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Linear parabolic problems with dynamic boundary conditions in spaces of Hölder continuous functions. (English) Zbl 1338.35211

The author considers a mixed linear parabolic problem with a strongly elliptic operator and with dynamic boundary conditions \[ \begin{cases} D_t u(t,\xi)-A(\xi,D_\xi)u(t,\xi)=f(t,\xi), & t\in(0,T), \xi\in\Omega,\cr D_t u(t,\xi')+B(\xi',D_\xi)u(t,\xi')=h(t,\xi'), & t\in(0,T), \xi'\in\partial\Omega,\cr u(0,\xi)=u_0(\xi), & \xi\in\Omega \end{cases} \] in a smooth domain \(\Omega\). For each \(\beta\in(0,1)\) the author determines necessary and sufficient assumptions on the data, so that there exist a unique solution \(u\in C^{1+\beta/2, 2+\beta}((0,T)\times \Omega)\), with \(D_t u|_{(0,T)\times \partial\Omega}\) bounded with values in \(C^{1+\beta}(\partial\Omega)\).

MSC:

35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
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