Some remarks on the regularity of anisotropic variational problems.
Some remarks on the regularity of anisotropic variational problems.
The author studies elliptic equations of the type \(\text{div}(a(x,u,Du))=\text{div}(g)\), \(u\in W^{1,(q_i)}_0(\Omega)\), and integral functionals of the calculus of variations of the form \(F(u)=\int_\Omega f(x,Du)dx\) when \(a\) or \(f\) satisfy the anisotropic coerciveness condition \(f(x,\xi)\geq m\sum^N_{i=1}|\xi_i|^{q_i}\). Various regularity results are obtained in Lorentz spaces, Orlicz spaces and in the Lebesgue space \(L^\infty(\Omega)\). The estimates are obtained by using test functions defined thanks to the nonincreasing rearrangement of \(u\). These results extend previous results obtained in this direction [cf., e.g., \textit{N. Fusco} and \textit{C. Sbordone}, Commun. Partial Differ. Equations 18, No. 1-2, 153-167 (1993; Zbl 0795.49025); \textit{B. Stroffolini}, Boll. Unione Mat. Ital., VII. Ser. A5, No. 3, 345-352 (1991; Zbl 0754.46026)].
Variational methods for second-order elliptic equations, integral functionals, regularity, Regularity of solutions in optimal control, elliptic equations, anisotropic coerciveness condition, Existence theories for free problems in two or more independent variables
Variational methods for second-order elliptic equations, integral functionals, regularity, Regularity of solutions in optimal control, elliptic equations, anisotropic coerciveness condition, Existence theories for free problems in two or more independent variables
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