Math 6410 • ODE

Consider the space of real-valued functions \(H^2(0,1)\) and define the quadratic form \[ a(u,v)=\int _0^1 u''v''\,dx+b\int _0^1 u'v'\,dx+c\int _0^1 uv\,dx, \] where \(b\) and \(c\) are positive constants. Show that \(a\) is a form to which the Lax-Milgram theorem may be applied and state the result thus obtained. Let \(f\in L^2(0,1)\), and restrict the form \(a\) to the space \(H^2_0(0,1)\). Show that \(f\) may be thought of as a continuous linear functional on \(H^2_0(0,1)\), and apply the Lax-Milgram theorem to the pair \(a,f\). Determine a boundary value problem which is solved in a distributional sense by the Lax-Milgram solution and show that this solution is at least three times continuously differentiable.