DIKUL - logo
(UL)
  • Subsolution estimates and Harnacks inequality for Schrödinger operators
    Hinz, Andreas M., 1954- ; Kalf, Hubert
    For the following result a new nonprobabilistic proof is presented. Let ▫$u \in C^0 \cap H^1_{\text{loc}}$▫ be a nonnegative subsolution of ▫$(\Delta + V)u \geq 0$▫ with a nonnegative perturbation ... ▫$V \in K^n_{\text{loc}}$▫; essentially ▫$V$▫ is form bounded with bound zero. Then ▫$u$▫ satisfies the mean value property (*) ▫$u(x) \leq C(\oint_{B_R(x)} u^p)^{1/p}$▫ for ▫$p=2$▫. The proof is based on Green's representation and a singular integral estimate of Simader. Applying this theorem, Kato's inequality and an interesting reverse Hölder inequality of Dahlberg and Kenig reducing (*) on smaller ▫$p$▫, the eigenfunction decay ▫$u(x) \to 0$▫ (▫$\vert x \vert \to \infty$▫) is derived for solutions ▫$u \in L^p$▫, ▫$1 \leq p < \infty$▫, of the Schrödinger equation ▫$(-\Delta + V)u = 0$▫ with ▫$V \in K^n$▫ outside of a ball. Furthermore, for such nonnegative solutions ▫$u$▫ the authors also obtain Harnack's inequality ▫$\sup u \leq C \inf u$▫.
    Source: Journal für die Reine und Angewandte Mathematik. - ISSN 0075-4102 (Band 404, 1990, str. 118-134)
    Type of material - article, component part ; adult, serious
    Publish date - 1990
    Language - english
    COBISS.SI-ID - 16858713

    Link(s):

    http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0404&DMDID=DMDLOG_0007

Shelf entry