On the regularity of solutions of the Dirichlet problem for elliptic integro-differential operators: A counterexample (Q1091563)

This work is concerned with the regularity of solutions of the integrodifferential equation \[ Au+\int \{\phi (x+z)-\phi (x)-\nabla \phi (x)\cdot z\mathbf{1}_{| z| \leq 1}\}\mu (x,dz)=f\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] where A is a second order linear elliptic operator on \(\Omega\) in \(\mu\) (x,.) is a nonnegative measure on \(R^ N\setminus 0\) such that \(\int_{| z| \leq 1}| z|^ 2\mu (x,dz)+\int_{| z| >1}| z| \mu (x,dz)<\infty.\) More precisely it is shown in the case \(N=1\) that there do not exist solutions in \(W^{2,p}\cap C^ 1_ b\).