| Volume 7, Number 4, 2017, Pages 1323-1335 DOI：10.11948/2017081 |
| Phragm\''{e}n-Lindel\"{o}f alternative for the Laplace equation with dynamic boundary conditions |
| Mari Carme Leseduarte,Ramon Quintanilla |
| Keywords:Phragm\''{e}n-Lindel\"{o}f alternative, dynamic boundary conditions, Laplace equation. |
| Abstract: |
| This paper investigates the spatial behavior of the solutions of the Laplace equation on a semi-infinite cylinder when dynamical nonlinear boundary conditions are imposed on its lateral side. We prove a Phragm\''{e}n-Lindel\"{o}f alternative for the solutions. To be precise, we see that the solutions increase in an exponential way or they decay as a polynomial. To give a complete description of the decay in this last case we also obtain an upper bound for the amplitude term by means of the boundary conditions. In the last section we sketch how to generalize the results to a system of two elliptic equations related with the heat conduction in mixtures. |
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