Timeline for A question about Gauss-Green formula - a weaker assumption

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Apr 27, 2023 at 7:40	comment	added	user378654		That said, that does not help OP with their actual question, where they would need to assume that $\Delta f_n \rightarrow \Delta h$ in $L^2$ to use this approach, in which case the conclusion is trivial anyway.
Apr 27, 2023 at 7:36	comment	added	user378654		It's true that you need $u \in H^2$ in order to give a ($L^2$) meaning to $\nabla u$ on the boundary, but you do not need that to discuss only the normal derivative on the boundary. Indeed, from the formula in the OP we have that $\partial_\nu u$ defines a bounded linear functional on $H^{1/2}(\partial \Omega)$ (acting by extending the $H^{1/2}$ function to $v \in H^1$), and the map $u \mapsto \partial_\nu u$ is bounded from $H^1 \cap \{\Delta u \in L^2\}$ to the dual of $H^{1/2}(\partial \Omega)$. This is common in the fluids literature, where they have vector fields with controlled div.
Apr 23, 2023 at 14:37	history	edited	Daniele Tampieri	CC BY-SA 4.0	Minor formatting
Apr 22, 2023 at 23:49	comment	added	Giorgio Metafune		You need $u \in H^2$ to write such a formula, in particular to give a meaning to $\nabla u$ at the boundary.
Apr 22, 2023 at 17:47	history	edited	Bogdan	CC BY-SA 4.0	edited tags
Apr 22, 2023 at 17:41	history	edited	Bogdan		edited tags
Apr 22, 2023 at 15:45	history	asked	Bogdan	CC BY-SA 4.0