A few natural extensions of the regularity of a very weak solution

Abstract

We consider a linear operator $L_m$ with variable coefficients of order $2m$ and we study the regularity of the very weak solution $u$ integrable in a bounded open smooth set $\Omega$, \[ \int_\Omega uL^*_m {\varphi} \,{\rm dx} = \int_\Omega {\varphi} d\mu\quad\forall\,{\varphi}\in C^{2m}( \overline \Omega) \] with $ \frac{{{\partial}}^j{\varphi}}{{{\partial}}\nu^j}=0$ on the boundary ${{\partial}}\Omega$ for $j{\leqslant} m-1$, where $L^*_m$ is the adjoint operator of $L_m$ and $\mu$ is in the space of weighted bounded Radon measures $M^1(\Omega,dist(x,{{\partial}}\Omega)^m)$. In particular, we show that the solution $u$ and all its derivatives of order $|{\gamma}|,\ |{\gamma}|{\leqslant} m-1,$ are in Lorentz spaces. If the measure on the right-hand side belongs to a smaller space such as $$ M^1(\Omega, dist(x,{{\partial}}\Omega)^{m-1+a}), \quad 0{\leqslant} a<1, $$ then all its derivatives o f order $|{\gamma}|,\ | {\gamma}|{\leqslant} m,$ are in Lorentz spaces.