Boundary limits of locally n-precise functions (Q923762)

This paper deals with boundary limits of Sobolev-Orlicz functions u satisfying a condition of the form: \[ \int_{G}| \text{grad} u(x)|^ n \psi (| \text{grad} u(x)|)\omega (\rho (x))dx<\infty, \] where G is a bounded Lipschitz domain in \({\mathbb{R}}^ n\), \(\rho\) (x) denotes the distance of x from the boundary \(\partial G\), \(\psi\) is a positive nondecreasing function on the interval \([0,\infty)\) satisfying \((\Delta_ 2)\) condition and \(\omega\) is a positive monotone function on the interval (0,\(\infty)\) satisfying \((\Delta_ 2)\) condition. The first aim in the present paper is to show that if \[ (1)\quad \int^{1}_{0}[\psi (r^{-1})]^{1/(1-n)}(1/r)dr<\infty, \] then u is continuous on G. This follows also from Theorem 5.4 in \textit{V. G. Maz'ya}'s book: Sobolev spaces (1985; Zbl 0692.46023). Under the same assumption on \(\psi\), the existence of (tangential) limits \(\lim_{x\to \xi,x\in T_{\phi}(\xi,a)}u(x)\) is discussed for \(\xi\in \partial G\) and \(T_{\phi}(\xi,a)=\{x\in G\); \(\phi (| x-\xi |)<a\rho (x)\}\). In this discussion, it is necessary to evaluate the size of the exceptional set (the set of boundary points at which u fails to have boundary limits), by use of capacities and Hausdorff measures. If \(G=\{x=(x_ 1,...,x_ n)\); \(0<x_ 1<1\), \(\sqrt{x^ 2_ 2+...+x^ 2_ n}<x^ a_ 1\}\) and \(0<a<1\), then G is not a Lipschitz domain. However, u is shown to have a (tangential) limit at the origin. Further, weighted boundary limits are also discussed. The problem is to find a function \(\kappa\) satisfying \(\lim_{x\to \xi,x\in T_{\phi}(\xi,a)}\kappa (x)u(x)=0.\) If (1) does not hold, then we will be concerned with fine-type limits, as usual.