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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1996 • Question 7

Question

Math 6420 • PDE Fall 1996 • Question 7

Compute the Euler equation for the minimizer of the following variational problem. Check Legendre’s necessary condition. Discuss the dependence of the minimizer on the parameter \(a\). \begin{equation} \min _{w\in W}\int _0^1\left ((xw')^2+aw^2w'-x\ln w\right )\,dx, \qquad W=\{w:\ w\in H_2(0,1);\ w(0)=1\}. \end{equation}

Raw LaTeX

Compute the Euler equation for the minimizer of the following
variational problem. Check Legendre's necessary condition. Discuss the dependence of the
minimizer on the parameter \(a\).
\begin{equation}
  \min_{w\in W}\int_0^1\left((xw')^2+aw^2w'-x\ln w\right)\,dx,
  \qquad W=\{w:\ w\in H_2(0,1);\ w(0)=1\}.
\end{equation}

Topic Tags

Finite Speed of Propagation and Domain of Dependence Linear Programming and Optimization Zero-Stability and Root Conditions