Analysis of a singular elliptic problem by wavelets. (Q811652)

Consider the equation \(-\nabla(a\nabla u)+u=f\) where the function \(a\) may vanish in some regions. If \(a\) is smooth and has zeros of order strictly larger than 2, the following optimal estimates of Green's function are proved. \(|\partial_ x^ \alpha\partial_ x^ \beta G(x,y)|\leq C/| x-y|^{n- 2+\alpha+\beta}\sup(\sqrt{a(x)a(y)}| x-y|^ 2)\). They are obtained through an ``almost diagonalization'' of the operator on a wavelet basis.