Boundary regularity for quasilinear elliptic systems with super quadratic controllable growth condition

where Ω is a bounded domain in $R^n$, $n\ge 2$, $N>1$, and u and $B_i$ take values in $R^N$. Here each $A_{ij}^{\alpha \beta }$ maps $\mathrm{\Omega }\times R^N$ into R, and each $B_i$ maps $\mathrm{\Omega }\times R^N\times R^{nN}$ into R. For $m>2$, we have the following.

for all $x,y\in \overline{\mathrm{\Omega }}$, $u,v\in R^N$.

where $r=\frac{nm}{n-m}$ if $n>m$, or any exponent if $n=m$; for all $x\in \overline{\mathrm{\Omega }}$, $\xi \in R^N$ and $\nu \in R^{nN}$.

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

If the domain we consider is an upper half unit ball $B^{+}$, the boundary condition is the following.

Here we write $B_{\rho }(x_0)=\{x\in R^n:|x-x_0|<\rho \}$, and further $B_{\rho }=B_{\rho }(0)$, $B=B_1$. For $x_0\in R^{n-1}\times \{0\}$ we write $B_{\rho }^{+}(x_0)$ for $\{x\in R^n:x_n>0,|x-x_0|<\rho \}$, and we set $B_{\rho }^{+}=B_{\rho }^{+}(0)$, $B^{+}=B_1^{+}$. We further write $D_{\rho }(x_0)=\{x\in R^n:x_n=0,|x-x_0|<\rho \}$, and we set $D_{\rho }=D_{\rho }(0)$, $D=D_1$. For bounded $X\subset R^n$ with $L^n(X)>0$ we denote the average of a given function $g\in L^1(X)$ by ${⨏}_{X}gdx$, i.e. ${⨏}_{X}gdx=\frac{1}{L^n(X)}{\int }_{X}gdx$. For $\nu \in L^1(\partial \mathrm{\Omega })$, $x_0\in \partial \mathrm{\Omega }$ we set ${\nu }_{x_0,R}^{\prime }={⨏}_{\partial \mathrm{\Omega }\cap {\overline{B}}_R(x_0)}\nu d{H}^{n-1}$. In particular, for $\nu \in L^1(D_{\rho }(x_0))$, $x_0\in D$, we write ${\nu }_{x_0,\rho }^{\prime }={⨏}_{D_{\rho }(x_0)}\nu d{H}^{n-1}$.

Now we can definite weak solutions to systems (1.1). Because there is a very large literature on the existence of weak solutions [1, 2], we assume that a weak solution exists [3] and deal with the problem of regularity directly.

for $x_0\in D$, $\rho \le 1-|x_0|$. Obviously, the inequality remains true if we replace ${\parallel g\parallel }_{H^{1,s}(B_{\rho }^{+}(x_0),R^N)}$ by ${\parallel g\parallel }_{H^{1,s}(B^{+},R^N)}$, which we will henceforth abbreviate simply as ${\parallel g\parallel }_{H^{1,s}}$.

for a constant $C_p$ depending only on n.

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

Under such assumptions, one cannot expect that weak solutions to (1.1) will be classical [4]. This was first shown by De Giorgi [5]. Thus, our goal is to establish a partial regularity for weak solutions of systems (1.1).

There are some previous partial regularity results at boundary for inhomogeneous quasilinear systems. Arkhipova has studied regularity up to the boundary for nonlinear and quasilinear systems [6–8]. For systems in diagonal form, boundary regularity was first established by Wiegner [9], and the proof was generalized and extended by Hildebrandt-Widman [10]. Jost-Meier [11] established full regularity in a neighborhood of boundary for minima of functionals with the form ${\int }_{\mathrm{\Omega }}A(x,u){|Du|}^{2}dx$.

The results which are most closely related to that given here were shown in [12] and [13]. In this paper, we would get the desired conclusions by the method of A-harmonic approximation. The A-harmonic approximation technique is a natural extension of harmonic approximation technique. In [14] Simon used harmonic approximation method to simplify Allard’s [15] regularity theorem and later on Schoen and Uhlenbeck’s [16] regularity result for harmonic maps. The idea was generalized to more general linear operators by Duzaar and Steffen [17], in order to deal with the regularity of almost minimizers to elliptic variational integrals in the setting of geometric measure theory. As a by-product Duzaar and Grotowski [18] were able to use the idea of A-harmonic approximation to deal with elliptic systems under quadratic growth, even to the boundary points for nonlinear elliptic systems [19] and variational problems [20].

In this context, we use an A-harmonic approximation method to establish boundary regularity results.

for some $R\in (0,R_1]$, $x_0\in \partial \mathrm{\Omega }$, which implies $u\in C^{0,\gamma }({\overline{B}}_{\frac{R}{2}}(x_0)\cap \overline{\mathrm{\Omega }},R^N)$.

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

A standard covering argument [3] allows us to obtain the following.

Corollary 1.1 Under the assumptions of Theorem  1.1 the singular set of the weak solution u has $(n-m)$-dimensional Hausdorff measure zero in $\overline{\mathrm{\Omega }}$.

If the domain of the main step in proving Theorem 1.1 is a half ball, the result then is the following.

for some $R\in (0,R_0]$, $x_0\in \partial \mathrm{\Omega }$, which implies $u\in C^{0,\gamma }({\overline{B}}_{\frac{R}{2}}(x_0),R^N)$.

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

In this section we present an A-harmonic approximation lemma [12], and some standard results due to Campanato [21, 22].

Lemma 2.1 (A-harmonic approximation lemma)

Next we recall a characterization of Hölder continuous functions with a slight modification [21].

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

for all $x,z\in {\overline{B}}_R^{+}(x_0)$, with the constant $C_{\kappa }$ 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 [22].

In this section we prove Caccioppoli’s inequality.

Theorem 3.1 (Caccioppoli inequality)

where $C_1$, $C_2$, $C_3$ depend only on λ, L, m, and ${\parallel g\parallel }_{L^{\mathrm{\infty }}(B,R^N)}$, and $C_2$ additionally on $C_p$, $C_s$, and also on s.

Proof Now we consider a cut off function $\eta \in C_0^{\mathrm{\infty }}(B_{\rho }^{+}(x_0))$, satisfying $0\le \eta \le 1$, $\eta \equiv 0$ on $B_{\rho }^{+}(x_0)$, and $|\mathrm{\nabla }\eta |<\frac{1}{R-\rho }$. Then the function $\phi =(u-g){\eta }^2$ is in $W_0^{1,m}(B_{\rho }^{+}(x_0),R^N)$, and thus it can be taken as a test-function.

here we have used the fact that $|D\eta |<\frac{1}{R-\rho }$ and $0<\rho <R<1$.

Fixing ε small enough yields the desired inequality immediately. □

In this section we proceed to the proof of partial regularity result.

for $z\in D$, $r\in (0,1-|z|)$.