Estimates of the derivatives of solutions of integral equations of potential theory (Q1823379)

The solution of the elliptic operator \[ (Lu)(n)=(\partial /\partial x_ i)(a_{ij}(n)\partial u/\partial x_ j)+a_ i(n)\partial u/\partial x_ i+c(n)u,\quad u\in C^ 2(D), \] can be reduced to the solution of two types of equations, corresponding to the Dirichlet and Neumann's conditions for the given operator. Notations: \(r=(x,y)\) is a point in \(R_ 2\); D is the definition domain for the solution u; D is bounded by a curve \(\Sigma \in C^{\infty}\), which can have a finite number of points with large curvature. The author studies the behaviour of the solution and its derivatives in the neighborhood of such points. His conclusion is that both solutions and their derivatives have a large growth around the points of the boundary with a large curvature. This result is in good agreement with the numerical data.