Elliptic operators with unbounded diffusion coefficients in \(L^p\) spaces (Q2915216)

The paper is concerned with elliptic operators with unbounded diffusion coefficients of the form \(Lu = a\cdot \Delta u\) acting on \(\mathbb{R}^N\). Of main interest is solvability of the corresponding parabolic and elliptic problems \(u_t - L u= 0\), \(u(0)=f\), resp., \((\lambda I-L)u =f\). Here, \(a(x) = m(x)\cdot (1+|x|^\alpha)\) with \(\alpha >0\) and \(m(\cdot)>0\) bounded and bounded away from \(0\). The emphasis is laid on \(\alpha >2\). For \(\alpha\leq 2\), it is known (cf. \textit{S. Fornaro} and \textit{L. Lorenzi} [``Generation results for elliptic operators with unbounded diffusion coefficients in \(L^p\) and \(C_b\)-spaces'', Discrete Contin. Dyn. Syst. 18, No. 4, 747--772 (2007; Zbl 1152.47031)]) that \(L\) generates an analytic semigroup on \(L^p\), \(1\leq p\leq \infty\). The paper is mainly concerned with solutions in \(L^p\)-spaces, however, basic results concerning continuous function spaces are needed. For the action on \(C_b\), see the first named author, \textit{D. Pallara} and \textit{M. Wacker} [``Feller semigroups on \(\mathbb{R}^N\)'', Semigroup Forum 65, No. 2, 159--205 (2002; Zbl 1014.35050)]: \(L\) generates a positive semigroup \((T(t))\) on \(C_b\). \(C_0(\mathbb{R}^N)\) need not be \((T(t))\)-invariant. However, for \(\alpha>2\), \(N\geq 3\), it follows that \(T(t)C_b\subseteq C_0\) (Proposition 2.2). In general, uniqueness of solutions cannot be proved. The concentration on positive functions defines a `minimal' semigroup \((T_{\min}(t))\) with generator \((L,D)\) on a domain \(D\) contained in the maximal domain \(D_{\max}\).NEWLINENEWLINELet, for \(p\geq 1\), \(\widehat{L}_p\) denote the operator \(L\) restricted to a domain \(\widehat{D}_p\subseteq D_{p, \max}:=\{u\in L^p\cap W_{\mathrm{loc}}^{2,p}: L u\in L^p\}\). Indeed, we have (Proposition 3.1) that \(D_{p, \max} =\{u\in W^{2,p}: (1+|x|^\alpha)\Delta u\in L^p\}\). Let \(\rho(\widehat{L}_p)\) denote the resolvent set of \(\widehat{L}_p\). Section 3 is concerned with existence, resp., non-existence of solutions of the elliptic problem for \(\lambda >0\), determining values of \((N,\alpha, p)\) for which \(\rho(\widehat{L}_p)\cap \mathbb{R}_+\neq\emptyset\), resp., \(= \emptyset\) (for \(m(\cdot)\equiv 1\)), Section 4 considers solvability again for the general case \(a(x)=m(x)\cdot (1+|x|^\alpha)\). The results rely on weighted estimates of the operator \(T = (-L)^{-1}\) (represented as integral operator with Newtonian potential) and lead to the main result of this section (Theorem 4.6): For \(\alpha >2\), \(p>N/(N-2)\), \(\lambda \geq 0\), the resolvent \((\lambda I-L)^{-1}\) exists on \(D_{p, \max}(L)\) and is positive.NEWLINENEWLINESection 5 is concerned with characterizations of the domains \(\widehat{D}_p\), \(D_{p, \max}(L)\) as weighted Sobolev spaces for different values \((N, \alpha, p)\) and \(m(\cdot)\), whereas the following section considers the restriction of \(L\) to \(C_0\), i.e., the generator of \((T_{\min}(t))\). In the subsequent Section 7, the spectrum of \(L\) on \(L^p\) is considered (with the convention that \(p =\infty \) corresponds to \(C_0(\mathbb{R}^N)\)). For \(\alpha >2\), \(p \geq N/(N-2)\), the spectrum of \((L, D_{p, \max})\) is independent of \(p\) and consists of a sequence \(\{\lambda_n\}\subseteq (-\infty, 0]\) of simple poles of the (compact) resolvent. These results are obtained embedding \(C_c^\infty\) into a suitable Hilbert space and investigating the corresponding operator semigroup there.NEWLINENEWLINESection 8 is concerned with special functions \(a(x)=s^\alpha + |x|^\alpha\), \(s>0\), showing that, for suitable values \((N, \alpha, p)\), the semigroup generated by \((L, D_{p, \max})\) is positive and analytic (on \(L^p)\); these results are extended (Theorem 9.8) to functions \(a(\cdot)\) satisfying \(a(x)/(1+|x|^\alpha)\to \ell >0\) for \(|x|\to\infty\). The proof relies on a sequence of lemmas proving estimates and analyticity of semigroups generated by operators corresponding to particular functions \(a(\cdot)\).