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Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations. (English) Zbl 0568.47035
Let X be a Banach space and let \(A: D(A)\subset X\to X\) be a generator of a semigroup \(e^{tA}\) in X. For \(0<\theta <1\), the interpolation spaces \(D_ A(\theta,\infty)\) and \(D_ A(\theta)\) are defined by \[ D_ A(\theta,\infty)=\{x\in X;\quad \sup_{0<t<1}\| t^{- \theta}(e^{tA}x-x)\|_ X<+\infty \} \] \[ D_ A(\theta)=\{x\in X;\quad \lim_{t\to 0^+}t^{-\theta}(e^{tA}x-x)=0\}. \] Here \(D_ A(\theta,\infty)\) and \(D_ A(\theta)\) are characterized when A is a second order elliptic differential operator with regular coefficients in \({\bar \Omega}\) and \(X=C_ 0({\bar \Omega})=\{f\in C({\bar \Omega});\quad f_{| \partial \Omega}=0\}\) (\(\Omega\) is a bounded open set in \({\mathbb{R}}^ n\) with regular boundary \(\partial \Omega)\). The result is \[ D_ A(\theta,\infty)=C_ 0^{2\theta}({\bar \Omega})=\{f\in C^{2\theta}({\bar \Omega});\quad f_{| \partial \Omega}=0\} \] \[ D_ A(\theta)=h_ 0^{2\theta}({\bar \Omega})=\{f\in h^{2\theta}({\bar \Omega});\quad f_{| \partial \Omega}=0\} \] for each \(\theta\in]0,1[\) with \(\theta\) \(\neq 1/2\). \(h^{2\theta}({\bar \Omega})\) is the closure of \(C^{\infty}({\bar \Omega})\) in the norm of \(C^{2\theta}({\bar \Omega}).\)
These characterizations are used to prove that each second order elliptic operator with reular coefficients and Dirichlet boundary condition generates an analytic semigroup in \(C_ 0^{\alpha}({\bar \Omega})\) and in \(h_ 0^{\alpha}({\bar \Omega})\), \(0<\alpha <1.\)
Moreover, the above results are applied in order to study fully nonlinear second order parabolic p.d.e. in spaces of continuous functions.

MSC:
47D03 Groups and semigroups of linear operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
46M35 Abstract interpolation of topological vector spaces
35K55 Nonlinear parabolic equations
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