Representação de soluções homogêneas contínuas de campos vetoriais no plano
Resumo
In this work we study conditions for the validity of the analogue of Mergelyan’s
theorem for continuous solutions of a type of locally integrable vector field.
On a domain in the plane, we consider a vector field L that has a first
integral on of the form Z(x, t) = x + i'(x, t), where '(x, t) is a smooth, realvalued
function. Given a continuous solution u of Lu = 0 on
, our first objective was to find conditions on
and Z for the validity of the factorization
u = U Z,
where U 2 C0(Z ()) \ H(int{Z ()}).
We will next study this factorization on the closure of . We assume that
u 2 C0( ) and that the boundary of is real analytic, then we show in which
cases the condition Z(p1) = Z(p2) implies that u(p1) = u(p2), for p1, p2 2 . The
cases are divided according to the geometry of the boundary in the points p1 and
p2. When is a compact set and u = U Z on , we obtain that u is uniformly
approximated by polynomials of Z on .