# 臨時解析セミナー（2月18日）

（曜日が通常と異なりますので，ご注意ください．）

$Au=f\text{ in } X,\qquad Lu=g \text{ on }\partial X,$
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 Htlder 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)

【 場所 】 自然学系Ｄ棟　５０９教室