Uniqueness of Lipschitz function satisfying differential equation - MathOverflow

Uniqueness of Lipschitz function satisfying differential equation

2012-09-07T15:00:38Z

I have a Lipschitz function $X=X(t)$ with the property that, at all points $t$, the right derivative $\lim_{\epsilon \downarrow 0} \epsilon^{-1}(X(t+\epsilon)-X(t))$ exists and is given by $f(X)$ for some (discontinuous) function $f$. (Of course, at all regulat points, i.e. almost everywhere, this is then the ordinary derivative.) Is it true that such $X$ is necessarily unique?

Of course, if I just specify that a Lipschitz function satisfies $X'(t) = f(X(t))$ at all regular points, the solution doesn't have to be unique; but I'm requiring that this hold for the right derivative at all times, which intuitively seems like it ought to work.

Answer by Thomas Richard for Uniqueness of Lipschitz function satisfying differential equation

2012-09-07T15:18:14Z

I suppose you want uniqueness assuming some initial condition.

Even when $F$ is continuous and $X$ is $C^1$ you don't have uniqueness. For $t\geq 0$ $X(t)=0$ and $X(t)=t^2$ are both solution of $X'=2\sqrt{X}$.

Maybe with stronger assumptions on $F$ you can do something assuming only that $X$ is Lipschitz but I'm not sure. Without the Lipschitz condition you don't have uniqueness assuming that the differential equation is satisfied a.e., Cantor stair is a counterexample.

Edit : The uniqueness statement you want is true when $F$ is Lipschitz. In this case, with your assumptions, $X$ has a right derivative which is continuous. From that it's not too complicated to show that $X$ has a derivative everywhere which is continuous. And then the classical Cauchy-Lipschitz applies.

Answer by Bazin for Uniqueness of Lipschitz function satisfying differential equation

2012-09-08T07:58:49Z

You have a logarithmic extension of the classical Cauchy-Lipschitz theorem: the equation $$\dot x=f(x),\quad x(0)=x_0\tag{ODE}$$ has a unique solution if $f$ has a modulus of continuity $\omega$ ($ \vert f(x+h)-f(x)\vert\le \omega(\vert h\vert) $) such that $$ \int_0^a\frac{dr}{\omega(r)}=+\infty,\quad \text{for some positive $a$}. $$ On the top of this $\omega$ should be positive increasing, $\omega(0_+)=0$. This is of course satisfied by $\omega(r)=Cr$ (Lipschitz case) but also by $$ \omega(r) =Cr\ln(1/r),\quad\text{the so-called Log-Lipschitz case.} $$ You have as well $r\ln(1/r)\ln(\ln(1/r))$ and so on.

Let me add a remark on the 1D case. When $x(t)\in \mathbb R^1$, $f$ continuous and $f(x_0)\not=0$, then (ODE) has the uniqueness property (just separate the variables). This is of course compatible with the counterexample in the previous answer where $f(x_0)=0$.