### Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional ${\int}_{\Omega}F(x,u,\nabla u)dx$. The proofs are based on arguments from the theory of Young measures.

We study the functional ${I}_{f}\left(u\right)={\int}_{\Omega}f\left(u\left(x\right)\right)dx$, 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.

We prove several results concerning density of ${C}_{0}^{\infty}$, behaviour at infinity and integral representations for elements of the space ${L}^{m,p}=\u2a0d|{\nabla}^{m}\u2a0d\in {L}^{p}$.

We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla {u}_{k}\right\}$, bounded in ${L}^{p}(\xd8;{\mathbb{R}}^{m\times n})$ if $p\>1$ and $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain with the extension property in ${W}^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega $ are required and links with lower semicontinuity results...

We consider the lower semicontinuous functional of the form ${I}_{f}\left(u\right)={\int}_{\Omega}f\left(u\right)\mathrm{d}x$ where $u$ satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar’s $\Lambda $-convexity condition for the integrand $f$ extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex...

We consider the lower semicontinuous functional of the form ${I}_{f}\left(u\right)={\int}_{\Omega}f\left(u\right)\mathrm{d}x$ where satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's -convexity condition for the integrand extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to...

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla {u}_{k}\right\}$, bounded in ${L}^{p}(\Omega ;{\mathbb{R}}^{m\times n})$ if and $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain with the extension property in ${W}^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of are required and links with lower semicontinuity results...

We obtain interpolation inequalities for derivatives: $\int {M}_{q,\alpha}\left(\right|\nabla f\left(x\right)\left|\right)dx\le C[\int {M}_{p,\beta}\left(\Phi \u2081\right(x,\left|f\right|,|{\nabla}^{\left(2\right)}f\left|\right))dx+\int {M}_{r,\gamma}\left(\Phi \u2082\right(x,\left|f\right|,|{\nabla}^{\left(2\right)}f\left|\right)\left)dx\right]$, 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 ${\left|\right|\xb7\left|\right|}_{(s,\kappa )}$ is the Orlicz norm relative to the function ${M}_{s,\kappa}\left(t\right)={t}^{s}{\left(ln(2+t)\right)}^{\kappa}$. 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...

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 ${\int}_{\mathbb{R}\u208a}M\left(\omega \left(x\right)\right|u\left(x\right)\left|\right)exp(-\phi \left(x\right))dx\le C{\int}_{\mathbb{R}\u208a}M\left(\right|{u}^{\text{'}}\left(x\right)\left|\right)exp(-\phi \left(x\right))dx$, where u belongs to some set of locally absolutely continuous functions containing $C{\u2080}^{\infty}\left(\mathbb{R}\u208a\right)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

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.

We obtain Hardy type inequalities $${\int}_{0}^{\infty}M\left(\omega \left(r\right)\left|u\left(r\right)\right|\right)\rho \left(r\right)dr\u2a7d{C}_{1}{\int}_{0}^{\infty}M\left(\left|u\left(r\right)\right|\right)\rho \left(r\right)dr+{C}_{2}{\int}_{0}^{\infty}M\left(\left|{u}^{\text{'}}\left(r\right)\right|\right)\rho \left(r\right)dr,$$ and their Orlicz-norm counterparts $${\u2225\omega u\u2225}_{{L}^{M}({\mathbb{R}}_{+},\rho )}\u2a7d{\tilde{C}}_{1}{\u2225u\u2225}_{{L}^{M}({\mathbb{R}}_{+},\rho )}+{\tilde{C}}_{2}{\u2225{u}^{\text{'}}\u2225}_{{L}^{M}({\mathbb{R}}_{+},\rho )},$$ 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.

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.

**Page 1**