Analytic semigroups generated by elliptic operators in \(L^ 1\) and parabolic equations
Let \(\Omega\subset\mathbb{R}^ n\) be a bounded set of class \(C^ 2\) and let \(E\) be the second order elliptic operator given by \[ Eu=\sum^ n_{i,j=1} {\mathcal D}_ i(a_{ij}(x) {\mathcal D}_ ju)+\sum^ n_{i=1}b_ i(x){\mathcal D}_ iu+c(x)u, \] where \({\mathcal D}_ i=\partial/ \partial x_ i\), \(a_{ij} \in C^ 1(\overline\Omega)\); \(b_ i,c \in C(\overline\Omega)\). Let us denote \(X=L_ 1 (\Omega)\) and let \(A:{\mathcal D} (A) \subset X \to X\) be the operator defined by \[ \begin{cases} {\mathcal D}(A)=\{u \in C^ 2(\overline\Omega):u(x)=0 \text{ for } x \in \partial \Omega\} \\ Au=Eu. \end{cases} \] We denote by \(A_ 1\) the closure of \(A\) in \(L_ 1(\overline\Omega)\), by \(S(t)\) the semigroup generated by \(A_ 1\) and by \({\mathcal D}_{A_ 1}(\theta,1)\) the interpolation space \[ {\mathcal D}_{A_ 1}(\theta,1)=\{u \in X:\int^ \infty_ 0 \| AS(t)u \|_ Xt^{-\theta} dt\}<\infty. \] It is proved that \(S(t)\) satisfies the following estimates for some \(M'\), \(M''>0\) and \(\omega \in \mathbb{R}\): \[ \sqrt t \| {\mathcal D}_ iS(t)\|_{L(X)}\leq M' \exp(\omega t),\quad t \| {\mathcal D}_ i {\mathcal D}_ kS(t) \|_{L(X)} \leq M'' \exp(\omega t). \] These estimates are used to prove the following characterization of the space \({\mathcal D}_{A_ 1}(\theta,1)\): \[ {\mathcal D}_{A_ 1} (\theta,1) = \begin{cases} W^{2\theta,1}(\Omega),\;0<\theta<{1 \over 2}\\ u \in B^{1,1} (\Omega): \int_ \Omega (d(x,\partial \Omega))^{-1} | u(x) | dx<+ \infty,\theta={1\over 2} \\ W^{2\theta,1} (\Omega) \cap W_ 0^{1,1}(\Omega),\;{1 \over 2}<\theta<1. \end{cases} \] Here \(W^{2 \theta,1} (\Omega)\) denotes the Sobolev space of fractional order, \(B^{1,1}\) denotes the Besov space and \(d(x,\partial \Omega)\) the distance from \(x\) to \(\partial \Omega\). Then the description of the regularity of the solutions of the problem \[ u_ t(t,x)= Eu(t,x) +f(t,x),\;t>0,\;x \in \Omega, \] \[ u(t,x)=0,\;t>0,\;x \in \partial \Omega,\quad u(0,x)=u_ 0(x),\;x \in \Omega, \] where \(f \in L_ 1(]0,T[\times \Omega)\) and \(u_ 0 \in L_ 1(\Omega)\), is given.
scientific article; zbMATH DE number 59007