Topic
Calculus of Variations
Courses
Courses whose questions include this topic tag.
Questions
This topic is tagged for 5 questions
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Fall 2025 • Question 1
Let \(U\) be a bounded domain in \(\mathbb {R}^d\) and \(\Gamma \) be a closed subset of \(\partial U\). Consider the energy
\[ I[v]=\frac 12\int _U \left (|Dv|^2+u^2\right )\,dx \]on the admissible class
\[ \mathcal {A}=\{v\in C^2(\overline {U})\cap C^1(\overline {U}): u|_{\Gamma }=g\},\quad g\in C(\partial U). \]Suppose that \(u\) is a minimizer for the calculus of variations problem
\[ I[u]=\min _{v\in \mathcal {A}} I[v] \]find and justify the PDE boundary value problem solved by \(u\). Show that there is at most one solution of this PDE in the admissible class \(\mathcal {A}\).
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Fall 2009 • Question 5
- Let \(f(x)=|x|\). Find its first and second derivatives using operational calculus, that is, the theory of distributions.
- Using Green’s functions, find the solution to \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=u(1)=0. \] Use the reproducing property of the delta distribution to formally verify that your solution satisfies the equation.
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Fall 1999 • Question 7
Derive the variation of constants formula (using the fundamental solution matrix \(\Phi \)) for solutions of \[ x'=A(t)x+h(t). \tag {2.5} \]
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Fall 1999 • Question 8
Let \(\Phi (0)=\text {identity}=\Psi (0)\). Give the variation of constants formula derived above in terms of \(\Phi \) and \(\Psi \). Do you see any computational advantages in this form of the variation of constants formula.
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Fall 1993 • Question 5
Assume the conditions of d) and let \(B(t)\) be a continuous matrix function with \[ \int _0^\infty \lVert A(t)-B(t)\rVert \,dt <+\infty . \] Let \(\Psi \) be a solution of \[ \Psi '=B(t)\Psi . \] Show that \(\Psi \) must be uniformly bounded on \([0,\infty )\).
Hint: \(\Psi \) also solves \[ \Psi '=A(t)\Psi +(B(t)-A(t))\Psi . \] Use variation of constants.