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Question

Math 6410 • ODE

Fall 2001 • Question 8

Question

Math 6410 • ODE Fall 2001 • Question 8

Let \(G\) be a bounded open set in \(\mathbb {R}^N\) with smooth boundary. State and indicate a proof of Poincaré’s inequality for functions \(u\in H^1_0(G)\). Using the inequality show that \[ \norm {u}_0^2=\int _G \lvert \nabla u\rvert ^2\,dx \] defines a norm equivalent to the \(H^1\) norm. Consider the form \[ a(u,v)=\int _G \nabla u\cdot \overline {\nabla v} +\int _G c(x)\cdot \nabla u\,\overline {v}\,dx, \tag {7} \] where the components of the vector function \(c\) belong to \(L^\infty (G)\).

Provide a condition on the function \(c\) in order that \(a\) is a sesquilinear form satisfying the conditions of the Lax-Milgram theorem. Determine a boundary value problem solved by the Lax-Milgram solution.

Let \(G\) be a bounded open set in \(\R^N\) with smooth boundary. State and indicate a proof
of Poincar\'e's inequality for functions \(u\in H^1_0(G)\). Using the inequality show that
\[
  \norm{u}_0^2=\int_G \lvert\nabla u\rvert^2\,dx
\]
defines a norm equivalent to the \(H^1\) norm. Consider the form
\[
  a(u,v)=\int_G \nabla u\cdot\overline{\nabla v}
  +\int_G c(x)\cdot\nabla u\,\overline{v}\,dx,
  \tag{7}
\]
where the components of the vector function \(c\) belong to \(L^\infty(G)\).

Provide a condition on the function \(c\) in order that \(a\) is a sesquilinear form
satisfying the conditions of the Lax-Milgram theorem. Determine a boundary value problem
solved by the Lax-Milgram solution.

Topic Tags

Boundary Value Problems Boundary Value Problems and Sturm-Liouville Theory Poincare Maps and Periodic Solutions Poincare-Bendixson Theorem