Abstract
A weak solution of the conservation law ut+j(ux)=0 is a locally Lipschitzian function u which satisfies the equation almost everywhere. We treat a boundary value problem and also a mixed initial-boundary value problem associated with the equation where the initial and boundary data are convex functions. The convexity hypothesis makes it possible to apply the Fenchel theory of conjugate convex functions to the problem. This leads to a construction of solutions rather than to a proof of their existence; the solutions so constructed turn out to be stable.
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