# zbMATH — the first resource for mathematics

Sobolev spaces and elliptic theory on unbounded domains in $$\mathbb{R}^n$$. (English) Zbl 1301.46015
Let $$\Omega$$ 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.

##### 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
Full Text: