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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 19

Question

Math 6410 • ODE Fall 1993 • Question 19

Let \(G=(0,1)\), \(V=H_0^1(G)\), and \(H=L^2(G)\). Define \[ a(u,v)=\int _0^1 \partial u\,\partial \overline {v}\,dx. \] Use what you know about coercive and elliptic sesquilinear forms and the spectral theorem for operators defined by them to find a basis for \(L^2(G)\). Be sure to present a fairly complete discussion.

Raw LaTeX

Let \(G=(0,1)\), \(V=H_0^1(G)\), and \(H=L^2(G)\). Define
\[
a(u,v)=\int_0^1 \partial u\,\partial\overline{v}\,dx.
\]
Use what you know about coercive and elliptic sesquilinear forms and the spectral
theorem for operators defined by them to find a basis for \(L^2(G)\). Be sure to
present a fairly complete discussion.

Topic Tags

Elliptic Equations Orthonormal Bases and Total Sets Sobolev Spaces and Energy Estimates Spectral Theory of Operators