Timeline for Existence of Green's function and the Dirichlet problem
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 24, 2013 at 7:25 | comment | added | Gatz' | I understand. But does the existence of Green's function for $U$ is equivalent to the existence of a barrier for the Dirichlet problem at every point of the boundary $\partial U$ ? My concern was whether there can be an open set $U$ (that does not satisfy the exterior sphere condition) for which Green's function exists but for which the associated Dirichlet problem has no solution ? | |
| Oct 24, 2013 at 5:52 | comment | added | username | @Gatz': The Barrier Postulate as you call it implies the unique solvability of the Dirichlet Problem, Thm 2.14 in Gilbard-Trudinger, i.e. it is solvable and the solution is unique. | |
| Oct 23, 2013 at 19:33 | comment | added | Gatz' | I'm considering a general open set $U$ that does not satisfy the exterior sphere condition at every point of its boundary. I'm looking for a solution $u \in C^2(U) \cap C(\overline{U})$ such that $\Delta u = 0$ in $U$ and $u_{\left| \partial U \right.} = g$ for some $g \in C(\overline{U})$. If the Barrier Postulate is satisfied, then there exists a solution to the Dirichlet problem, which I assume can be written in terms of $g$ and Green's function for $U$. Now, does the existence of Green's function for $U$ implies the existence of a solution to Dirichlet problem ? Thanks. | |
| Oct 23, 2013 at 9:30 | history | edited | username | CC BY-SA 3.0 |
added a reference to another question
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| Oct 23, 2013 at 8:08 | history | answered | username | CC BY-SA 3.0 |