# Quasiharmonic Fields: a Higher Integrability Result

• Volume: 10-B, Issue: 3, page 843-851
• ISSN: 0392-4041

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## Abstract

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In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $\operatorname{div}(A(x)\nabla u(x))=0$, where the symmetric matrix $A(x)$ satisfies the condition $|\xi|^{2}+|A(x)\xi|^{2}\leq K(x)\langle A(x)\xi,\xi\rangle$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp(\beta K(x))$ is integrable for some $\beta>0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L^{2}_{\text{loc}}\log^{\alpha-1}L$, $0<\alpha=\alpha(\beta)$. Moreover we show exactly how the degree of improved regularity depends on $\beta$.

## How to cite

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Di Gironimo, Patrizia. "Quasiharmonic Fields: a Higher Integrability Result." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 843-851. <http://eudml.org/doc/290398>.

@article{DiGironimo2007,
abstract = {In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $\operatorname\{div\}(A(x)\nabla u(x))= 0$, where the symmetric matrix $A(x)$ satisfies the condition $|\xi|^2+|A(x)\xi|^2 \leq K(x)\langle A(x)\xi,\xi\rangle$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp(\beta K(x))$ is integrable for some $\beta >0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L^2_\{\text\{loc\}\} \log^\{\alpha - 1\}L$, $0 < \alpha = \alpha (\beta)$. Moreover we show exactly how the degree of improved regularity depends on $\beta$.},
author = {Di Gironimo, Patrizia},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {843-851},
publisher = {Unione Matematica Italiana},
title = {Quasiharmonic Fields: a Higher Integrability Result},
url = {http://eudml.org/doc/290398},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Di Gironimo, Patrizia
TI - Quasiharmonic Fields: a Higher Integrability Result
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 843
EP - 851
AB - In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $\operatorname{div}(A(x)\nabla u(x))= 0$, where the symmetric matrix $A(x)$ satisfies the condition $|\xi|^2+|A(x)\xi|^2 \leq K(x)\langle A(x)\xi,\xi\rangle$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp(\beta K(x))$ is integrable for some $\beta >0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L^2_{\text{loc}} \log^{\alpha - 1}L$, $0 < \alpha = \alpha (\beta)$. Moreover we show exactly how the degree of improved regularity depends on $\beta$.
LA - eng
UR - http://eudml.org/doc/290398
ER -

## References

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9. MIGLIACCIO, L. - MOSCARIELLO, G., Higher integrability of div-curl products, Ricerche di Matematica, (1) XLIX (2000), 151-161. MR1795037
10. MOSCARIELLO, G., On the integrability of finite energy solutions for p-harmonic equations, Nodea, 11 (2004) 393-406. Zbl1102.35039MR2090281DOI10.1007/s00030-004-2020-6
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