On the boundedness and compactness of weighted Green operators of second-order elliptic operators

zbMATH Open1461.47023arXiv1601.01464MaRDI QIDQ5355546

Yehuda Pinchover

Publication date: 7 September 2017

Abstract: For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(ph{i}_p)$ with their dual spaces, where $1leqpleqinfty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(ph{i}_p)$ for any $1leqpleqinfty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$ on $C_0^{i}nfty$, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if $W$ is a (semi)small perturbation of $L$, then for any $1leqpleqinfty$, the associated Green operator is compact on $L^p(ph{i}_p)$, and the corresponding spectrum is $p$-independent.

Full work available at URL: https://arxiv.org/abs/1601.01464

zbMATH Keywords

elliptic operatorsprincipal eigenfunctionweighted Green operators

Mathematics Subject Classification ID

Spaces of measurable functions ((L^p)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) (46E30) General topics in linear spectral theory for PDEs (35P05) Integral operators (47G10) Green's functions for elliptic equations (35J08) Elliptic operators and their generalizations (47F10)

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