Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights

Filomena Feo (), Antonia Passarelli di Napoli () and Maria Rosaria Posteraro ()
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Filomena Feo: Università degli Studi di Napoli “Parthenope”
Antonia Passarelli di Napoli: Università degli Studi di Napoli “Federico II”
Maria Rosaria Posteraro: Università degli Studi di Napoli “Federico II”

Journal of Optimization Theory and Applications, 2024, vol. 201, issue 3, No 14, 1313-1332

Abstract: Abstract We study the local boundedness of minimizers of non uniformly elliptic integral functionals with a suitable anisotropic $$p,q-$$ p , q - growth condition. More precisely, the growth condition of the integrand function $$f(x,\nabla u)$$ f ( x , ∇ u ) from below involves different $$p_i>1$$ p i > 1 powers of the partial derivatives of u and some monomial weights $$|x_i|^{\alpha _i p_i}$$ | x i | α i p i with $$\alpha _i \in [0,1)$$ α i ∈ [ 0 , 1 ) that may degenerate to zero. Otherwise from above it is controlled by a q power of the modulus of the gradient of u with $$q\ge \max _i p_i$$ q ≥ max i p i and an unbounded weight $$\mu (x)$$ μ ( x ) . The main tool in the proof is an anisotropic Sobolev inequality with respect to the weights $$|x_i|^{\alpha _i p_i}$$ | x i | α i p i .

Keywords: Degenerate anisotropic functionals; Local boundedness; Weighted Sobolev inequalities; Monomial weights; 35J87; 49J40; 47J20; 35A23 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02432-3

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