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Kaushik Mohanta

Publications and source records attributed to Kaushik Mohanta.

14 recordsLinked to original sources
arXiv2026

How to recognise extension domains

Let $Ω\subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We prove that there is a bounded extension operator $\dot{W}^{1,p}(Ω)\to \dot{W}^{1,p}(\mathbb{R}^n)$ if and only if $Ω$ satisfies the measure density condition and a Bourgain-Brezis-Mironescu type inequality (or limiting formula). As a key ingredient, we establish a fractional Poincaré-type inequality under the assumption of Ahlfors regularity alone, improving a result of Ponce (2004). We also prove that, under a mild Hausdorff measure condition on the boundary $\partial Ω$, fractional extension (from $\dot{W}^{1,p}(Ω)$ to $\dot{W}^{s,p}(\mathbb{R}^n)$) at a single exponent $s > 1/p$ self-improves to full first-order Sobolev extension (from $\dot{W}^{1,p}(Ω)$ to $\dot{W}^{1,p}(\mathbb{R}^n)$). These results clarify the role of nonlocal estimates in the geometry of Sobolev extension domains.

math.FA↗
arXiv2026

Ciarlet Nečas condition in fractional Sobolev spaces

Let $s\in(\frac{n}{n+1},1)$, $Ω\subset\mathbb{R}^n$ be an open set and let $f\in W^{s,n/s}(Ω,\mathbb{R}^n)$ be mapping with positive distributional Jacobian $\mathcal{J}_f>0$ which models some deformation in fractional Nonlinear Elasticity. We show change of variables formula in this class and as a consequence we show that the analogue of Ciarlet-Nečas condition $\mathcal{J}_f(Ω)=|f(Ω)|$ implies that our mapping is one-to-one a.e.

math.FA↗
arXiv2025

Note on injectivity in second-gradient Nonlinear Elasticity

Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $Ω\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(Ω,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(Ω)$ are homeomorphism if they agree with a homeomorphism on the boundary. The condition $(1-\frac{1}{q})a\geq 1$ is sharp. We also have a new sharp result about the $\mathcal{H}^{n-1}$ measure of the projection of the set $\{J_f=0\}$ in $\mathbb{R}^n$.

math.AP↗
arXiv2024

Traces of vanishing Hölder spaces

For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the Hölder space $\dot{C}^{0,ω}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/ω(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions $\dot{C}^{0,ω}(E)$ by Lipschitz and boundedly supported functions.

math.CA↗
arXiv2024

Fractional Hardy inequalities and capacity density

We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting.

math.CA↗
arXiv2024

Classifying Triebel-Lizorkin capacities in metric spaces

We study non-local or fractional capacities in metric measure spaces. Our main goal is to clarify the relations between relative Hajlasz-Triebel-Lizorkin capacities, potentional Triebel-Lizorkin capacities, and metric space variants of Riesz capacities. As an application of our results, we obtain a characterization of a Hajlasz-Triebel-Lizorkin capacity density condition, which is based on an earlier characterization of a Riesz capacity density condition in terms of Hausdorff contents.

math.CA↗
arXiv2024

Bourgain-Brezis-Mironescu formula for $W^{s,p}_q$-spaces in arbitrary domains

Under certain restrictions on $s,p,q$, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces $W^{s,p}_q$. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for $W^{s,p}_q$-seminorms in arbitrary domain. This addresses an open question raised by Brazke-Schikorra-Yung in [Bourgain-Brezis-Mironescu convergence via Triebel-Lizorkin spaces; Calc. Var. Partial Differential Equations; 2023].

math.FA↗
arXiv2023

On the singular problem involving $g$-Laplacian

In this paper, we show that the existence of a positive weak solution to the equation $(-Δ_g)^s u=f u^{-q(x)}\;\mbox{in}\; Ω,$ where $Ω$ is a smooth bounded domain in $R^N$, $q\in C^1(\overlineΩ)$, and $(-Δ_g)^s$ is the fractional $g$-Laplacian with $g$ is the antiderivative of a Young function and $f$ in suitable Orlicz space subjected to zero Dirichlet condition. This includes the mixed fractional $(p,q)-$Laplacian as a special case. The solution so obtained is also shown to be locally Hölder continuous.

math.AP↗
arXiv2023

Magnetic fractional Poincaré inequality in punctured domains

We study Poincaré-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb-Seiringer-Yngvason [E. Lieb, R. Seiringer, and J. Yngvason, Poincaré inequalities in punctured domains, Ann. of Math., 2003] showed that, if a bounded domain $Ω$ is the union of two disjoint sets $Γ$ and $Λ$, then the $L^p$-norm of a function calculated on $Ω$ is dominated by the sum of magnetic seminorms of the function, calculated on $Γ$ and $Λ$ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete.

math.FA↗
arXiv2023

Improved Hardy inequalities on Riemannian Manifolds

We study the following version of Hardy-type inequality on a domain $Ω$ in a Riemannian manifold $(M,g)$: $$ \intΩ|\nabla u|_g^pρ^αdV_g \geq \left(\frac{|p-1+β|}{p}\right)^p\intΩ\frac{|u|^p|\nabla ρ|_g^p}{|ρ|^p}ρ^αdV_g +\intΩ V|u|^pρ^αdV_g, \quad \forall\ u\in C_c^\infty (Ω). $$ We provide sufficient conditions on $p, α, β,ρ$ and $V$ for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, $\mathbb{R}^N$ with $p<N$, $\mathbb{R}^N\setminus \{0\}$ with $p\geq N$, $\mathbb{H}^N$, etc.

math.AP↗
arXiv2021

Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces

We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions on open sets are provided for fractional Orlicz Poincaré inequality and regional fractional Orlicz Poincaré inequality to hold.

math.AP↗
arXiv2021

On the best constant in fractional $p$-Poincaré inequalities on cylindrical domains

We investigate the best constants for the regional fractional $p$-Poincaré inequality and the fractional $p$-Poincaré inequality in cylindrical domains. For the special case $p=2$, the result was already known due to Chowdhury-Csató-Roy-Sk [Study of fractional Poincaré inequalities on unbounded domains, Discrete Contin. Dyn. Syst., 41(6), 2021]. We addressed the asymptotic behaviour of the first eigenvalue of the nonlocal Dirichlet $p$-Laplacian eigenvalue problem when the domain is becoming unbounded in several directions.

math.AP↗
arXiv2020

Bourgain-Brezis-Mironescu Domains

Bourgain et al.(2001) proved that for $p>1$ and smooth bounded domain $Ω\subseteq\mathbb{R}^N$, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y \rvert^{N+sp}}dx dy=κ\int \limits_Ω\lvert \nabla f(x) \rvert^p dx \end{equation*} for all $f\in L^p(Ω)$. This gives a characterization of $W^{1,p}(Ω)$ by means of $W^{s,p}(Ω)$ seminorms only. For the case $p=1$, Dávila(2002) proved that when $Ω$ is a bounded domain with Lipschitz boundary, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert}{\lvert x-y \rvert^{N+s}}dx dy=κ[f]_{BV(Ω)} \end{equation*} for all $f\in L^1(Ω)$. This characterizes $BV(Ω)$ in terms of $W^{s,1}(Ω)$ seminorm. In this paper we extend the first result and partially extend the second result to extension domains.

math.AP↗