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On very weak solutions of a class of nonlinear elliptic systems

Menita CarozzaAntonia Passarelli di Napoli — 2000

Commentationes Mathematicae Universitatis Carolinae

In this paper we prove a regularity result for very weak solutions of equations of the type - div A ( x , u , D u ) = B ( x , u , D u ) , where A , B grow in the gradient like t p - 1 and B ( x , u , D u ) is not in divergence form. Namely we prove that a very weak solution u W 1 , r of our equation belongs to W 1 , p . We also prove global higher integrability for a very weak solution for the Dirichlet problem - div A ( x , u , D u ) = B ( x , u , D u ) in Ω , u - u o W 1 , r ( Ω , m ) .

Model problems from nonlinear elasticity: partial regularity results

Menita CarozzaAntonia Passarelli di Napoli — 2007

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we prove that every weak and strong local minimizer u W 1 , 2 ( Ω , 3 ) of the functional I ( u ) = Ω | D u | 2 + f ( Adj D u ) + g ( det D u ) , where u : Ω 3 3 , grows like | Adj D u | p , grows like | det D u | q and , is C 1 , α on an open subset Ω 0 of such that 𝑚𝑒𝑎𝑠 ( Ω Ω 0 ) = 0 . Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case p = q 2 is also treated for weak local minimizers.

Regularity results for an optimal design problem with a volume constraint

Menita CarozzaIrene FonsecaAntonia Passarelli di Napoli — 2014

ESAIM: Control, Optimisation and Calculus of Variations

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with -power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (), Hölder continuity of the function is proved as well as partial regularity of the boundary of the minimal set . Moreover, full regularity of the boundary of the minimal set is obtained...

Linear elliptic equations with BMO coefficients

Menita CarozzaGioconda MoscarielloAntonia Passarelli di Napoli — 1999

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove an existence and uniqueness theorem for the Dirichlet problem for the equation div a x u = div f in an open cube Ω R N , when f belongs to some L p Ω , with p close to 2. Here we assume that the coefficient a belongs to the space BMO( Ω ) of functions of bounded mean oscillation and verifies the condition a x λ 0 > 0 for a.e. x Ω .

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