The Wiener test and potential estimates for quasilinear elliptic equations (Q1327358)

In this important paper the authors nail down the last part of the boundary regularity for the Dirichlet problem of quasilinear elliptic equations. The problem has a long history starting with the fundamental work by N. Wiener in 1924. He solved completely the boundary regularity problem in the case of harmonic functions. The so-called Wiener integral criterion (the Wiener test) came out of these studies. \textit{V. Maz'ya} [Vestn. Leningr. Univ. Math. 3, 225-242 (1976); transl. from Vestn. Leningr. Univ. 25, No. 13, 42-55 (1970; Zbl 0252.35024)] showed that a corresponding \(p\)-Wiener test gives a sufficient condition for the solvability of the Dirichlet problem associated with the quasilinear elliptic equation \(\text{div} A(t, \nabla u(x)) = 0\) where \(A(x,\xi) \approx | \xi |^ p\) for some \(p\), \(1 < p \leq n\), and \(n\) is the dimension of the underlying Euclidean space. In [\textit{P. Lindqvist} and \textit{O. Martio}, Acta Math. 155, 153-171 (1985; Zbl 0607.35042)] it was shown that the \(p\)-Wiener test gives a necessary condition for \(n - 1 < p \leq n\) and this paper provides this result for the full range of \(p\), \(1 < p \leq n\). A new and efficient tool in this connection is the Wolff potential [\textit{L. I. Hedberg} and \textit{Th. Wolff}, Ann. Inst. Fourier 33, 161-187 (1983; Zbl 0525.31005)]. As a byproduct the authors generalize many known results of thinness for ordinary superharmonic functions to the nonlinear situation.