We study the functional $I_{f}(u)=∫_{Ω} f(u(x))dx$, where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$'s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j=1,...,m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.
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We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional $\int_{Ω} F(x,u,∇u)dx$. The proofs are based on arguments from the theory of Young measures.
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We obtain interpolation inequalities for derivatives: $∫ M_{q,α}(|∇f(x)|)dx ≤ C[∫M_{p,β}(Φ₁(x,|f|,|∇^{(2)}f|))dx + ∫M_{r,γ}(Φ₂(x,|f|,|∇^{(2)}f|))dx]$, and their counterparts expressed in Orlicz norms: ||∇f||²_{(q,α)} ≤ C||Φ₁(x,|f|,|∇^{(2)}f|)||_{(p,β)} ||Φ₂(x,|f|,|∇^{(2)}f|)||_{(r,γ)}$, where $||·||_{(s,κ)}$ is the Orlicz norm relative to the function $M_{s,κ}(t) = t^{s}(ln(2+t))^{κ}$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher order gradients are also considered.
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We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.
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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.
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Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities $∫_{ℝ₊} M(ω(x)|u(x)|) exp(-φ(x)) dx ≤ C ∫_{ℝ₊} M(|u'(x)|) exp(-φ(x)) dx$, where u belongs to some set 𝓡 of locally absolutely continuous functions containing $C₀^{∞}(ℝ₊)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set 𝓡. The set 𝓡 may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.
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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.
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We prove several results concerning density of $C_{0}^{∞}$, behaviour at infinity and integral representations for elements of the space $L^{m,p} = {⨍ | ∇^{m}⨍ ∈ L^p}$.
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