Sobolev spaces and elliptic theory on unbounded domains in \(\mathbb{R}^n\).

*(English)*Zbl 1301.46015Let \(\Omega\) be an open subset of \(\mathbb R^n\) and \(\varphi:\Omega\to\mathbb R\) be \(C^{\infty}\). The authors denote by \(L^p(\Omega,\varphi)\) the space of functions \(f:\Omega\to\mathbb C\) such that \(\int_{\Omega}|f|^pe^{-\varphi} dV<\infty,\) where \(dV\) is the Lebesgue measure on \(\mathbb C^n.\) The boundary \(b\Omega\) of \(\Omega\) is assumed to be at least Lipschitz. Six hypotheses on \(\Omega\) and \(\varphi\) are listed in Section 2: suitable subsets of them are used in the paper. Weighted differential operators are introduced (Section 2.2), namely:
\[
X_j=\frac{\partial}{\partial x_j}-\frac{\partial\varphi}{\partial x_j}=e^{\varphi }\frac{\partial}{\partial x_j }e^{-\varphi}, \quad 1\leq j\leq n,\, \nabla_X=(X_1,\dots,X_n).
\]
Let \(Y_j=X_j \) or \(D_j(=\frac{\partial}{\partial x_j})\), take a nonnegative \(k\in\mathbb Z\). The weighted Sobolev spaces are defined as follows:
\[
W^{k,p}(\Omega,\varphi;Y)=\{f\in L^p(\Omega,\varphi):Y^{\alpha}f\in L^p(\Omega,\varphi) \text{ for } |\alpha|\leq k\},
\]
where \(\alpha=(\alpha_1,\dots,\alpha_n)\) is an \(n\)-tuple of non negative integers, \(Y^{\alpha}= Y_1^{\alpha_1},\dots,Y_n^{\alpha_n}\), associated with the norm
\[
||f||^p_{W^{k,p}(\Omega,\varphi;Y)}=\underset{|\alpha|\leq k}{\sum}||Y^{\alpha}f||^p_{L^p(\Omega,\varphi)}.
\]
The closure of the Schwartz space \(\mathcal D(\Omega),\) in the normed space \(W^{k,p}(\Omega,\varphi;Y),\) is denoted by \(W_o^{k,p}(\Omega,\varphi;Y).\) In the same way, weighted Sobolev spaces \(W^{k,p}(M,\varphi;T)\) are defined on the boundary \(M=b\Omega\). Fractional Besov spaces of the form
\[
B^{s;p,q}(\Omega,\varphi,X)=(L^p(\Omega),W^{m,p}(\Omega,\varphi;X))_{s/m,q;J}
\]
and
\[
B^{s;p,q}(M,\varphi,T)=(L^p(M),W^{m,p}(M,\varphi;T))_{s/m,q;J}
\]
are defined by means of the \(J\)-interpolation method (whose definition is recalled in the appendix of the paper), where \(0<s<\infty\), \(1\leq p,q\leq\infty\), and \(m\) is the smallest integer larger than \(s\) (Definition 6.2). Trace and extension theorems are stated. Embeddings results are also studied.

In the second part of the paper, differential operators of the form \[ L=\sum_{j,k=1}^n X^*_j\, a_{jk}X_k+\sum_{j=1}^n (b_jX_j+X^*_jb^{\prime}_j)+b \] are considered, when \(p=2\). Here, \(a_{j,k},b_j, b^{\prime} _j,b\) are bounded functions on a neighborhood of \(\bar{\Omega}\). Associated to \(L\) is the Dirichlet form \(\mathfrak D\) given by \(\mathfrak D(v,u)=(v,Lu)_{\varphi}\) for all \(u,v\in\mathcal D(\Omega)\). Let \(\chi\) be a closed subspace of \(W^{1,2} (\Omega,\varphi;X)\) that contains \(W^{1,2}_o (\Omega,\varphi;X)\), \(\mathfrak D\) be a Dirichlet form that is coercive on \(\chi\) and \(f\in L^2(\Omega,\varphi).\) Existence, uniqueness, regularity of solutions \(u\in\chi\) of \(\mathfrak D(v,u)=(v,f)_{\varphi}\) for all \(v\in\chi\) are studied. When \(\mathfrak D\) is self-adjoint, it is proved that \(L^2(\Omega,\varphi)\) has a basis of eigenvectors. Traces of \(L\)-harmonic functions are also investigated.

In the second part of the paper, differential operators of the form \[ L=\sum_{j,k=1}^n X^*_j\, a_{jk}X_k+\sum_{j=1}^n (b_jX_j+X^*_jb^{\prime}_j)+b \] are considered, when \(p=2\). Here, \(a_{j,k},b_j, b^{\prime} _j,b\) are bounded functions on a neighborhood of \(\bar{\Omega}\). Associated to \(L\) is the Dirichlet form \(\mathfrak D\) given by \(\mathfrak D(v,u)=(v,Lu)_{\varphi}\) for all \(u,v\in\mathcal D(\Omega)\). Let \(\chi\) be a closed subspace of \(W^{1,2} (\Omega,\varphi;X)\) that contains \(W^{1,2}_o (\Omega,\varphi;X)\), \(\mathfrak D\) be a Dirichlet form that is coercive on \(\chi\) and \(f\in L^2(\Omega,\varphi).\) Existence, uniqueness, regularity of solutions \(u\in\chi\) of \(\mathfrak D(v,u)=(v,f)_{\varphi}\) for all \(v\in\chi\) are studied. When \(\mathfrak D\) is self-adjoint, it is proved that \(L^2(\Omega,\varphi)\) has a basis of eigenvectors. Traces of \(L\)-harmonic functions are also investigated.

Reviewer: Denise Huet (Nancy)

##### MSC:

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46B70 | Interpolation between normed linear spaces |

35J15 | Second-order elliptic equations |

35B25 | Singular perturbations in context of PDEs |