On the continuity of degenerate n-harmonic functions

Flavia Giannetti; Antonia Passarelli di Napoli

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 621-642
  • ISSN: 1292-8119

Abstract

top
We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ∫ 1 ∞ P ( t ) t 2   d t = ∞ .

How to cite

top

Giannetti, Flavia, and Passarelli di Napoli, Antonia. "On the continuity of degenerate n-harmonic functions." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 621-642. <http://eudml.org/doc/244353>.

@article{Giannetti2012,
abstract = {We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition \begin\{equation\} \int\_1^\infty\frac\{P(t)\}\{t^2\}\,\{\rm d\}t=\infty. \end\{equation\}},
author = {Giannetti, Flavia, Passarelli di Napoli, Antonia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Orlicz classes; degenerate elliptic equations; continuity; n-harmonic functions; regularity of solution; finite energy solutions},
language = {eng},
month = {11},
number = {3},
pages = {621-642},
publisher = {EDP Sciences},
title = {On the continuity of degenerate n-harmonic functions},
url = {http://eudml.org/doc/244353},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Giannetti, Flavia
AU - Passarelli di Napoli, Antonia
TI - On the continuity of degenerate n-harmonic functions
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 621
EP - 642
AB - We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition \begin{equation} \int_1^\infty\frac{P(t)}{t^2}\,{\rm d}t=\infty. \end{equation}
LA - eng
KW - Orlicz classes; degenerate elliptic equations; continuity; n-harmonic functions; regularity of solution; finite energy solutions
UR - http://eudml.org/doc/244353
ER -

References

top
  1. E. Acerbi and N. Fusco, An approximation lemma for W1,p functions, in Material Instabilities in Continuum Mechanics, J.M. Ball Ed. (Edinburgh, 1985–1986). Oxford University Press, New York (1988).  
  2. M. Carozza, G. Moscariello and A. Passarelli di Napoli, Regularity for p-harmonic equations with right hand side in Orlicz-Zygmund classes. J. Differ. Equ.242 (2007) 248–268.  
  3. F. Gehring, Rings and quasiconformal mapping in the space. Trans. Amer. Math. Soc.103 (1962) 353–393.  
  4. F. Giannetti and A. Passarelli di Napoli, Isoperimetric type inequalities for differential forms on manifolds. Indiana Univ. Math. J.54 (2005) 1483–1497.  
  5. F. Giannetti and A. Passarelli di Napoli, On very weak solutions of degenerate equations. NoDEA14 (2007) 739–751.  
  6. F. Giannetti, L. Greco and A. Passarelli di Napoli, The self-improving property of the Jacobian determinant in Orlicz spaces. Indiana Univ. Math. J.59 (2010) 91–114.  
  7. F. Giannetti, L. Greco and A. Passarelli di Napoli, Regularity of solutions of degenerate A-harmonic equations. Nonlinear Anal.73 (2010) 2651–2665.  
  8. T. Iwaniec and J. Onninen, Continuity estimates for n-harmonic equations. Indiana Univ. Math. J.56 (2007) 805–824.  
  9. T. Iwaniec and C. Sbordone, Quasiharmonic fields. Ann. Inst. Henri Poincaré Anal. non Linéaire18 (2001) 519–572.  
  10. T. Iwaniec, L. Migliaccio, G. Moscariello and A. Passarelli di Napoli, A priori estimates for nonlinear elliptic complexes. Advances Difference Equ.8 (2003) 513–546.  
  11. J. Kauhanen, P. Koskela, J. Maly, J. Onninen and X. Zhong, Mappings of finite distortion : sharp Orlicz conditions. Rev. Mat. Iberoamericana19 (2003) 857–872.  
  12. P. Koskela and J. Onninen, Mappings of finite distortion : the sharp modulus of continuity. Trans. Amer. Math. Soc.355 (2003) 1905–1920.  
  13. P. Koskela, J. Manfredi and E. Villamor, Regularity theory and traces of 𝒜-harmonic functions. Trans. Amer. Math. Soc.348 (1996) 755–766.  
  14. M.A. Krasnosel’skii and Ya.B. Rutickii, Convex Functions and Orlicz Spaces. P. Noordhoff LTD, Groningen, The Netherlands (1961).  
  15. J. Lewis, On very weak solutions of certain elliptic systems. Commun. Partial. Differ. Equ.18 (1993) 1515–1537.  
  16. J. Manfredi, Weakly monotone functions. J. Geom. Anal.4 (1994) 393–402.  
  17. G. Moscariello, On the integrability of “finite energy” solutions for p-harmonic equations. NoDEA11 (2004) 393–406.  
  18. M.M. Rao and Z.D. Ren, Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics146. Marcel Dekker, Inc., New York (1991).  
  19. G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus. Semin. de Math. Supérieures 16, Univ. de Montréal (1966). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.