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New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

100%

Kałamajska A. , Pietruska-Pałuba K.

Open Mathematics

2012

tom 10

nr 6

2033-2050

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands

100%

Tachago J. F. , Nnang H. , Zappale E.

Opuscula Mathematica

2021

tom Vol. 41, no. 1

113--143

Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.

On a Variant of the Gagliardo–Nirenberg Inequality Deduced from the Hardy Inequality

100%

Kałamajska A. , Pietruska-Pałuba K.

2011

tom Vol. 59, no 2

133--149

We obtain new variants of weighted Gagliardo–Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

100%

Černý R.

nr 1

114-127

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}} {{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u} {{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ℝn.

Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities

75%

Černý R.

Open Mathematics

2012

tom 10

nr 2

590-602

Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the exponent concerning the Concentration-Compactness Principle for the embedding of the Orlicz-Sobolev space W 01 L n logα L(Ω) into the Orlicz space corresponding to a Young function that behaves like exp t n/(n−1−α) for large t. We also give the result for the case of the embedding into double and other multiple exponential spaces.

Existence of solutions for unilateral problems in L1 involving lower order terms in divergence form in Orlicz spaces

75%

Aharouch L. , Azroul E. , Rhoudaf M.

Journal of Applied Analysis

2007

tom Vol. 13, nr 2

151-181

This article is concerned with the existence result of the unilateral problem associated to the equations of the type Au - divφ(u) = ∫ ∈ L1 (Ω), where A is a Leray-Lions operator having a growth not necessarily of polynomial type and Φ ∈ C0 (R,RN).