On the singular boundary value problem for elliptic equations

Abstract:

The operator $\mathcal {L}$ is elliptic and of second order in a domain $\Omega$ in ${R^N}$. We consider the following boundary value problem: $\mathcal {L}u = f$ in $\Omega$ and $\mathcal {B}u = 0$ on $\partial \Omega$, where $\mathcal {B} = ad/dn + \beta$ (d/dn is the conormal derivative on $\partial \Omega$). The coefficient $\alpha$ is assumed to be nonnegative. However, $\alpha$ may vanish partly on $\partial \Omega$. Then the regularity of the weak solutions for the above problem is shown by the variational method.