Regularity for anisotropic solutions to some nonlinear elliptic system

Abstract

This paper deals with anisotropic solutions u\in W^{\mathrm{1},(pi)}(\Omega ,{ℝ}^N) to the nonlinear elliptic system

-\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}Di\left(a_i^{\alpha }(\chi ,Du\left(\chi \right))\right)=-\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}Di{F}_i^{\alpha }\left(\chi \right),\alpha =1,2,\mathrm{...},N,

We present a monotonicity inequality for the matrix a=({{\displaystyle a}}_i^{\alpha })\in {ℝ}^{N\times n},whichguarantees global pointwise bounds for anisotropic solutionsu.

Key words： Regularity; anisotropic solution; nonlinear elliptic system

References

1 Gao H.Regularity for solutions to anisotropic obstacle problems. Sci China Math, 2014, 57: 111–122

2 Gao H, Chu Y.Quasiregular Mappings and A-Harmonic Equation. Beijing: Science Press, 2013 (in Chinese)

3 Gao H, Di Q, Ma D.Integrability for solutions to some anisotropic obstacle problems. Manuscripta Math (to appear)

4 Gao H, Huang Q.Local regularity for solutions of anisotropic obstacle problems. Nonlinear Anal, 2012, 75: 4761–4765

5 Gao H,Liu C, Tian H. Remarks on a paper by Leonetti and Siepe. J Math Anal Appl, 2013, 401: 881–887

6 Giachetti D, Porzio M M. Local regularity results for minima of functionals of the calculus of variation. Nonlinear Anal, 2000, 39: 463–482

7 Giaquinta M.Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Ann of Math Stud, Vol 105. Princeton: Princeton University Press, 1983

8 De Giorgi E.Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll Unione Mat Ital, 1968, 4: 135–137

9 Leonetti F, Mascolo E.Local boundedness for vector valued minimizers of anisotropic functionals. Z Anal Anwend, 2012, 31: 357–378

10 Leonetti F, Petricca P V.Regularity for vector valued minimizers of some anisotropic integral functionals. J Inequal Pure Appl Math, 2006, 7(3): Art 88

11 Leonetti F, Petricca P V.Existence of bounded solutions to some nonlinear degenerate elliptic systems. Discrete Contin Dyn Syst Ser B, 2009, 11: 191–203

12 Leonetti F, Siepe F.Integrability for solutions to some anisotropic elliptic equations. Nonlinear Anal, 2012, 75: 2867–2873

13 Leonetti F, Siepe F.Global integrability for minimizers of anisotropic functionals. Manuscripta Math, 2014, 144: 91–98

14 Mingione G.Regularity of minima: an invitation of dark side of the calculus of variations. Appl Math, 2006, 51: 355–426

15 Stampacchia G.Equations Elliptiques du second ordre a coefficientes discontinus. Semin de Math Superieures, Univ de Montreal, 1996

16 Stroffolini B.Global boundedness of solutions of anisotropic variational problems. Boll Unione Mat Ital, 1991, 5A: 345–352

17 Tang Q.Regularity of minimizers of non-isotropic integrals of the calculus of variations. Ann Mat Pura Appl, 1993, 164: 77–87

18 Zhou S.A note on nonlinear elliptic systems involving measure. Electron J Differential Equations, 2000, (08): 1–6