The behavior of generalized solutions of the Dirichlet problem for quasilinear elliptic divergence form equations of second order near a conical point (Q810735)

Consider the Dirichlet problem \[ \sum^{n}_{i=1}\frac{d}{dx_ i}a_ i(x,u,u_ x)=a(x,u,u_ x),\quad x\in G;\quad u(x)=0,\quad x\in \partial G, \] where \(G\subset {\mathbb{R}}^ n\) is an open region with boundary \(\partial G\) which is smooth everywhere except at the origin which is a conical point. Assuming that the equation is elliptic and the coefficients \(a_ i(x,u,p)\) satisfy some minimal smoothness conditions and growth conditions with respect to p, the author shows that a continuous generalized solution of the problem behaves in the neighbourhood of the origin as \(u(x)=O(| x|^{\lambda})\), \(\nabla u(x)=O(| x|^{\lambda -1})\) giving an exact value of \(\lambda >0\), and the second derivatives are square integrable with an exact weight function.