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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1997 • Question 6

Question

Math 6420 • PDE Fall 1997 • Question 6

Derive the boundary value problem for a minimizer \(w_0\) of the functional \[ \min _{\substack {w\in C^2[0,1]\\ w(0)=0}} \frac {\int _0^1\left (\phi (x)(w'')^2+w^2\right )\,dx} {\int _0^1\phi (x)(w')^2\,dx+\beta w(1)^2} \] where \(\phi (x)>0\), \(\forall x\in [0,1]\), \(\beta \) is a real parameter.

Raw LaTeX

Derive the boundary value problem for a minimizer \(w_0\) of the
functional
\[
  \min_{\substack{w\in C^2[0,1]\\ w(0)=0}}
  \frac{\int_0^1\left(\phi(x)(w'')^2+w^2\right)\,dx}
       {\int_0^1\phi(x)(w')^2\,dx+\beta w(1)^2}
\]
where \(\phi(x)>0\), \(\forall x\in[0,1]\), \(\beta\) is a real parameter.

Topic Tags

Boundary Value Problems Dual Spaces and Bounded Linear Functionals Finite Difference Boundary Value Problems Linear Programming and Optimization