Weighted Sobolev spaces for Laplace's equation in \(\mathbb{R}^ n\)
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: \[ \begin{aligned} \Delta : W_\ell^{1,p} (\mathbb{R}^n)/P_{[1 - \ell - n/p]} & \mapsto W_\ell^{-1, p} (\mathbb{R}^n) \perp P^\Delta_{[\ell + 1 - n/p']}, \\ \Delta : W_\ell^{2,p} (\mathbb{R}^n)/P_{[2 - \ell - n/p]} & \mapsto W_\ell^{0,p} (\mathbb{R}^n) \perp P^\Delta_{[\ell - n/p']}, \\ \Delta : W_{- \ell}^{2,p'} (\mathbb{R}^n)/P^\Delta_{[2 + \ell - n/p']} & \mapsto W_{-\ell}^{0,p'} (\mathbb{R}^n). \end{aligned} \] {}.
scientific article; zbMATH DE number 811456