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On the continuity of degenerate \(n\)-harmonic functions. (English) Zbl 1258.35044

This paper is con concerned with the regularity of finite energy solutions to the elliptic equation \(\text{div}A(x,Du)=\text{div} f\) in a bounded domain \(\Omega\subset \mathbb R^n\), \(n\geq 2\). Here \(A:\Omega\times \mathbb R^n\rightarrow \mathbb R^n\) satisfies the growth condition \[ |\xi|^n+|A(x,\xi)|^{n/(n-1)}\leq K(x) \langle A(x,\xi),\xi\rangle, \] for a.e. \(x\in \Omega\), \(\xi\in \mathbb R^n\), where \(K(x)=k(x)[k(x)^{n/(n-1)}+1]\).
The main result established by the authors is the following: Let \(u\) be a finite energy solution of the above equation where \(\exp(P(K))\in L^{1}_{loc}(\Omega)\), where \(P\) satisfies \[ \int_1^\infty \frac{P(t)}{t^2}dt=\infty; \] and \(\exp(P(t^{(1-\nu)/\nu})\) is convex for some \((n-1)/n<\nu<1\), and \(fK^{1/n}\in L^q_{loc}(\Omega)\) for some \(q>n/(n-1)\).
Then \(u\) is continuous on a subset \(\Omega_0\subset\Omega\) with full measure. Furthermore, there exist positive constants \(\alpha, \sigma, c>0\) and a radius \(R>0\) such that \[ |u(x)-u(y)|^n\leq \frac{c}{{\mathcal A}(4^{-n}|x-y|^{-n})}+c|x-y|^{n\sigma}, \] for all Lebesgues points \(x,y\) of \(u\) with \(x,y\in B_R\subset\subset\Omega\). Here \[ {\mathcal A}(t)=\exp\Big[\int_{t_0}^t\frac{1}{\tau P^{-1}(\log \tau)}d\tau \Big]. \]

MSC:

35B65 Smoothness and regularity of solutions to PDEs
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35J62 Quasilinear elliptic equations
35D30 Weak solutions to PDEs
35J70 Degenerate elliptic equations
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