parabolic holder space. Need some explanation of paper.

2012-10-13T15:25:22Z

Hello all,

I'm reading a paper by Almgren, Taylor, Wang called "Curvature-driven flows: a variational approach".

They define $X = \tilde{C}^{2, \alpha}(S^1 \times [0,T])$ and $Y= \tilde{C}^{0, \alpha}(S^1 \times [0,T])$ to be parabolic Holder spaces as defined in Ladyschenkaja etc.

They consider the map $F:X \to Y$ defined $$F(u) = u_t - G(x,t,u,u_x,u_{xx})$$ with $G(a,b,c,d,e)$ smooth.

They let the point $u^0 = G(x,0,0,0,0)t \in X$.

Firstly, how to know that $u^0 \in X$? Since $G$ is smooth and we're taking supremums over a compact set, the $C^2$ part of the norm is fine, but for the Holder seminorm part, how do we know that the supremums exist?

Secondly, the authors state that $F(u^0)$ converges to 0 in $Y=\tilde{C}^{0, \alpha}(S^1 \times [0,T])$ as $T \to 0$. We get $$F(u^0) = G(x,0,0,0,0) - G(x,t,G(x,0,0,0,0)t, G_{x}(x,0,0,0,0)t, G_{xx}(x,0,0,0,0)t).$$ And want to show the $Y$ norm of this tends to zero. The $Y$ norm involves supremums of $[0,T]$ and I thought since $T$ is getting smaller the supremum must get smaller but we might have a constant in time function so this is not right.

Thanks for any help.

parabolic holder space. Need some explanation of paper. - MathOverflow

parabolic holder space. Need some explanation of paper.