Question

Define the seminorm on the space $S=[0,1]\times[0,T]$ $$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$ Define the norms on the same space $$\lVert u \rVert_{C^{0, \alpha}} = \lVert u \rVert_{C^0} + \mid u\mid_{\alpha}$$ and $$\lVert u \rVert_{C^{2, \alpha}} = \lVert u \rVert_{C^0} +\lVert u_x \rVert_{C^0}+\lVert u_{xx} \rVert_{C^0}+\lVert u_t \rVert_{C^0}+ \mid u_{xx}\mid_{\alpha} + \mid u_t\mid_{\alpha}.$$

Suppose that $\lVert u \rVert_{C^2, \alpha} \leq C$ where $C$ is a constant. Let $a, b, c \in C^{0, \alpha}$. How can I show that $$\lVert au_{xx} + bu_x + cu\rVert_{C^{0, \alpha}} \leq K\lVert u \rVert_{C^{2, \alpha}}$$ for some constant $K$?

Or equivalently, want to show that $$\sup_{\lVert u \rVert_{C^{2,\alpha}} \leq C_1}\lVert au_{xx} + bu_x + cu\rVert_{C^{0, \alpha}} \leq K_1$$

(ALL the above norms are over the compact set $S$).

Thanks for any help

Answer 1

If $f,g\in C^{\alpha}(S)$, then for all $(x,t),(x',t')\in S$, we have \begin{align} |f\cdot g(x,t)-f\cdot g(x',t')|&=|f(x,t)(g(x,t)-g(x',t'))+g(x',t')f(x,t)-f(x',t')g(x',t')|\\ &\leq \lVert f\rVert_{\infty}|g(x,t)-g(x',t')|+\lVert f\rVert_{\infty}|f(x,t)-f(x',t')|, \end{align} hence \begin{equation}[f\cdot g]{\alpha}\leq \lVert f\rVert{\infty}[g]{\alpha}+\lVert g\rVert{\infty} [f]{\alpha}. \end{equation} We deduce that \begin{align} \lVert au{xx}+bu_x+cu\rVert_{C^{0,\alpha}}&\leq \max(\lVert a\rVert_{\infty},\lVert b\rVert_{\infty},\lVert c\rVert_{\infty})(\lVert u_{xx}\rVert_{\infty}+\lVert u_x\rVert_{\infty}+\lVert u\rVert_{\infty})\\ &+\max(\lVert a\rVert_{\infty},\lVert b\rVert_{\infty},\lVert c\rVert_{\infty})([ u_{xx}]_{\alpha}+[u_x]{\alpha}+[u]{\alpha}) \\ &+\max([a]{\alpha},[b]{\alpha},[c]{\alpha})(\lVert u{xx}\rVert_{\infty}+\lVert u_x\rVert_{\infty}+\lVert u\rVert_{\infty})\\ &\leq \max(\max([a]{\alpha},[b]{\alpha},[c]{\alpha}),\max(\lVert a\rVert{\infty},\lVert b\rVert_{\infty},\lVert c\rVert_{\infty}))\lVert u\rVert_{C^{2,\alpha}}. \end{align} We can get an equivalent norm on $C^{2,\alpha}(S)$ defining \begin{equation}\lVert u\rVert:=[u]{\alpha}+\lVert u\rVert{\infty}+[u_t]_{\alpha}+\lVert u_t\rVert_{\infty}+[u_x]_{\alpha}+\lVert u_x\rVert_{\infty}+[u_{xx}]{\alpha}+\lVert u{xx}\rVert_{\infty}. \end{equation} This can be shown using mean value theorem.