By Kazuaki Taira

This cautious and available textual content specializes in the connection among interrelated matters in research: analytic semigroups and preliminary boundary worth difficulties. This semigroup method may be traced again to the pioneering paintings of Fujita and Kato at the Navier-Stokes equation. the writer experiences nonhomogeneous boundary price difficulties for second-order elliptic differential operators, within the framework of Sobolev areas of Lp type, which come with as specific circumstances the Dirichlet and Neumann difficulties, and proves that those boundary price difficulties supply an instance of analytic semigroups in Lp.

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Let s > 0. t - s 27n Jr µ - s R(µ)dµ - 2iri Jr 1 1 1 -R(s) - 1 27ri j (µ-s)T R([i)s dµ. µ- e But we find that the second term in the last line tends to zero as T -> oo: Jr e(µ-s)T R(µ) dy /L - s

17). Then we can define the fractional power (-A)a for 0 < a < 1. The operator (-A)a is a closed linear, invertible operator with domain D((-A)a) 3 D(A). We let Ea =the space D((-A)') endowed with the graph norm of (-A)a where lixiJa = (IIxJJ' + II(-A)axII2)1/2 Then we have the following: x E D((-A)a). 17. (i) The space Ea is a Banach space. (ii) The graph norm IIxIIa is equivalent to the norm JJ(-A)axJJ. (iii) If 0 < a < Q < 1, then we have EQ C Ea with continuous injection. Proof. (i) Assume that {xj} is a Cauchy sequence in Ea, that is, {xj} is a Cauchy sequence in E, { {(-A)axj} is a Cauchy sequence in E.

II. SOBOLEV IMBEDDING THEOREMS 48 If fI = R', we simply write I ' Ik,p = I ' Ik,P,R^. Furthermore, we introduce a norm 1/p if 1 < p < oo, I D°u(x)IP dx) IIuIIm,P,S1= maxi«j