Following work of K.\ Taira we consider the boundary value problem
$$Au=f\text{ in } X,\qquad Lu=g \text{ on }\partial X,$$
where $X$ is a compact manifold with boundary,$A$ is a strongly elliptic second order operator which in local coordinates is of the form
$$A=\sum_{jk}a^{jk}\partial_{x_j}\partial_{x_k}+\sum b^j\partial_{x_j} + c$$
with real coefficients $a^{jk}=a^{jk}, b^j,c$ in the H?der class $C^\tau$, $\tau>2$.
We require that $\sum a^{jk}\xi_j\xi_k\ge \alpha |\xi|^2$ for some $\alpha>0$ and $0\not\equiv c\le0$.
The boundary condition $L$ is assumed to be of the form
$$Lu = \mu_0\gamma_0u + \mu_1\gamma_1u,$$
where $\gamma_0$ is the evaluation map at the boundary and $\gamma_1$ is the evaluation of the exterior normal derivative at the boundary.
The $C^\tau$-functions $\mu_0$ and $\mu_1$ on $\partial X$ are supposed to be nonnegative with $\mu_0+\mu_1$ strictly positive everywhere.
Using the calculus of pseudodifferential operators with symbols of limited regularity we then show the solvability of the boundary value problem
in various classes of Sobolev and Zygmund spaces with regularity depending on the smoothness $\tau$ of the coefficients.
We also study the resolvent in suitable sectors of the complex plane.
\hfill (joint work with M. Hassan Zadeh)
Seminar on Analysis at University of Tsukuba