arXiv subjects

Ujjal Das

Publications and source records attributed to Ujjal Das.

22 records · Page 2Linked to original sources

arXiv2022

The space of Hardy-weights for quasilinear equations: Maz'ya-type characterization and sufficient conditions for existence of minimizers

Let $p \in (1,\infty)$ and $Ω\subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(Ω; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|ξ|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) ξ_i ξ_j$, $ξ\in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(ϕ):=\displaystyle \int_Ω [|\nabla ϕ|_A^p + V|ϕ|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(Ω)\cap C_c(Ω)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all Hardy-weights for the functional $Q_{p,A,V}$, extending Maz'ya's well known characterization of the space of Hardy-weights for the $p$-Laplacian. In addition, we provide various sufficient conditions on the potential $V$ and the Hardy-weight $g$ such that the best constant of the corresponding variational problem is attained in an appropriate Beppo-Levi space.

math.AP↗

arXiv2021

The compactness and the concentration compactness via $p$-capacity

For $p \in (1,N)$ and $Ω\subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$ is the completion of $C_c^{\infty}(Ω)$ with respect to the norm $\left( \int_Ω|\nabla u|^p \right)^ \frac{1}{p}.$ Using the $p$-capacity, we define a norm and then identify the Banach function space $\mathcal{H}(Ω)$ with the set of all $g$ in $L^1_{loc}(Ω)$ that admits the following Hardy-Sobolev type inequality: \begin{eqnarray*} \int_Ω |g| |u|^p \leq C \int_Ω |\nabla u|^p, \forall\; u \in \mathcal{D}^{1,p}_0(Ω), \end{eqnarray*} for some $C>0.$ Further, we characterize the set of all $g$ in $\mathcal{H}(Ω)$ for which the map $G(u)= \int_Ω g |u|^p$ is compact on $\mathcal{D}^{1,p}_0(Ω)$. We use a variation of the concentration compactness lemma to give a sufficient condition on $g\in \mathcal{H}(Ω)$ so that the best constant in the above inequality is attained in $\mathcal{D}^{1,p}_0(Ω)$.

math.AP↗

arXiv2021

On the Generalized Hardy-Rellich Inequalities

In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: $ \int_Ω g(x) u^2 dx \leq C \int_Ω |Δu|^2 dx, \forall u \in \mathcal{D}^{2,2}_0(Ω), $ for some constant $C>0$, where $Ω$ is an open set in $\mathbb{R}^N$ with $N\ge 1$. We find various classes of such weight functions, depending on the dimension $N$ and the geometry of $Ω.$ Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of $\mathcal{D}^{2,2}_0(Ω)$ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

math.AP↗

arXiv2020

On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

For $N \geq 3$ and $p \in (1,N)$, we look for $g \in L^1_{loc}(\mathbb{R}^N)$ that satisfies the following weighted logarithmic Sobolev inequality: \begin{equation*} \int_{\mathbb{R}^N} g |u|^p \log |u|^p \ dx \leq γ\log \left( C_γ \int_{\mathbb{R}^N} |\nabla u|^p \ dx \right) \,, \end{equation*} for all $u \in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$ with $\int_{\mathbb{R}^N} g|u|^p=1$, for some $γ,C_γ>0$. For each $r \in(p,\frac{Np}{N-p}]$, we identify a Banach function space $\mathcal{H}_{p,r}(\mathbb{R}^N)$ such that the above inequality holds for $g \in \mathcal{H}_{p,r}(\mathbb{R}^N)$. For $γ> \frac{r}{r-p}$, we also find a class of $g$ for which the best constant $C_γ$ in the above inequality is attained in $\mathcal{D}^{1,p}_0(\mathbb{R}^N)$. Further, for a closed set $E$ with Assouad dimension $=d 0$, where $δ_E(x)$ is the distance between $x$ and $E$. The second order extension of the logarithmic Hardy inequality is also obtained.

math.AP↗