On a Class of Weakly Coupled Systems of Elliptic Operators with Unbounded Coefficients

Abstract

We consider a class of weakly coupled systems of elliptic operators A with unbounded coefficients defined in R N . We prove that a semigroup (T (t)) t≥0 of bounded linear operators can be associated with A, in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T (t)f , when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator A. Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L p -spaces over R N , related to the Lebesgue measure, when p ∈ [1, ∞). We also provide sufficient conditions for this semigroup to be analytic when p ∈ (1, ∞). Finally, we prove some L p -L q -estimates.Mathematics Subject Classification (2010). Primary 47D06; Secondary 35B45, 35B65, 35K40, 60J35.

‐Theory for Schrödinger systems

“…The strategy in these references is quite different from that in . Namely, in solutions to the parabolic equation are at first constructed in the space of bounded and continuous functions. Afterwards the semigroup is extrapolated to the ‐scale.…”

“…Note that the results obtained here do not follow from [18] by setting = 0. Following [18], there were some other publications [1,2,11] with less restrictive assumptions on the coefficients; in particular, also unbounded diffusion coefficients were considered. The strategy in these references is quite different from that in [18].…”

On a polynomial scalar perturbation of a Schrödinger system in L-spaces

“…To our knowledge one of the first papers dealing with such kind of systems is [8]. Subsequently, there were some other publications [1,2,5]. Here the strategy in these references is quite different from that in [8].…”

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

“…The first step will be to turn (3.1) into a coupled system of real equations and then using the results of [DL11].…”