Interpolation theorem for the p-harmonic transform

Luigi D'Onofrio, and Tadeusz Iwaniec. "Interpolation theorem for the p-harmonic transform." Studia Mathematica 159.3 (2003): 373-390. <http://eudml.org/doc/284627>.

@article{LuigiDOnofrio2003,
abstract = {We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces $ℒ^\{s\}(ℝⁿ)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation $div|∇u|^\{p-2∇\} u = div $. In this example the p-harmonic transform is essentially inverse to $div(|∇|^\{p-2\}∇)$. To every vector field $ ∈ ℒ^\{q\}(ℝⁿ,ℝⁿ)$ our operator $ℋ_\{p\}$ assigns the gradient of the solution, $ℋ_\{p\} = ∇u ∈ ℒ^\{p\}(ℝⁿ,ℝⁿ)$. The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments require substantial innovations as compared with the classical interpolation theory of Riesz, Thorin and Marcinkiewicz. The subject is largely motivated by recent developments in geometric function theory.},
author = {Luigi D'Onofrio, Tadeusz Iwaniec},
journal = {Studia Mathematica},
keywords = {-harmonic equation; interpolation theorem; Lebesgue spaces},
language = {eng},
number = {3},
pages = {373-390},
title = {Interpolation theorem for the p-harmonic transform},
url = {http://eudml.org/doc/284627},
volume = {159},
year = {2003},
}

TY - JOUR
AU - Luigi D'Onofrio
AU - Tadeusz Iwaniec
TI - Interpolation theorem for the p-harmonic transform
JO - Studia Mathematica
PY - 2003
VL - 159
IS - 3
SP - 373
EP - 390
AB - We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces $ℒ^{s}(ℝⁿ)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation $div|∇u|^{p-2∇} u = div $. In this example the p-harmonic transform is essentially inverse to $div(|∇|^{p-2}∇)$. To every vector field $ ∈ ℒ^{q}(ℝⁿ,ℝⁿ)$ our operator $ℋ_{p}$ assigns the gradient of the solution, $ℋ_{p} = ∇u ∈ ℒ^{p}(ℝⁿ,ℝⁿ)$. The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments require substantial innovations as compared with the classical interpolation theory of Riesz, Thorin and Marcinkiewicz. The subject is largely motivated by recent developments in geometric function theory.
LA - eng
KW - -harmonic equation; interpolation theorem; Lebesgue spaces
UR - http://eudml.org/doc/284627
ER -