Regularity of weak solutions for aclass of elliptic PDEs in Orlicz-Sobolev spaces
Keywords
Regularity, elliptic equations, Orlicz-Sobolev spacesAbstract
We consider the elliptic partial differential equation in the divergence form \[ -\mathrm{div}(\nabla G(\nabla u(x)))+ F_u(x,u(x))=0, \] where $G$ is a convex, anisotropic function satisfying certain growth and ellipticity conditions. We prove that weak solutions in $W^{1,G}$ are in fact of class $W^{2,2}_{\loc}\cap W^{1,\infty}_{\loc}$.References
A. Alberico, Boundedness of solutions to anisotropic variational problems, Comm. Partial Differential Equations 36 (2010).
G. Barletta and A. Cianchi, Dirichlet problems for fully anisotropic elliptic equations, Proc. Roy. Soc. Edinburgh 147 (2017), 25–60.
P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. 57 (2018).
M. Bulı́ček, G. Cupini, B. Stroffolini and A. Verde, Existence and regularity results for weak solutions to (p, q)-elliptic systems in divergence form, Adv. Calc. Var. 11 (2017).
M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Pointwise bounds for minimizers of some anisotropic functionals, Nonlinear Anal. 177 (2018).
A. Cianchi, Local boundedness of minimizers of anisotropic functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000).
G. Cupini, P. Marcellini and E. Mascolo, Regularity under sharp anisotropic general growth conditions, Discrete Contin. Dyn. Syst. Ser. B (2009).
G. Cupini, P. Marcellini and E. Mascolo, Local boundedness of solutions to quasilinear elliptic systems, Manuscripta Math. 137 (2012).
G. Cupini, P. Marcellini and E. Mascolo, Existence and regularity for elliptic equations under p,q-growth, Advances in Differential Equations 19 (2014).
G. Cupini, P. Marcellini and E. Mascolo, Regularity of minimizers under limit growth conditions, Nonlinear Anal. 153 (2017).
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003.
A. Kovalevsky, Integrability and boundedness of solutions to some anisotropic problems, J. Math. Anal. Appl. 432 (2015).
P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991), 1–30.
P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), 296–333.
L. Pick, A. Kufner, O. John, S. Fučı́k, Function Spaces 1, De Gruyter (2013).
F. Siepe, On the Lipschitz regularity of minimizers of anisotropic functionals, J. Math. Anal. Appl. 263 (2001), 69–94.
N.S. Trudinger, An imbedding theorem for H0 (G, Ω) spaces. Studia Math. 50 (1974), 17–30.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
