On the solvability of the Neumann problem for a planar domain with a peak
The paper under review is a well-written overview of the Neumann problem, that in addition serves to characterize solvability when the planar domain \(\Omega \subset\mathbb R^n\) contains an inward or outward peak. The authors start with the introduction of the Neumann problem, showing that the solvability of the problem is equivalent to describing the dual of the boundary trace space. They then proceed to give that description for domains with a peak; in particular this is done in terms of embeddings of functional spaces. To state the problem mathematically, assume that \(p \in (1,\infty)\), \(0 < \delta \leq a \in L_\infty(\Omega)\), and let \(\nu\) be the unit outward normal vector along \(\partial\Omega\). The Neumann problem is given by \[ \begin{aligned} -\nabla \cdot \left( | \nabla u| ^{p-2} \nabla u \right) + a | u| ^{p-2} u &= 0 \quad\text{in }\Omega,\\ | \nabla u | ^{p-2} D_\nu &= f \quad\text{on }\partial \Omega, \end{aligned} \] where \(f = \langle f, \cdot \rangle\) is a homogeneous additive functional on \({\mathcal V} := W_p^1(\Omega) \cap L_\infty(\Omega) \cap C^\infty(\Omega)\), vanishing on the compactly supported functions \(C_0^\infty(\Omega)\). A solution is defined as a function \(u \in W_p^1(\Omega)\) such that \[ \int_\Omega ( | \nabla u | ^{p-2} \nabla u \cdot \nabla v + a | u| ^{p-2} u v)\,dx = \langle f, v \rangle \quad\text{for all }v \in {\mathcal V}. \] While the main results depend on the precise behaviour of the peak -- and therefore on the notation introduced in the paper -- the following provides a flavour of the results. Let \(q \in[1, p/ (2-p)]\) for \(p < 2\); \(q \in [1,\infty)\) for \(p=2\); \(q \in [1,\infty]\) for \(p > 2\); and \(1/q + 1/q^\prime = 1\). The following propositions are then equivalent: {\parindent=6mm \begin{itemize}\item[i)] The Neumann problem is solvable for all \(f \in L_{q^\prime}(\partial\Omega)\). \item[ii)] \(TW_p^1(\Omega)\) is continuously embedded in \(L_q(\partial\Omega)\). \item[iii)] For \(f \in L_{q^\prime}(\partial\Omega)\) the functional \(TW_p^1(\Omega) \ni v \mapsto \int_{\partial\Omega} f v \,ds\) is continuous. \end{itemize}} The authors then show that this is equivalent to embeddings between certain functional spaces on the real interval \((0,1)\).
scientific article; zbMATH DE number 5345576