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Weighted Sobolev spaces for Laplace’s equation in \(\mathbb{R}^ n\). (English) Zbl 0836.35038
The purpose of this paper is to solve the problem \(\Delta u = f\) in \(\mathbb{R}^n\). The problem is set in spaces similar to the usual Sobolev spaces \(W^{m,p} (\mathbb{R}^n)\), with weights that control the behaviour of the function at infinity. \(D (\mathbb{R}^n)\) denotes the space of infinitely differentiable functions with compact support and \(D' (\mathbb{R}^n)\) denotes its dual space. Let \(\rho = (1 + |x |^2)^1/2\), \(r = |x |\), \(x \in \mathbb{R}^n\). \[ W^{m, p}_{ \alpha, \beta} (\mathbb{R}^n) = \{u \in D' (\mathbb{R}^n);\;\forall \lambda \in \mathbb{N}^n : 0 \leq |\lambda |\leq k,\;\rho^{\alpha - m + |\lambda |} (\ln (2 + r^2))^{\beta - 1} D^\lambda u \in L^p (\mathbb{R}^n); \]
\[ \forall \lambda \in \mathbb{N}^n : k + 1 \leq |\lambda |\leq m,\;\rho^{\alpha - m + |\lambda |} (\ln (2 + r^2))^\beta D^\lambda u \in L^p (\mathbb{R}^n)\}, \] which is a reflexive Banach space with natural norm \(|u |_{W^{m, p}_{\alpha, \beta}} (\mathbb{R}^n)\), and the seminorm:
\(|u |_{W^{m,p}_{\alpha, \beta}} (\mathbb{R}^n) = (\sum_{|\lambda |= m} |\rho ^\alpha (\ln (2 + r^2))^\beta D^\lambda u |^p_{L^p (\mathbb{R}^n)} )^{1/p}\). The main results in this paper are:
1) Let \(\alpha\) and \(\beta\) be two real numbers and \(m \geq 1\) an integer not satisfying simultaneously \((n/p) + \alpha \in \{1,2, \dots, m\}\) and \((\beta - 1) p = - 1\). Let \(q' = \inf (q,m - 1)\), where \(q\) is the highest degree of the polynomials contained in \(W^{m, p}_{\alpha, \beta} (\mathbb{R}^n)\). Then the seminorm \(|\cdot |_{W^{m,p}_{\alpha, \beta}} (\mathbb{R}^n)\) defines on \(W^{m,p}_{\alpha, \beta} (\mathbb{R}^n)/P_{q'}\) a norm which is equivalent to the quotient norm.
2) Let \(\ell\) be a nonnegative integer and assume that \(n/p\), does not belong to \(\{1, \dots, \ell\}\), with the conventions that this set is empty when \(\ell = 0\). The following Laplace operators are isomorphisms: \[ \begin{aligned} \Delta : W_\ell^{1,p} (\mathbb{R}^n)/P_{[1 - \ell - n/p]} & \mapsto W_\ell^{-1, p} (\mathbb{R}^n) \perp P^\Delta_{[\ell + 1 - n/p']}, \\ \Delta : W_\ell^{2,p} (\mathbb{R}^n)/P_{[2 - \ell - n/p]} & \mapsto W_\ell^{0,p} (\mathbb{R}^n) \perp P^\Delta_{[\ell - n/p']}, \\ \Delta : W_{- \ell}^{2,p'} (\mathbb{R}^n)/P^\Delta_{[2 + \ell - n/p']} & \mapsto W_{-\ell}^{0,p'} (\mathbb{R}^n). \end{aligned} \] {}.

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B03 Isomorphic theory (including renorming) of Banach spaces