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Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains
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- Journal:
- Canadian Journal of Mathematics / Volume 66 / Issue 2 / 01 April 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 429-452
- Print publication:
- 01 April 2014
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- Article
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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\,<\,p\,<\,\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 
$p\,>\,1$, the initial ${{L}^{p}}$ Dirichlet problem associated with 
$Lu\,=\,0$ over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.
