Existence of nontangential limits of solutions of nonlinear Laplace equation
Yoshihiro Mizuta · Hiroshima Mathematical Journal · 1980
Our aim in this note is to study the boundary behavior of (weak) solutions of the non-linear Laplace equationwhere Ω is a domain in the n-dimensional Euclidean space R n .We say that ξedΩ satisfies the interior cone condition if there is an open truncated cone Γ in Ω with vertex at ξ. Let F be the set of all ξedΩ satisfying the interior cone condition.We can show that F is an F^-set**.A function u on Ω is said to have a non-tangential limit at ξ eF if for any open truncated cone Γcί3 with vertex at ξ, lim u(x)x-*ξ,xeΓ' exists and is finite whenever Γ' is a cone with vertex at ξ whose closure Γ' is included in Γl){ξ}.In this note let l<jp<oo and let p(x) denote the distance of x from R n ~ Ω. THEOREM. Let l<p^n and let u be a function satisfying the following properties: i) u is continuous on Ω; ii) u is p-precise**) on any relatively compact open subset ofΩ; iii) u satisfies (1) in the weak sense (cf[4]); iv) \ |gradw(x)| i 'p(x) α dx<oo for α < p. Then there exists a set EcdΩ such that B xa/PtP (E)=0 and u has a non-tangential limit at each point ofF-E.Here B xa/PtP denotes the Bessel capacity of index (1-α/p, p) (see [1]).In case p=2, our theorem is shown in [3; Theorem 2'].*) This fact was pointed out by Professor Makoto Sakai.**) For the definition of/^-precise functions, see Ziemer [5].