For the solution of the ODE $dy/dt=f(t,y)$, $f$ has to be Lipschitz for all $t$. So, if $f$ is a function that is not differentiable with respect to $y$ but Lipschitz, what can I say about $f_y$? Can I estimate some norm of it? I can't say $||f_y||_C$ because $f$ is not $C^1$, however, I can say that $f_y$ is defined in a weak sense and then this weak derivative is bounded (by the Lipschitz property) and I can estimate $||f_y||_{\mathbb{L}^{\infty}}$ and have it bounded. Is that correct?
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Yes, every Lipschitz function (on an open subset $\Omega\subset \mathbb R^n$) has a weak derivative which belongs to $L^\infty(\Omega)$. The $L^\infty$ norm of the derivative is bounded by the Lipschitz constant of the function. Under appropriate assumptions on the geometry of $\Omega$ one can reverse this implication and obtain the global Lipschitz condition from the boundedness of the derivative. |
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