Abstract We prove an existence (and regularity) result of weak solutions u ∈W1,p0( )∩W1,qloc( ), to a Dirichlet problem for a second order elliptic equation in divergence form, under general and p, q−growth conditionsof the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with q=p. We found a way to treat the general context with explicit dependence on (x,u), other than on the gradient variable ξ=Du; these aspects require particular attention due to the p, q-context, with some differences and new difficulties compared to the standard case p=q.
The Leray-Lions existence theorem under general growth conditions / Giovanni Cupini, Paolo Marcellini, Elvira Mascolo. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 1090-2732. - STAMPA. - 416:(2024), pp. 1405-1428. [10.1016/j.jde.2024.10.025]
The Leray-Lions existence theorem under general growth conditions
Paolo Marcellini
;Elvira Mascolo
2024
Abstract
Abstract We prove an existence (and regularity) result of weak solutions u ∈W1,p0( )∩W1,qloc( ), to a Dirichlet problem for a second order elliptic equation in divergence form, under general and p, q−growth conditionsof the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with q=p. We found a way to treat the general context with explicit dependence on (x,u), other than on the gradient variable ξ=Du; these aspects require particular attention due to the p, q-context, with some differences and new difficulties compared to the standard case p=q.File | Dimensione | Formato | |
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