Gradient estimates for elliptic systems in Carnot-Carathéodory spaces

Giuseppe Di Fazio; Maria Stella Fanciullo

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 4, page 605-618
  • ISSN: 0010-2628

Abstract

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Let X = ( X 1 , X 2 , ⋯ , X q ) be a system of vector fields satisfying the Hörmander condition. We prove L X 2 , λ local regularity for the gradient X u of a solution of the following strongly elliptic system - X α * ( a i j α β ( x ) X β u j ) = g i - X α * f i α ( x ) ∀ i = 1 , 2 , ⋯ , N , where a i j α β ( x ) are bounded functions and belong to Vanishing Mean Oscillation space.

How to cite

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Di Fazio, Giuseppe, and Fanciullo, Maria Stella. "Gradient estimates for elliptic systems in Carnot-Carathéodory spaces." Commentationes Mathematicae Universitatis Carolinae 43.4 (2002): 605-618. <http://eudml.org/doc/249009>.

@article{DiFazio2002,
abstract = {Let $X=(X_1,X_2,\dots ,X_q)$ be a system of vector fields satisfying the Hörmander condition. We prove $L^\{2,\lambda \}_X$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system \[ -X^\{*\}\_\{\alpha \}(a^\{\alpha \beta \}\_\{ij\}(x)X\_\{\beta \} u^\{j\})= g\_\{i\}-X^\{*\}\_\{\alpha \} f^\{\alpha \}\_\{i\}(x) \quad \forall i=1,2,\dots ,N, \] where $a^\{\alpha \beta \}_\{ij\}(x)$ are bounded functions and belong to Vanishing Mean Oscillation space.},
author = {Di Fazio, Giuseppe, Fanciullo, Maria Stella},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {elliptic systems; Morrey space regularity; Carnot-Carathéodory metric; regularity of weak solutions of elliptic systems; VMO spaces; Carnot-Carathéodory spaces; Hörmander vector fields; Morrey spaces},
language = {eng},
number = {4},
pages = {605-618},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Gradient estimates for elliptic systems in Carnot-Carathéodory spaces},
url = {http://eudml.org/doc/249009},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Di Fazio, Giuseppe
AU - Fanciullo, Maria Stella
TI - Gradient estimates for elliptic systems in Carnot-Carathéodory spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 605
EP - 618
AB - Let $X=(X_1,X_2,\dots ,X_q)$ be a system of vector fields satisfying the Hörmander condition. We prove $L^{2,\lambda }_X$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system \[ -X^{*}_{\alpha }(a^{\alpha \beta }_{ij}(x)X_{\beta } u^{j})= g_{i}-X^{*}_{\alpha } f^{\alpha }_{i}(x) \quad \forall i=1,2,\dots ,N, \] where $a^{\alpha \beta }_{ij}(x)$ are bounded functions and belong to Vanishing Mean Oscillation space.
LA - eng
KW - elliptic systems; Morrey space regularity; Carnot-Carathéodory metric; regularity of weak solutions of elliptic systems; VMO spaces; Carnot-Carathéodory spaces; Hörmander vector fields; Morrey spaces
UR - http://eudml.org/doc/249009
ER -

References

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