Maximal L _p Regularity for Second Order Elliptic Operators with Uniformly Continuous Coefficients on Domains

Abstract

Maximal regularity of type L _p is an important tool when dealing with quasi-linear equations of parabolic type (see, e.g., [1], [3]). If the closed linear operator A is the generator of a bounded analytic C _o-semigroup (T _t) in a Banach space X and p ∈ (1, ∞) the A is sais to have maximal L _p -regularity (which we denote by A ∈ MR _p (X)) if for any f ∈ L _p ((0,∞), X) the solution u=T * f of the equation u′ = Au + f, u(0) = (0) satisfies u′ ∈ L _p ((0,∞), X) and Au ∈ L _p ((0,∞), X).By the closed graph theorem this is equivalent to the existence of a C > 0 such that

$$ \left\| {u\prime } \right\|_{L_p \left( {\left( {{\text{0,}}\infty } \right),X} \right)} + \left\| {Au} \right\|_{L_p \left( {\left( {{\text{0,}}\infty } \right),X} \right)} \leqslant C\left\| f \right\|_{L_p \left( {\left( {{\text{0,}}\infty } \right),X} \right)} . $$

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