Discussion Board Lecture 14

Dear Jamil,

oh yes, it should be "Section 1.1" and not "Section 11" – thank you!

As for the conditions on page 164, we did mean "condition (ii)", although you're right, of course, that condition (iv) is also needed. We thought that the latter is obvious, so we only meantioned the former, but probably we should just have mentioned both conditions.

The point is this: In order to show measurability of $ t \mapsto a(t,u(t),v(t)) $, we need to approximate $ u $ and $ v $ by simple functions. Condition (iv) yields measurability for simple $ u,v $, and condition (ii) makes the approximation procedure work.

Best wishes,  Hendrik

Dear ISem team,

in Theorem 14.16, p. 165, the inequality

$ |E(v,w)|\leq \lVert v\rVert|_{\mathcal{V}}\lVert w'\rVert|_{L_{2}\left(0,\tau;\mathcal{V}^{\ast}\right)}+M \lVert v\rVert|_{\mathcal{V}} \lVert w\rVert|_{\mathcal{V}} $

should be replaced by

$ |E(v,w)|\leq C^{2}\lVert v\rVert|_{\mathcal{V}}\lVert w'\rVert|_{L_{2}\left(0,\tau;\mathcal{V}^{\ast}\right)}+M \lVert v\rVert|_{\mathcal{V}} \lVert w\rVert|_{\mathcal{V}} $

where $ C\geq 0 $ is the embedding constant from $ V\mathop{\hookrightarrow}\limits^{d}H $ due to scalar product of $ H $ which appears in the definition of $ E. $

Best regards, Karsten

Dear Karsten,

I don't think that this is a misprint. We consider elements of $ H $ as elements of the antidual $ V^* $ via $ H\ni f\mapsto(f|\,\cdot\,)_H=:\eta_f $. We then obtain in detail

$ \left|\int_0^\tau(v(t)|w'(t))_H\,\mathrm dt\right|\leq\int_0^\tau|(w'(t)|v(t))_H|\,\mathrm dt =\int_0^\tau|\langle\eta_{w'(t)},v(t)\rangle|\,\mathrm dt $

$ \leq\int_0^\tau\|\eta_{w'(t)}\|_{V^*}\cdot\|v(t)\|_{V}\,\mathrm dt \leq\left(\int_0^\tau\|\eta_{w'(t)}\|_{V^*}^2\,\mathrm dt\right)^{\frac{1}{2}}\cdot\left(\int_0^\tau\|v(t)\|_{V}^2\,\mathrm dt\right)^{\frac{1}{2}} =\|v\|_{\mathcal V}\cdot\|w'\|_{L_2(0,\tau;V^*)} $.

Moreover, I don't see how to derive the second inequality displayed in your posting. Of course, you may use Cauchy-Schwarz and the embedding $ V\hookrightarrow H $ with constant $ C $ to deduce $ |(v(t)|w'(t))_H|\leq C\|v(t)\|_V\cdot\|w'(t)\|_H $. But then I am stucked and I don't see how to proceed as we only know $ \|w'(t)\|_{V^*}\leq C\|w'(t)\|_H $. Further note that the estimate $ \|w'(t)\|_H\leq C\|w'(t)\|_V=C\|(w'(t)|\,\cdot\,)_V\|_{V^*} $ does not help since we identify $ w'(t) $ with $ (w'(t)|\,\cdot\,)_H $ and not with $ (w'(t)|\,\cdot\,)_V $ (or put it another way, the canonical identification of $ V $ with its antidual is in general not compatible with the Gelfand triple $ (V,H,V^*) $).

Best wishes, Heiko

Dear Heiko,

you are right. I identified $ w'(t)\in V $ with $ \left( w'(t)|\cdot\right)_{V}\in V^{\ast} $ by habbit and not with $ \left( w'(t)|\cdot\right)_{H}\in V^{\ast} $. Thanks for pointing that out.

Best wishes, Karsten