  The Oleinik inequality for conservation laws and Aronson-Benilan type
inequalities for porous medium or p-Laplacian equations are one-sided
inequalities that provide the fundamental features of the solution such as the
uniqueness and sharp regularity. In this paper such one-sided inequalities are
unified and generalized for a wide class of first and second order equations in
the form of $$ u_t=\sigma(t,u,u_x,u_{xx}),\quad u(x,0)=u^0(x)\ge0,\quad
t>0,\,x\in\bfR, $$ where the non-strict parabolicity ${\partial\over\partial q}
\sigma(t,z,p,q)\ge0$ is assumed. The generalization or unification of one-sided
inequalities is given in a geometric statement that the zero level set $$
A(t;m,x_0):=\{x:\rhom(x-x_0,t)-u(x,t)>0\} $$ is connected for all $t,m>0$ and
$x_0\in\bfR$, where $\rhom$ is the fundamental solution with mass $m>0$. This
geometric statement is shown to be equivalent to the previously mentioned
one-sided inequalities and used to obtain uniqueness and TV boundedness of
conservation laws without convexity assumption. Multi-dimensional extension for
the heat equation is also given.
