| Contents |
| We are concerned with the following Dirichlet problem in a separable Hilbert space $H$
$$
\left\{ \begin{array}{l}
u_t(t,x)=\frac12\;\mbox{\rm Tr}\;[u_{xx}(t,x)]+\langle Ax,u_x(t,x) \rangle,\quad t>0,\;x\in\mathcal O,\\
\\
u(t,x)=0, \quad t>0,\;x\in\partial\mathcal O,\\
\\
u(0,x)=\varphi(x),\quad x\in \overline{\mathcal O},
\end{array}\right.\eqno{(1)}
$$
where $\mathcal O$ is an open subset of $H$ of the form $\mathcal O=G^{-1}((-\infty,0))$ of boundary $\partial\mathcal O=G^{-1}(0)$, where $G$ is a given Borel function.
We shall present some representation formulas for the gradient of $u(t,x)$ which only involves $\varphi$. |
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