Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains
Printed: Apr 2014
For parabolic linear operators $L$ of second order in divergence form,
we prove that the solvability of initial $L^p$ Dirichlet problems for
the whole range $1\lt p\lt \infty$ is preserved under appropriate small
perturbations of the coefficients of the operators involved.
We also prove that if the coefficients of $L$ satisfy a suitable
controlled oscillation in the form of Carleson measure conditions,
then for certain values of $p\gt 1$, the initial $L^p$ Dirichlet problem
associated to $Lu=0$ over non-cylindrical domains is solvable.
The results are adequate adaptations of the corresponding results for
initial $L^p$ Dirichlet problem, second order parabolic equations in divergence form, non-cylindrical domains, reverse Hölder inequalities
35K20 - Initial-boundary value problems for second-order parabolic equations