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Prosenjit Roy

Publications and source records attributed to Prosenjit Roy.

26 records · Page 2Linked to original sources
arXiv2020

Study of fractional Poincaré inequalities on unbounded domains

The central aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various conditions on domains and on the range of $s \in (0,1)$. The best constant in both regional fractional and fractional Poincaré inequality is characterized for strip like domains $(ω\times \mathbb{R}^{n-1})$, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [\textit{Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2}].

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↗
arXiv2016

Extremal function for Moser-Trudinger type Inequality with Logarithmic weight

On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $β\in [0,1)$ and $w_0(x) = |\log |x||^β$ then $$ \sup_{\int_B |\grad u|^2w_0 \leq 1 , u \in H_{0,rad}^1(w_0,B)} \int_B e^{αu^{\frac{2}{1-β}}} dx < \infty,$$ if and only if $α\leq α_β= 2\left[2π(1-β) \right]^{\frac{1}{1-β}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $α= α_β$ thereby generalizing the result of Carleson-Chang who proved the case when $β=0$.

math.AP↗
arXiv2016

On some Variational Problems set on domains tending to infinity

Let $Ω_\ell = \ellω_1 \times ω_2$ where $ω_1 \subset \R^p$ and $ω_2 \subset \R^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{Ω_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(Ω_\ell)}E_{Ω_\ell}(u)$$ where $E_{Ω_\ell}(u) = \int_{Ω_\ell}F(\grad u)-fu$, $F$ is a convex function and $f\in L^{q'}(ω_2)$. We are interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.

math.AP↗
arXiv2015

On the Asymptotic Analysis of Problems Involving Fractional Laplacian in Cylindrical Domains Tending to Infinity

The article is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second order elliptic problems by Chipot and Rougirel, where the force functions are considered on the cross section of domains, we prove the non-local counterpart of their result. Furthermore, recently Yeressian established a weighted estimate for solutions of nonlocal Dirichlet problems which exhibit the asymptotic behavior. The case whens= 1=2 was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this article, we extend this result to each order between 0 and 1.

math.AP↗
arXiv2015

The singular Moser- Trudger Inequality on simply connected domains

In this paper the authors complete their study of the singular Moser-Trudinger embedding [G. Csato and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-015-0867-5], abbreviated [CR]. The proof in [CR] is however far too technical and complicated for simply connected domains. Here we give a much simpler and more self-contained proof using complex analysis, which also generalizes the corresponding proof given by Flucher for such domains. This should make [CR] more easily accessible.

math.AP↗