Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2 : an a priori estimate

Andrea Gentile

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche (2018)

  • Volume: 85, Issue: 1, page 185-200
  • ISSN: 0370-3568

Abstract

top
We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form ℱ ⁢ ( ν , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ ν ⁢ ( x ) ) ⁢ 𝑑 x with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

How to cite

top

Gentile, Andrea. "Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 < p < 2$: an a priori estimate." Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche 85.1 (2018): 185-200. <http://eudml.org/doc/296735>.

@article{Gentile2018,
abstract = {We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form \begin\{equation*\}\mathcal\{F\}(\nu, \Omega) = \int\_\{\Omega\} f(x, D\nu(x))\, dx \end\{equation*\} with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.},
author = {Gentile, Andrea},
journal = {Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche},
keywords = {Local minimizers; A priori estimate; Sobolev coefficients},
language = {eng},
month = {12},
number = {1},
pages = {185-200},
publisher = {Società Nazione di Scienze, Lettere e Arti in Napoli; Giannini},
title = {Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 < p < 2$: an a priori estimate},
url = {http://eudml.org/doc/296735},
volume = {85},
year = {2018},
}

TY - JOUR
AU - Gentile, Andrea
TI - Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1 < p < 2$: an a priori estimate
JO - Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche
DA - 2018/12//
PB - Società Nazione di Scienze, Lettere e Arti in Napoli; Giannini
VL - 85
IS - 1
SP - 185
EP - 200
AB - We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form \begin{equation*}\mathcal{F}(\nu, \Omega) = \int_{\Omega} f(x, D\nu(x))\, dx \end{equation*} with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
LA - eng
KW - Local minimizers; A priori estimate; Sobolev coefficients
UR - http://eudml.org/doc/296735
ER -

References

top
  1. Acerbi, E. and Fusco, N. (1989), Regularity for minimizers of nonquadratic functionals: The case 1 < p < 2 , J. Math. Anal. Appl., 140, no. 1, 115-135. Zbl0686.49004MR997847DOI10.1016/0022-247X(89)90098-X
  2. Baisón, A. L., Clop, A., Giova, R., Orobitg, J. and Passarelli di Napoli, A. (2017), Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal., 46, no. 3, 403-430. Zbl1362.35064MR3630402DOI10.1007/s11118-016-9585-7
  3. Carozza, M., Kristensen, J. and Passarelli di Napoli, A. (2011), Higher differentiability of minimizers of convex variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire28, no. 3, 395-411. Zbl1245.49052MR2795713DOI10.1016/j.anihpc.2011.02.005
  4. Clop, A., Faraco, D., Mateu, J., Orobitg, J. and Zhong, X. (2009), Beltrami equations with coefficient in the Sobolev space W 1 , p , Publ. Mat., 53, no. 1, 197-230. Zbl1189.30053MR2474121DOI10.5565/PUBLMAT_53109_09
  5. Clop, A., Giova, R. and Passarelli di Napoli, A. (2017), Besov regularity for solutions of p -harmonic equations, Adv. Nonlinear Anal., DOI: 10.1515/anona-2017-0030 Zbl1418.35149MR3918404DOI10.1515/anona-2017-0030
  6. Cruz-Uribe, D., Moen, K. and Rodney, S. (2016), Regularity results for weak solutions of elliptic PDEs below the natural exponent, Ann. Mat. Pura Appl., (4), 195 , no. 3. Zbl1348.35040MR3500302DOI10.1007/s10231-015-0486-y
  7. Diening, L., Stroffolini, B. and Verde, A. (2009), Everywhere regularity of functionals with ϕ -growth, Manu. Math., 129, 449-481. Zbl1168.49035MR2520895DOI10.1007/s00229-009-0277-0
  8. Diening, L., Stroffolini, B. and Verde, A. (2011), Lipschitz regularity for some asymptotically convex problems. ESAIM Control Optim. Calc. Var., (1), 17, 178-189. Zbl1231.35031MR2775192DOI10.1051/cocv/2009046
  9. Eleuteri, M., Marcellini, P., Mascolo, E. (2016), Lipschitz estimates for systems with ellipticity conditions at infinity, Ann. Mat. Pura e Appl., (4), 1951575-1603. Zbl1354.35035MR3537963DOI10.1007/s10231-015-0529-4
  10. Eleuteri, M., Marcellini, P., Mascolo, E. (2016), Lipschitz continuity for energy integrals with variable exponents, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 2761-87. Zbl1338.35169MR3470676DOI10.4171/RLM/723
  11. Fusco, N. and Hutchinson, J. E. (1985). C 1 , α partial of function minimizing quasiconvex integrals. Manuscripta Math., 54, 121–143. Zbl0587.49005MR808684DOI10.1007/BF01171703
  12. Giaquinta, M. (1983), Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud., 105, Princeton University Press. Zbl0516.49003MR717034
  13. Giaquinta, M. and Modica, G. (1986), Remarks on the regularity of the minimizers of certain degenerate functionals. Manu. Math., 57, 55-99. Zbl0607.49003MR866406DOI10.1007/BF01172492
  14. Giova, R., Passarelli Di Napoli, A. (2017), Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients, Adv. Calc. Var.. Zbl1406.49040MR3898187DOI10.1515/acv-2016-0059
  15. Giova, R. (2015), Higher differentiability for n-harmonic systems with Sobolev coefficients, J. Differential Equations, 259, no. 11, 5667-5687. Zbl1326.35155MR3397304DOI10.1016/j.jde.2015.07.004
  16. Giusti, E. (2003), Direct Methods in the Calculus of Variations, World Scientific Publishing. Zbl1028.49001MR1962933DOI10.1142/9789812795557
  17. Hajlasz, P. (1996), Sobolev Spaces on an Arbitrary Metric Space, Potential Anal.5 , 403-415. Zbl0859.46022MR1401074DOI10.1007/BF00275475
  18. Kristensen, J. and Mingione, G. (2010), Boundary Regularity in Variational Problems, Arch Rational Mech. Anal., 198, 369-455. Zbl1228.49043MR2721587DOI10.1007/s00205-010-0294-x
  19. Kuusi, T. and Mingione, G. (2012), Universal potential estimates, Journal of Functional Analysis, 262, 4205-4269. Zbl1252.35097MR2900466DOI10.1016/j.jfa.2012.02.018
  20. Passarelli di Napoli, A. (2014), Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Adv. Calc. Var.7 , no. 1, 59-89. Zbl1280.49058MR3176584DOI10.1515/acv-2012-0006
  21. Passarelli di Napoli, A. (2014), Higher differentiability of solutions of elliptic systems with Sobolev coefficients: The case p = n = 2 , Potential Anal.41, no. 3, 715-735. Zbl1315.35048MR3264817DOI10.1007/s11118-014-9390-0

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.