The Neumann problem in weighted Sobolev spaces. (English) Zbl 0579.35021

Linear operators of the form \[ (Lu)(x)=\sum_{| \alpha |,| \beta | \leq m}(-1)^{| \alpha |}D^{\alpha}(a_{\alpha \beta}(x)D\quad^{\beta}u(x)) \] are considered on a domain \(\Omega\) of \({\mathbb{R}}^ n\), where the coefficients \(a_{\alpha \beta}\in L^{\infty}(\Omega)\). The operator is assumed to be elliptic in the sense that for a suitable constant \(c>0\) \[ <Lu,u>\geq c\| u\|^ 2_{W^{m,2}(\Omega)}\quad for\quad every\quad u\in W^{m,2}(\Omega). \] Given \(M\subseteq \partial \Omega\) and a function \(s:{\mathbb{R}}_+\to {\mathbb{R}}_+\) the authors consider the weighted Sobolev space \(W^{m,2}(\Omega,M,s)\) defined as the set of all functions u such that \[ \| u\|_{W^{m,2}(\Omega,M,s)}=[\sum_{| \alpha | \leq m}\int_{\Omega}| D^{\alpha}u|^ 2\quad s(dist(x,M))dx]^{1/2}<+\infty. \] When \(s(t)=t^{\epsilon}\) the notation \(W^{m,2}(\Omega,M,\epsilon)\) is used. The following result about Neumann problem is announced in the case \(m=1.\)
Theorem. Assume \(\Omega\) is a bounded Lipschitz domain of \({\mathbb{R}}^ n\), and \(M\subseteq \partial \Omega\) is a k-dimensional manifold with the (rather restrictive) condition n-k\(\geq 3\). Then, for a suitable \(\epsilon_ 0>0\) and for every \(\epsilon <\epsilon_ 0\) there exists one and only one weak solution of \(Lu=f\) in \(\Omega\) with the Neumann boundary condition \[ \sum_{| \alpha | =| \beta | =1}a_{\alpha \beta}\nu_{\alpha}D^{\beta}u=g\quad on\quad \partial \Omega, \] where f and g belong to the dual space of \(W^{1,2}(\Omega,M,\epsilon)\).
Reviewer: G.Buttazzo


35J40 Boundary value problems for higher-order elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems