary condition (H5), where the structure conditions (H1)-(H3) hold for $A_{ij}^{\alpha \beta },$, and (H4) holds for Bⁱ. Consider a fixed γ ∈ (0, σ]. Then there exist positive R₁ and ε₀(depending only on n, N, λ, L, ω(·) and γ) with the property that

$${–\int }_{B_R\left(x_0\right)\cap \Omega }|u-u_{x_0,R}^{\prime }{|}^{2}dx+\parallel g{\parallel }_{H^{1,s}}^2{R}^{2\left(1-\frac{n}{s}\right)}+R^2\le {\epsilon }_0^2,$$

for some R ∈ (0, R₁] for a given x₀ ∈ ∂ Ω implies $u\in C^{0,\gamma }\left({\bar{B}}_{\frac{R}{2}}\left(x_0\right)\cap \bar{\Omega },R^N\right).$

Note in particular that the boundary condition (H5) means that $u_{x_0,R}^{\prime }$ makes sense: in fact, we have $u_{x_0,R}^{\prime }=g_{x_0,R}^{\prime }.$

Combining this result with the analogous interior [18] and a standard covering argument allows us to obtain immediately the following bound on the size of the singular set:

Corollary 1.2. Under the assumptions of Theorem 1.1, the singular set of the weak solution u has (n − 2)-dimensional Hausdorff measure zero in $\bar{\Omega }$.

If the domain of the main step in proving Theorem 1.1 is a half ball; the result then follows from a relatively standard transformation argument.

Theorem 1.2. Consider a weak solution of (1.1) on the upper half unit ball B⁺ which satisfies the boundary condition (H 5)′, where the structure conditions (H1)-(H3)hold for$A_{ij}^{\alpha \beta },$ and (H4) holds for Bⁱ. Then there exist positive R₀ and ε₀(depending only on n, N, λ, L, ω(·) and γ) with the property that

$${–\int }_{B_R^{+}\left(x_0\right)}|u-u_{x_0,R}^{\prime }{|}^{2}dx+|\left|g\right|{|}_{H^{1,s}}^2{R}^{2\left(1-\frac{n}{s}\right)}+R^2\le {\epsilon }_0^2,$$

for some R ∈ (0, R₀] for a given x₀ ∈ D implies $u\in C^{0,\gamma }\left({\bar{B}}_{\frac{R}{2}}\left(x_0\right),R^N\right)$.

Note that analogously to above, the boundary condition (H5)′ ensures that $u_{x_0,{}^R}^{\prime }$ exists, and we have indeed $u_{x_0,R}^{\prime }=g_{x_0,R}^{\prime }$.

We close this section by briefly summarizing the notation we use in this article. For a given set X we denote by Lⁿ(X) and H^k(X) its n-dimensional Lebesgue measure and k-dimensional Harsdorff measure, respectively. We write B _ρ (x₀) = {x ∈ Rⁿ: |x − x₀| < ρ}, and further B _ρ = B _ρ (0), B = B₁. Similarly we denote upper half balls as follows: for x₀ ∈ R^{n− 1}× {0} we write $B_{\rho }^{+}\left(x_0\right)$ for {x ∈ Rⁿ: x _n > 0, |x - x₀| < ρ}, and set $B_{\rho }^{+}=B_{\rho }^{+}\left(0\right),B^{+}=B_1^{+}$. For bounded X ⊂ Rⁿwith Lⁿ(X) > 0 we denote the average of a given g ∈ L¹ by ${–\int }_{X}gdx$, i.e. ${–\int }_{X}gdx=\frac{1}{L^n\left(X\right)}{\int }_{X}gdx.$ For v ∈ L¹ (∂Ω), x₀ ∈ ∂Ω we set ${\nu }_{x_0,R}^{\prime }={–\int }_{\partial \Omega \cap {\bar{B}}_R\left(x_0\right)}vd{H}^{n-1}$. In particular, for ν ∈ L¹(D _ρ (x₀)), x₀ ∈ D, we write ${\nu }_{x_0,\rho }^{\prime }={–\int }_{D_{\rho }\left(x_0\right)}vd{H}^{n-1}$. We let α _n denote the volume of the unit-ball in Rⁿ, i.e. α _n = Lⁿ(B). We write Bil(R^nN ) for the space of bilinear forms on the space R^nN .

2 The A-harmonic approximation technique and some preliminary results

In this section we present the A-harmonic approximation lemma [12], and some standard results due to Campanato [19, 20].

Lemma 2.1. (A-harmonic approximation lemma) Consider fixed positive λ and L, and n, N ∈ N with n ≥ 2. Then for any given ε > 0 there exists δ = δ(n, N, λ, L, ε) ∈ (0, 1] with the following property: for any $A_{ij}^{\alpha \beta }\in \text{Bil}\left(R^{nN}\right)$satisfying

$$A_{ij}^{\alpha \beta }\left(v,v\right)\ge \lambda |v{|}^2,\text{for all}v\in R^{nN},$$

(2.1)

and

$$|A_{ij}^{\alpha \beta }(v,\overline{v}|\le L|v\left|\right|\overline{v}|,\text{for all }v,\overline{v}\in R^{nN},$$

(2.2)

for any $w\in H^{1,2}\left(B_{\rho }^{+}\left(x_0\right),R^N\right)$ (for some ρ > 0, x₀ ∈ Rⁿ) satisfying

$${\rho }^{2-n}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|Dw{|}^{2}dx\le 1,$$

(2.3)

and

$$\left|{\rho }^{2-n}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }{A}_{ij}^{\alpha \beta }\left(Dg,D\phi \right)dx\right|\le \delta \rho \underset{B_{\rho }^{+}\left(x_0\right)}{\text{sup}}\left|D\phi \right|,$$

(2.4)

and

$$w{|}_{D_{\rho }\left(x_0\right)}=0,$$

(2.5)

for all $\phi \in C_0^1\left(B_{\rho }^{+}\left(x_0\right),R^N\right),$there exists an A-harmonic function

$$v\in \tilde{H}=\left\{\tilde{w}\in H^{1,2}\left(B_{\rho }^{+}\left(x_0\right),R^N\right)\left|{\rho }^{2-n}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|D\tilde{w}{|}^{2}dx\le 1,\tilde{w}{|}_{D_{\rho }\left(x_0\right)}\equiv 0\right.\right\},$$

with

$${\rho }^{-n}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|v-w{|}^{2}dx\le \epsilon .$$

(2.6)

Next we recall a slight modification of a characterization of Hölder continuous functions [19].

Lemma 2.2. Consider n ∈ N, n ≥ 2 and x₀ ∈ R^{n− 1}× {0}. Suppose that there are positive constants κ and α, with α ∈ (0, 1] such that, for some $v\in L^2\left(B_{6R}^{+}\left(x_0\right)\right),$ there hold the following:

$$\underset{\mu \in R}{\text{inf}}\left\{{\rho }^{-n}\underset{B_{\rho }^{+}\left(y\right)}{\int }|v-\mu {|}^{2}dx\right\}\le {\kappa }^2{\left(\frac{\rho }{R}\right)}^{2\alpha },$$

(2.7)

for all y ∈ D_2R (x₀) and ρ ≤ 4R; and

$$\underset{\mu \in R}{\text{inf}}{\rho }^{-n}\left\{\underset{B_{\rho }\left(y\right)}{\int }|v-\mu {|}^{2}dx\right\}\le {\kappa }^2{\left(\frac{\rho }{R}\right)}^{2\alpha },$$

(2.8)

for all $y\in B_{2R}^{+}\left(x_0\right)\text{and}{B}_{\rho }\left(y\right)\subset B_{2R}^{+}\left(x_0\right).$

Then there exists a Hölder continuous representatives of the L²-class of ν on ${\bar{B}}_R^{+}\left(x_0\right),$ and for this representative $\bar{v}$ there holds:

$$|\bar{v}\left(x\right)-\bar{v}\left(z\right)|\le C_{\kappa }{\left(\frac{|x-z|}{R}\right)}^{\alpha },$$

(2.9)

for all $x,z\in {\bar{B}}_R^{+}\left(x_0\right)$, for a constant C _κ depending only on n and α.

We close this section by a standard estimate for the solutions to homogeneous second order elliptic systems with constant coefficients, due originally to Campanato [20].

Lemma 2.3. Consider fixed positive λ and L, and n, N ∈ N with n ≥ 2. Then there exists C₀ depending only on n, N, λ and L (without loss of generality we take C₀≥ 1) such that for $A_{ij}^{\alpha \beta }\in \text{Bil}\left(R^{nN}\right)$ satisfying (2.1) and (2.2), any A-harmonic function h on $B_{\rho }^{+}\left(x_0\right)$ with $h{|}_{D_{\rho }\left(x_0\right)}\equiv 0$ satisfies

$${\rho }^2\underset{B_{\frac{\rho }{2}}^{+}\left(x_0\right)}{\text{sup}}|Dh{|}^2\le C_0{\rho }^{2-n}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|Dh{|}^{2}dx.$$

3 Caccioppoli inequality

In this section we prove the following Caccioppoli's inequality.

Theorem 3.1. Let $u\in W^{1,2}\left(\bar{\Omega },R^N\right)$ be a weak solution of systems (1.1) under (H1)-(H5). Then there exists ρ₀> 0, ρ₀ depending only on L, s and $\left|\right|g|{|}_{H^{1,s}},$ such that for all $B_{\rho }^{+}\left(x_0\right)\subset B^{+},$ with x₀ ∈ D⁺, ρ < R < ρ₀, there holds

$$\begin{array}{l}\underset{B_{\rho /2}^{+}\left(x_0\right)}{\int }|Du{|}^{2}dx\le \frac{C_1}{{\rho }^2}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|u\left(x\right)-{u^{\prime }}_{x_0,\rho }{|}^{2}dx+C_2{\alpha }_n\parallel g{\parallel }_{H^{1,s}}^2{\rho }^{n\left(1-\frac{2}{s}\right)}\\ +C_3{\left(\underset{B_{\rho }^{+}\left(x_0\right)}{\int }\left(|Du{|}^2+|u{|}^r+1\right)dx\right)}^{2\left(1-\frac{1}{r}\right)},\end{array}$$

(3.1)

where C₁, C₂, C₃ depending only on λ, L and ${∥g∥}_{L^{\infty }\left(B,R^N\right)},$ and C₂ additional on C _p , C _s , and also on s.

Proof. Now we consider a cut -off function $\eta \in C_0^{\infty }\left(B_{\rho /2}^{+}\left(x_0\right)\right)$, satisfying $0\le \eta \le 1,\eta \equiv 1\text{on}{B}_{\rho /2}^{+}\left(x_0\right)$ and $|\nabla \eta |<\frac{4}{\rho }$. Then the function (u − g)η² is in $W_0^{1,2}\left(B_{\rho /2}^{+}\left(x_0,R^N\right)\right)$, and thus can be taken as a test-function.

Using (1.2), (H1), (H4), (H5), for ε positive but arbitrary (to be fixed later), we have:

$$\begin{array}{l}{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}{A}_{ij}^{\alpha \beta }}(x,u)(Du,Du){\eta }^{2}dx\\ =-2{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}{A}_{ij}^{\alpha \beta }}(x,u)(Du,D\eta )(u-g)\eta dx\\ +{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}{A}_{ij}^{\alpha \beta }}(x,u)(Du,Dg){\eta }^{2}dx+{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}{B}^i}(x,u,Du)·{\phi }^{i}dx\\ \le 2L{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}Du\left|\right|D\eta \left|\right|u-g|\eta dx+L{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}Du|\left|Dg\right|{\eta }^{2}dx\\ +C{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}\left(|Du{|}^{2(1-\frac{1}{r})}+|u{|}^{r-1}+1\right)}·\phi dx\\ \le 2\epsilon {\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}Du{|}^2{\eta }^{2}dx+\frac{L^2}{\epsilon }{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}u-g{|}^2|D\eta {|}^{2}dx+\frac{L^2}{\epsilon }{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}Dg{|}^{2}dx\\ +C{\left({\displaystyle \underset{B_{\rho }^{+}(x_0)}{\int }(|Du{|}^2+|u{|}^r+1)dx}\right)}^{\left(1-\frac{1}{r}\right)}{\left({\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}\phi {|}^{r}dx\right)}^{\frac{1}{r}}\\ \le 2\epsilon {\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}Du{|}^2{\eta }^{2}dx+\frac{L^2}{\epsilon }{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}u-g{|}^2|D\eta {|}^{2}dx+\frac{L^2}{\epsilon }{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}Dg{|}^{2}dx\\ +C{\left({\displaystyle \underset{B_{\rho }^{+}(x_0)}{\int }(|Du{|}^2+|u{|}^r+1)}dx\right)}^{\left(1-\frac{1}{r}\right)}{\left({\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|}D\phi {|}^{2}dx\right)}^{\frac{1}{2}}\\ \le 3\epsilon {\displaystyle \underset{B_{\rho }^{+}(x_0)}{\int }|}Du{|}^2{\eta }^{2}dx+\left(\frac{L^2}{\epsilon }+C\left(\epsilon \right)\right){\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|u-g{|}^2|D\eta {|}^{2}dx+C(\epsilon )}{\displaystyle {\int }_{B_{\rho }^{+}(x_0)}|Dg{|}^{2}dx}\\ +C(\epsilon ){\left({\displaystyle \underset{B_{\rho }^{+}(x_0)}{\int }(|Du{|}^2+|u{|}^r+1)}dx\right)}^{2\left(1-\frac{1}{r}\right)}.\end{array}$$

(3.2)

Using (H2) and the fact that $u_{x_0,\rho }^{\prime }=g_{x_0,\rho }^{\prime }$, we thus have:

$$\begin{array}{l}\lambda {\int }_{B_{\rho }^{+}\left(x_0\right)}|Du{|}^2{\eta }^{2}dx\\ \le 3\epsilon {\int }_{B_{\rho }^{+}\left(x_0\right)}Du{|}^2{\eta }^{2}dx+2\left(\frac{L^2}{\epsilon }+C\left(\epsilon \right)\right){\int }_{B_{\rho }^{+}\left(x_0\right)}|u-{u^{\prime }}_{x_0,\rho }{|}^2|D\eta {|}^{2}dx+C\left(\epsilon \right){\int }_{B_{\rho }^{+}\left(x_0\right)}|Dg{|}^{2}dx\\ +2\left(\frac{L^2}{\epsilon }+C\left(\epsilon \right)\right){\int }_{B_{\rho }^{+}\left(x_0\right)}|g-{g^{\prime }}_{x_0,\rho }{|}^2|D\eta {|}^{2}dx+C\left(\epsilon \right){\left(\underset{B_{\rho }^{+}\left(x_0\right)}{\int }\left(|Du{|}^2+|u{|}^r+1\right)dx\right)}^{2\left(1-\frac{1}{r}\right)}.\end{array}$$

Thus, by (1.4), (1.5), we can get:

$$\begin{array}{l}\left(\lambda -3\epsilon \right){\int }_{B_{\rho }^{+}\left(x_0\right)}|Du{|}^2{\eta }^{2}dx\le 2\left(\frac{L^2}{\epsilon }+C\left(\epsilon \right)\right){\int }_{B_{\rho }^{+}\left(x_0\right)}|u-{u^{\prime }}_{x_0,\rho }{|}^2|D\eta {|}^{2}dx\\ +\left[2\left(\frac{L^2}{\epsilon }+\epsilon \right)C_p+C\left(\epsilon \right)\right]{\alpha }_n\left|\right|g|{|}_{H^{1,s}}^2{\rho }^{n\left(1-\frac{2}{s}\right)}+C\left(\epsilon \right){\left(\underset{B_{\rho }^{+}\left(x_0\right)}{\int }\left(|Du{|}^2+|u{|}^r+1\right)dx\right)}^{2\left(1-\frac{1}{r}\right)}.\end{array}$$

(3.3)

Fix ε small enough such that λ − 3ε > 0, thus (3.3) yields the desired inequality, where C₁, C₂, C₃ depending only on ${∥g∥}_{L^{\infty }\left(B,R^N\right)}$λ and L, and C₂ additional on C _p , C _s and also on s.

4 The proof of the main theorem

In this section we proceed to the proof of partial regularity result, and hence consider u ∈ W^1,2(Ω, Rⁿ) to be a weak solution of (1.1). For R < 1 − |x₀|, x₀ ∈ D, y ∈ D _R (x₀), D _ρ (y) ⊂⊂ D _R (x₀),and $\phi \in C_0^{\infty }\left(B_{\frac{\rho }{2}}^{+}\left(y\right),R^N\right)$ with $\underset{B_{\rho }^{+}\left(y\right)}{\text{sup}}\left|D\phi \right|\le 1$, we have

Lemma 4.1.

$$\left|{\left(\frac{\rho }{2}\right)}^{2-n}\underset{B_{\frac{\rho }{2}}^{+}\left(y\right)}{\int }A\left(y,u_{y,\rho }^{+}\right)\left(Du,D\phi \right)dx\right|\le C_8\frac{\rho }{2}\sqrt{I}\left(\sqrt{I}+{\omega }^{\frac{1}{2}}\left(I\right)\right)\underset{B_{\frac{\rho }{2}}^{+}\left(x_0\right)}{\text{sup}}\left|D\phi \right|,$$

(4.1)

here and hereafter, we define

$$I\left(z,r_0\right)={–\int }_{B_{r_0}^{+}\left(z\right)}|u-u_{z,r_0}^{\prime }{|}^{2}dx+|\left|g\right|{|}_{H^{1,s}}^2{r}_0^{2\left(1-n/s\right)}+r_0^2,$$

for z ∈ D, r ₀ ∈ (0, 1 - |z|).

Proof. Consider x₀ ∈ D and y ∈ D _R (x₀), D _ρ (y)⊂⊂ D _R (x₀), for R < 1 − |x₀|, and $\phi \in C_0^{\infty }\left(B_{\frac{\rho }{2}}^{+}\left(y\right),R^N\right)$ with $\underset{B_{\rho }^{+}\left(y\right)}{\text{sup}}\left|D\phi \right|\le 1$. From the definition of weak solution:

$$\begin{array}{l}{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}{A}_{ij}^{\alpha \beta }\left(y,{u^{\prime }}_{y,\rho }\right)\left(Du,D\phi \right)dx\\ \le C\left[{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}\left(|Du{|}^{2\left(1-\frac{1}{r}\right)}+|u{|}^{r-1}+1\right)dx\right]\rho \underset{B_{\frac{\rho }{2}}^{+}\left(y\right)}{\text{sup}}\left|D\phi \right|\\ +{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|A_{ij}^{\alpha \beta }\left(y,{u^{\prime }}_{y,\rho }\right)-A_{ij}^{\alpha \beta }\left(x,u\right)|\left|Du\right|dx\underset{B_{\frac{\rho }{2}}^{+}\left(y\right)}{\text{sup}}\left|D\phi \right|.\end{array}$$

(4.2)

By Sobolev' s and Hölder's inequalities, and then Young's inequality, we have

$$\begin{array}{l}C{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}\left(|Du{|}^{2\left(1-\frac{1}{r}\right)}+|u{|}^{r-1}+1\right)dx\\ \le C{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{1-\frac{1}{r}}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}+C\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right){\alpha }_n{\rho }^n/2\\ +C{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|u-{u^{\prime }}_{y,\rho /2}{|}^{r}dx\right)}^{\left(1-\frac{1}{r}\right)}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}\\ \le C{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{1-\frac{1}{r}}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}+C\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right){\alpha }_n{\rho }^n/2\\ +C{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{\frac{r}{2}\left(1-\frac{1}{r}\right)}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}\\ \le C{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{1-\frac{1}{r}}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}+C\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right){\alpha }_n{\rho }^n/2\\ +C{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{\left(\frac{r}{2}-1\right)\left(1-\frac{1}{r}\right)}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{1-\frac{1}{r}}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}\\ \le C\left(\left|\right|u|{|}_{W^{1,2}}\right){\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{1-\frac{1}{r}}{\left({\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}dx\right)}^{\frac{1}{r}}+C\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right)\frac{{\alpha }_n{\rho }^n}{2}\\ \le \epsilon {\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx+C\left(\epsilon ,\left|\right|u|{|}_{W^{1,2}}\right)\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right){\alpha }_n{\rho }^n/2,\end{array}$$

(4.3)

and

$$\begin{array}{l}{C}_3{\left({\int }_{B_{\rho }^{+}\left(x_0\right)}\left(|Du{|}^2+|u{|}^r+1\right)dx\right)}^{2\left(1-\frac{1}{r}\right)}\\ \le \frac{c_4}{2}{\alpha }_n{\rho }^{n+2}{\left({–\int }_{B_{\rho }^{+}\left(x_0\right)}\left(|Du{|}^2+|u-{u^{\prime }}_{x_0,\rho }{|}^r+\left|{u^{\prime }}_{x_0,\rho }\right|+1\right)dx\right)}^{2\left(1-\frac{1}{r}\right)}\\ \le C_5{\alpha }_n{\rho }^{n+2}\left[{\left({–\int }_{B_{\rho }^{+}\left(x_0\right)}|Du{|}^{2}dx\right)}^{2\left(1-\frac{1}{r}\right)}+{\left({–\int }_{B_{\rho }^{+}\left(x_0\right)}|Du{|}^{2}dx\right)}^{r\left(1-\frac{1}{r}\right)}+{\left(\left|{u^{\prime }}_{x_0,\rho }\right|+1\right)}^{2\left(1-\frac{1}{r}\right)}\right]\\ \le C_6\left(\left|\right|u|{|}_{W^{1,2}},|{u^{\prime }}_{x_0,\rho }|\right){\alpha }_n{\rho }^{n+2},\end{array}$$

where we have used $r=\frac{2n}{n-2}$ and $u\in W^{1,2}\left(\bar{\Omega },R^N\right)$.

Then Caccioppoli inequality yields

$$\underset{B_{\rho /2}^{+}\left(x_0\right)}{\int }|Du{|}^{2}dx\le \frac{C_1}{{\rho }^2}\underset{B_{\rho }^{+}\left(x_0\right)}{\int }|u\left(x\right)-u_{x_0,\rho }^{\prime }{|}^{2}dx+C_2{\alpha }_n\left|\right|g|{|}_{H^{1,s}}^2{\rho }^{n\left(1-\frac{2}{s}\right)}+C_6{\alpha }_n{\rho }^{n+2}.$$

(4.4)

Henceforth we restrict to ρ sufficiently small. Applying in turn Young's inequality, (H3), (4.4) and Jensen's inequality we calculate from (4.2):

$$\begin{array}{l}{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}{A}_{ij}^{\alpha \beta }\left(y,{u^{\prime }}_{y,\rho }\right)\left(Du,D\phi \right)dx\\ \le C\rho \left[{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx+\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right){\alpha }_n{\rho }^n/2\right]\\ +{\left[{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|A_{ij}^{\alpha \beta }\left(y,{u^{\prime }}_{y,\rho }\right)-A_{ij}^{\alpha \beta }\left(x,u\right){|}^{2}dx\right]}^{\frac{1}{2}}\cdot {\left[{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx\right]}^{\frac{1}{2}}\\ \le C\rho \left[{\int }_{B_{\frac{\rho }{2}}^{+}\left(y\right)}|Du{|}^{2}dx+\left(|{u^{\prime }}_{y,\rho /2}{|}^{r-1}+1\right){\alpha }_n{\rho }^n/2\right]\\ +{\omega }^{\frac{1}{2}}\left({\rho }^2+{–\int }_{B_{\rho }^{+}\left(y\right)}|u-{u^{\prime }}_{y,\rho }{|}^{2}dx\right)\left({\alpha }_n{\rho }^n/2\right){\left(f_{B_{\rho }^{+}\left(y\right)}|Du{|}^{2}dx\right)}^{\frac{1}{2}}.\end{array}$$

(4.5)

For z ∈ D, r₀ ∈ (0, 1 − |z|). We introduce the notation

$I\left(z,r_0\right)={–\int }_{B_{r_0}^{+}\left(z\right)}|u-u_{z,r_0}^{\prime }{|}^{2}dx+|\left|g\right|{|}_{H^{1,s}}^2{r}_0^{2\left(1-n/s\right)}+r_0^2,$

and further write I for I(y, ρ). We have from (4.5) and by Poincare's inequality:

$$\underset{B_{\frac{\rho }{2}}^{+}\left(y\right)}{\int }{A}_{ij}^{\alpha \beta }\left(y,{u^{\prime }}_{y,\rho }\right)\left(Du,D\phi \right)dx\le \frac{C_7}{2}{\alpha }_n{\rho }^{n-1}\left[I+{\omega }^{\frac{1}{2}}\left(I\right)I^{\frac{1}{2}}\right].$$

For arbitrary $\phi \in C_0^{\infty }(\Omega ,R^N)$ we thus have, by recalling:

$$\underset{B_{\frac{\rho }{2}}^{+}\left(y\right)}{\int }{A}_{ij}^{\alpha \beta }\left(y,u_{y,\rho }^{\prime }\right)\left(Du,D\phi \right)dx\le \frac{C_7}{2}{\alpha }_n{\rho }^{n-1}\left[I+{\omega }^{\frac{1}{2}}\left(I\right)I^{\frac{1}{2}}\right]\underset{B_{\frac{\rho }{2}}^{+}\left(x_0\right)}{\text{sup}}\left|D\phi \right|.$$

Multiplying through by ${\left(\frac{\rho }{2}\right)}^{2-n},$ this yields:

$$\left|{\left(\frac{\rho }{2}\right)}^{2-n}\underset{B_{\frac{\rho }{2}}^{+}\left(y\right)}{\int }{A}_{ij}^{\alpha \beta }\left(y,{u^{\prime }}_{y,\rho }\right)\left(Du,D\phi \right)dx\right|\le C_8\frac{\rho }{2}\left[I+{\omega }^{\frac{1}{2}}\left(I\right)I^{\frac{1}{2}}\right]\underset{B_{\frac{\rho }{2}}^{+}\left(x_0\right)}{\text{sup}}\left|D\phi \right|,$$

(4.6)

for C₈ define by C₈ = 2^{n− 3}C₇α _n .

Lemma 4.2. Consider u satisfying the condition of Theorem 1 and σ fixed, then we can find δ and s₀ together with positive constants C₁₁ such that the smallness conditions:

$$0<\omega \left(\left|u\right|,s_0\right)\le {\left(\frac{\delta }{2}\right)}^2,$$

and

$$I\left(x_0,R\right)\le C_{13}^{-1}\text{min}\left\{\frac{{\delta }^2}{4},s_0\right\},$$

together imply the growth condition

$$I\left(y,\theta \rho \right)\le {\theta }^{2\sigma }I\left(y,\rho \right).$$

Proof. Combining Lemma 2.1 and Theorem 3.1, and applying Lemma 2.2, we can conclude the desired Hö lder continuity. As the argument is relatively standard and similarly to [12], we omit it.