Let \( \Omega \subseteq \mathbb{R}^d \) be a bounded domain and \( u \in C^{(2,1)}(\Omega_T) \cap C^0(\overline{\Omega_T}) \) with
\[ \partial_t u - \Delta u = 0. \]If there is a \( (x_0, t_0) \in \Omega_T \) with
\[ \max_{\Omega_T} u(x,t) = u(x_0, t_0), \]then \( u \) is constant on \( \Omega_T \). [1, 2.3 Theorem 4]
Remarks
- Using the mean value property , \( u \) is constant on a heat ball . But it is complicated to deduce that it is constant everywhere.
See also Link to heading
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.