This work studies conditions that insure the existence of weak boundary values for solutions of a complex, planar, smooth vector field $L$. Applications to the F. and M. Riesz property for vector fields are discussed.

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem ${\partial }_{t}u+Xu=f,u_{{|}_{t=0}}=g,$ where $X$ is the vector fieldwith a boundedness condition on the divergence of each vector field $a_1,a_2$. This model was studied in the paper [LL] with a $W^{1,1}$ regularity assumption replacing our $BV$ hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of...

We construct travelling wave graphs of the form $z=-ct+\phi \left(x\right)$, $\phi :x\in {ℝ}^{N-1}\mapsto \phi \left(x\right)\in ℝ$, $N\ge 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\ge c_0$) with prescribed asymptotics. For any $1$-homogeneous function ${\phi }_{\infty }$, viscosity solution to the eikonal equation $|D{\phi }_{\infty }|=\sqrt{{(c/c_0)}^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by ${\phi }_{\infty }$. We also describe ${\phi }_{\infty }$ in terms of a probability measure on ${\S }^{N-2}$.