Just the simple ODE with Lipschitz coefficient $a$ \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} We know that the existence and uniqueness holds and the solution is in $C^1$. Does the second derivative of the solution exist (in distributional sense perhaps)? How do I show it? So the solution is in $W^{2,p}$, what's $p$?
second derivative of solutions to ODE with Lipschitz coefficients
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real-analysis
analysis
ordinary-differential-equations
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sobolev-spaces