arXiv subjects

Prasun Roychowdhury

Publications and source records attributed to Prasun Roychowdhury.

23 records · Page 2Linked to original sources

arXiv2021

On ergodic control problem for viscous Hamilton--Jacobi equations for weakly coupled elliptic systems

In this article we study ergodic problems in the whole space $\mathbb{R}^N$ for weakly coupled systems of viscous Hamilton-Jacobi equations with coercive right-hand sides. The Hamiltonians are assumed to have a fairly general structure and the switching rates need not be constant. We prove the existence of a critical value $λ^*$ such that the ergodic eigenvalue problem has a solution for every $λ\leqλ^*$ and no solution for $λ>λ^*$. Moreover, the existence and uniqueness of non-negative solutions corresponding to the value $λ^*$ are also established. We also exhibit the implication of these results to the ergodic optimal control problems of controlled switching diffusions.

math.AP↗

arXiv2021

Hardy-Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

We prove a family of Hardy-Rellich and Poincaré identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincaré inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived.

math.AP↗

arXiv2020

Generalized principal eigenvalues of convex nonlinear elliptic operators in $\mathbb{R}^N$

We study the generalized eigenvalue problem in $\mathbb{R}^N$ for a general convex nonlinear elliptic operator which is locally elliptic and positively $1$-homogeneous. Generalizing article of Berestycki and Rossi in [Comm. Pure Appl. Math. 68 (2015), no. 6, 1014-1065] we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

math.AP↗

arXiv2020

On Higher order Poincaré Inequalities with radial derivatives and Hardy improvements on the hyperbolic space

In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, \begin{equation*} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{k} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \geq \bigg(\frac{N-1}{2}\bigg)^{2(k-l)} \int_{\mathbb{H}^{N}} |\nabla_{r,\mathbb{H}^{N}}^{l} u|^2 \, {\rm d}v_{\mathbb{H}^{N}} \ \ \text{ for all } u\in H^k(\mathbb{H}^{N}), \end{equation*} where underlying space is $N$-dimensional hyperbolic space $\mathbb{H}^{N}$, $0\leq l<k$ are integers and the constant $\big(\frac{N-1}{2}\big)^{2(k-l)}$ is sharp. Furthermore we improve the above inequalities by adding Hardy-type remainder terms and the sharpness of some constants is also discussed.

math.FA↗

arXiv2020

On some strong Poincaré inequalities on Riemannian models and their improvements

We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed.

math.FA↗