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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

##### MSC:
 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