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Analytic semigroups generated in $$L^{1}(\Omega)$$ by second order elliptic operators via duality methods. (English) Zbl 1190.47043
Summary: Given an open domain (possibly unbounded) $$\Omega \subset \mathbb{R}^{n}$$, we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in $$L^{1}(\Omega)$$. We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 35J57 Boundary value problems for second-order elliptic systems 47F05 General theory of partial differential operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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