The~Boundary Behavior of Functions of Sobolev Spaces Defined on a~Planar Domain with a~Peak Vertex on the~Boundary

M. Yu. Vasil'chik

Matematičeskie trudy, Tome 6 (2003) no. 1, pp. 3-27

Résumé

Let $G$ be a domain with piecewise smooth boundary $\partial G$ and with vertices of exterior peaks on the boundary and let $k$ functions $f_1,\dots,f_k$ ($k$ is a nonnegative integer) be given on $\partial G$. We find necessary and sufficient conditions for existence of a function $F\in W_p^l(G)$, where $1$ and $l\geqslant k+1$ is an integer, such that $\frac{\partial^r F}{\partial N^r} \bigr\vert_{\partial G}=f_r$, $r=0,1,\dots,k$, with $N$ a unit vector field defined on $\partial G$ and nontangent to $\partial G$.