Interior \(C^{1,\alpha}\) regularity of weak solutions for a class of quasilinear elliptic equations (Q316912)

\noindent Let \(a:(0,+\infty)\rightarrow (0,+\infty)\) be a \(C^1\)-function such that NEWLINE\[NEWLINE-1<\inf_{t>0}\frac{ta'(t)}{a(t)}\leq \sup_{t>0}\frac{ta'(t)}{a(t)}<+\infty.NEWLINE\]NEWLINE The authors prove the \(C^{1,\alpha}\)-local regularity of local weak solutions \(u\in W_{\mathrm{loc}}^{1,B}(\Omega)\) to the equation NEWLINE\[NEWLINE\operatorname{div}(a(|\nabla u|)\nabla u)=0 \text{ in }\Omega,NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), and \(W_{\mathrm{loc}}^{1,B}(\Omega)\) is the Sobolev-Orlicz space corresponding to the Young function \(B(t):=\int_0^t\tau a(\tau)d\tau\).NEWLINENEWLINEThe method of proof is as follows: given a local weak solution \(u\in W_{\mathrm{loc}}^{1,B}(\Omega)\), the authors construct a sequence of approximation solutions whose gradients are locally uniformly bounded and converge almost everywhere to the gradient of \(u\). Then, using a uniform \(C^{1,\alpha}\)-estimate for the approximation solutions, the authors obtain the \(C^{1,\alpha}\)-regularity fo \(u\).