Timoshenko beam coercivity: $A((w,\beta),(v,\eta)):=\int^1_0\beta'\eta'dx+\int^1_0(w'-\beta)(v'-\eta)dx=\int^1_0fvdx \forall(v,\eta)\in H^1_0(0,1)^2$

Question

I want to show coercivity of the following bilinear form from Timoshenko beam theory:

For t<<1 and $(w,\beta)\in H^1_0(\Omega)$ with $\Omega=(0,1)\times(-t/2,t/2)$

$A((w,\beta),(v,\eta)):= \displaystyle\int^1_0\beta'\eta'dx+\int^1_0(w'-\beta)(v'-\eta)dx=\int^1_0fvdx \forall(v,\eta)\in H^1_0(0,1)^2$

Showing that $A((w,\beta),(w,\beta))\ge0$ is trivial as $A$ just consists of L2 norms but I need to show that A is coercive. I do not know how to continue.

Answer 1

Coercivity is shown e.g. in this paper: Discretization by finite elements of a model parameter dependent problem, Douglas N. Arnold, Numerische Mathematik 37, pages 405–421 (1981).

Hint: use Poincaré inequality (also known as Wirtinger's inequality in 1D) and the triangle inequality.