  The paper studies a PDE model for the growth of a tree stem or a vine, having
the form of a differential inclusion with state constraints. The equations
describe the elongation due to cell growth, and the response to gravity and to
external obstacles.
  The main theorem shows that the evolution problem is well posed, until a
specific "breakdown configuration" is reached. A formula is proved,
characterizing the reaction produced by unilateral constraints. At a.e. time t,
this is determined by the minimization of an elastic energy functional under
suitable constraints.
