ODE with non-continuous right hand side My brother asked me a question which I didn't know the answer to.
Are there theorems about existence, uniqueness and stability of solutions of ODEs of the followin type
$$
\frac{d^2 y}{dt^2} = f(t,y,\frac{dy}{dt})H(g(y)), 
$$
where $f$ and $g$ are Lipschitz functions and $H$ is the Heaviside function?
If there are no general theorems that could apply, how does one analyze these problems for given $f,g$?
 A: Here is one possible approach.
Choose    smooth approximations  of  $H$. For example   consider a smooth function
$$ \eta:\mathbb{R}\to [0,\infty)$$
such that
$$\eta (t)=0,\;\;\forall t\leq 0$$
$$\eta(t)=1,\;\;\forall t\geq 1.$$
Now define 
$$H_\varepsilon(t)= \eta(t/\varepsilon)$$
Then
$$\lim_{\varepsilon\searrow 0} H_\varepsilon (t) =H(t),\;\;\forall t\neq 0$$.
Now investigate the above equation with $H$ replaced with $H_\varepsilon$ an hope that you  understand what happens     when $\varepsilon \searrow 0$.
On the other hand, there is a large class of   evolution equations with discountinuous  right-hand side described by the so called maximally monotone operators.   For  2nd  order equations    such as yours, one of the most influential papers is

Barbu, Viorel Existence theorems for a
  class of two point boundary problems. 
  J. Differential Equations 17 (1975),
  236–257. 

For the  general theory of     evolution equations involving  maximal monotone operators the best sources are

Barbu, Viorel  Nonlinear
  differential equations of monotone
  types in Banach spaces.  Springer
  Monographs in Mathematics. Springer,
  New York, 2010. x+272 pp.
Brezis, Haim  Opérateurs maximaux
  monotones et semi-groupes de
  contractions dans les espaces de
  Hilbert. (French)  North-Holland
  Mathematics Studies, No. 5. Notas de
  Matemática (50). North-Holland
  Publishing Co., Amsterdam-London;
  American Elsevier Publishing Co.,
  Inc., New York, 1973. vi+183 pp

A: Your brother needs to provide more specific questions that he wants answered. But here are a few thoughts.
Suppose you want to prove existence and uniqueness of the initial value problem. The initial data will imply a value for $H(g(y))$. You can use the standard theorem to solve the initial value problem and extend the solution until $g(y) = 0$. At that point, check the sign of $g'(y)y'$. If it is nonzero, then you can just use the limiting values of $y$ and $y'$ to solve the initial value problem starting from that point. The fact that $y$ and $y'$ are being extended continuously implies that the ODE is being solved at least in the weak sense, and that's usually all that is needed.
If $g(y)$ does not change sign when it vanishes, things get more complicated. But that's something your brother should worry about only if he really does encounter it.
