Summary
We study the equation (A − λ) x + (B−λ)x=y, with unknown x, in a Banach space X. y ∈ Xis the datum, λ> 0, A and B are linear closed unbounded operators in X with domains DA, DB. In the non commutative case, under assumptions already considered in the literature (see [7]), we show that for large values of λ any solution x ∈ DA ∩ DB satisfies an a priori estimate ¦|x¦|≤cλ−1¦|y||and we prove that for any y ∈ X there exists a unique strong solution x, i.e. there exist xn∈DA∩ DB such that xn → x, (A−λ) xn+(B−λ) xn→y in X. We also study regularity properties of strong solutions and we show that they belong to suitable interpolation spaces between DA (or DB) and X.
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Fuhrman, M. Sums of linear operators of parabolic type: a priori estimates and strong solutions. Annali di Matematica pura ed applicata 164, 229–257 (1993). https://doi.org/10.1007/BF01759322
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DOI: https://doi.org/10.1007/BF01759322