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Boundary value problems and duality between $L^p$ Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains

Universitat de Barcelona
Publication Date
  • Mathematics


In this paper we consider general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders $\Omega = \{(x_0,x,t)\in \mathbb{R}\times \mathbb{R}^{n-1}\times \mathbb{R} : x_0 > A(x,t)\}.$ We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in $L^p$ and $L^p_{1,1/2}$ (the parabolic Sobolev space having tangential (spatial) gradients and half a time derivative in $L^p$) for $p\in (2 -\epsilon, 2 +\epsilon )$ assuming that $A(x, \cdot)$ is uniformly Lipschitz with respect to the time variable and that $\parallel D^t_{1/2}A\parallel_\ast\leq \epsilon_0< \int$ for $\epsilon_0$ small enough $\parallel D^t_{1/2}A\parallel_\ast$ is the parabolic BMO-norm of a half-derivative in time). We also prove a general structural theorem (duality theorem between Dirichlet and regularity problems) stating that if the Dirichlet problem is solvable in $L^p$ with the relevant bound on the parabolic non-tangential maximal function then the regularity problem can be solved with data in $L^q_{1,1/2}(\partial\Omega)$ with $q^-1 + p^-1 = 1$. As a technical tool, which also is of independent interest, we prove certain square function estimates for solutions to the system.

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