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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2007 • Question 7

Question

Math 6410 • ODE Winter 2007 • Question 7

Let \(U\subset \mathbb {R}^n\) be open and bounded, with smooth boundary. For \(T>0\), denote \(U_T=U\times (0,T]\). Let \(\Gamma _T=\overline {U_T}-U_T\) (closure taken in \(\mathbb {R}^n\times \mathbb {R}\)). Prove that functions \(u(x,t)\) satisfying \(u\in C^{2,1}(U_T)\cap C(\overline {U_T})\) and \[ \begin {aligned} \Delta u-u &\ge u_t, && \text {in } U_T,\\ u &\ge 0, && \text {in } \overline {U_T}, \end {aligned} \] also satisfy the maximum principle \[ \max _{\overline {U_T}} u=\max _{\Gamma _T} u. \]

Raw LaTeX

Let \(U\subset\R^n\) be open and bounded, with smooth boundary. For \(T>0\),
denote \(U_T=U\times(0,T]\). Let \(\Gamma_T=\overline{U_T}-U_T\) (closure taken in
\(\R^n\times\R\)). Prove that functions \(u(x,t)\) satisfying
\(u\in C^{2,1}(U_T)\cap C(\overline{U_T})\) and
\[
\begin{aligned}
  \Delta u-u &\ge u_t, && \text{in } U_T,\\
  u &\ge 0, && \text{in } \overline{U_T},
\end{aligned}
\]
also satisfy the maximum principle
\[
  \max_{\overline{U_T}} u=\max_{\Gamma_T} u.
\]

Topic Tags

Boundary Value Problems and Sturm-Liouville Theory Distributions and Weak Solutions Fundamental Solutions Maximum Principles