We study the local boundedness of minimizers of a nonuniformly energy integral of the form $int_{Omega }f(x,Dv),dx$ under $p,q$-growth conditions of the type [lambda (x)|\xi |^{p}leq f(x,\xi )leq mu (x)left( 1+|\xi |^{q} ight)] for some exponents $qgeq p>1$ and with nonnegative functions [$lambda ,mu $ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where $lambda (x)$ and $mu (x)$ had the role of the minimum and the maximum eigenvalues of an $n imes n$ symmetric matrix $left( a_{ij}left( x ight) ight) $ and [f(x,\xi )=displaystyle sum_{i,j=1}^{n}a_{ij}left( x ight) \xi _{i}\xi _{j}] was the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands $f(x,\xi)$ satisfying the general growth conditions above.

Nonuniformly elliptic energy integrals with p,q-growth

Cupini, Giovanni;
2018

Abstract

We study the local boundedness of minimizers of a nonuniformly energy integral of the form $int_{Omega }f(x,Dv),dx$ under $p,q$-growth conditions of the type [lambda (x)|\xi |^{p}leq f(x,\xi )leq mu (x)left( 1+|\xi |^{q} ight)] for some exponents $qgeq p>1$ and with nonnegative functions [$lambda ,mu $ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where $lambda (x)$ and $mu (x)$ had the role of the minimum and the maximum eigenvalues of an $n imes n$ symmetric matrix $left( a_{ij}left( x ight) ight) $ and [f(x,\xi )=displaystyle sum_{i,j=1}^{n}a_{ij}left( x ight) \xi _{i}\xi _{j}] was the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands $f(x,\xi)$ satisfying the general growth conditions above.
2018
Cupini, Giovanni; Marcellini, Paolo*; Mascolo, Elvira
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/649528
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