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Some imbeddings for weighted Sobolev spaces. (English) Zbl 0545.46021

Constructive function theory, Proc. int. Conf., Varna/Bulg. 1981, 400-407 (1983).
[For the entire collection see Zbl 0529.00021.]
The paper deals with estimates of the type \[ (*)\quad \int_{\Omega}| u(x)|^ pb_ 0(x)dx\leq c\sum^{N}_{i=1}\int_{\Omega}| \partial u/\partial x_ i|^ pa_ i(x)dx \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ N\) with a Lipschitz boundary \(\partial \Omega\). Conditions on the weight functions \(b_ 0,a_ 1,...,a_ N\) are derived which gurantee the validity of (*) for certain classes of functions u. In the first part, a theorem is proved which shows that in the case \(p=2\) the validity of (*) is connected with the solvability of a certain boundary value problem for the equation \[ \sum^{N}_{i=1}(\partial /\partial x_ i)(a_ i\partial v/\partial x_ i)+b_ 0v=0\quad on\quad \Omega. \] An analogous result (with a nonlinear equation) is formulated for general \(p>1\) (\(p\neq 2)\). [Remark: This last result is proved in an extended form in recent papers of the authors: Trudy Sem. S. L. Soboleva, No.1, 1983, 108-117; Proc. Int. Conf. Varna/Bulg. 1984 (to appear).] - The second part deals with a generalization of (*) for special weights of the type \(s_ i(dist(x,M))\) with \(M\subset \partial \Omega\) and \(s_ i=s_ i(t)\). Here, a mixed \(L^{(p,1)}\)-norm appears on the right hand side of (*). - Examples are given.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators

Citations:

Zbl 0529.00021