arXiv subjects

Indranil Chowdhury

Publications and source records attributed to Indranil Chowdhury.

13 recordsLinked to original sources

arXiv2026

Large Solutions for Fractional Laplacian on Infinite Cylindrical Domains

We investigate large solutions of linear and semi-linear equations involving the fractional Laplacian on domains that are becoming unbounded in some, but not all, directions. Solutions that blow up on the boundary of a domain are commonly called large solutions. For local operators, such behaviour arises only in the presence of lower-order nonlinear terms. However, for nonlocal operators, the existence of such solutions is more subtle. In contrast to the local case, the nonlocal nature of the operator gives rise to different boundary blow-up phenomena even in the case of linear equations. In this article, we study the existence and qualitative behaviour of boundary blow-up solutions to fractional linear and semi-linear equations on finite cylindrical domains and employ these properties to construct large solutions on infinite cylinders.

math.AP↗

arXiv2025

Large solutions to semilinear equations for subordinate Laplacians in $C^{1,1}$ bounded open sets

We study the existence of a large solution to a semilinear problem in a bounded open $C^{1,1}$ set for a class of nonlocal operators obtained by an appropriate subordination of the Laplacian. These operators are classical generalisations of the fractional Laplacian. The existence result is shown under a nonlocal version of the Keller-Osserman condition, stated in terms of the subordinator and the source term $f$.

math.AP↗

arXiv2024

Boundary Blow-up Solutions of Second Order Quasilinear Equation on Infinite Cylinders

This article studies large solutions, for a class of quasi-linear equations involving p-Laplacian on the infinite cylindrical domains. We study the wellposedness of weak large solutions on infinite cylinders by the convergence of large solutions on finite cylinders and observe that any such solution coincides with the large solution on its cross-section. Finally, the results are generalized to a class of operators involving non-linearity in the gradient.

math.AP↗

arXiv2024

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $σ\in(0,2)$ since they involve fractional Laplace operators $(-Δ)^{σ/2}$. They arise e.g. in control and game theory as dynamic programming equations -- HJB and Isaacs equation -- and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $σ$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $σ$ and are worse when $σ$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

math.NA↗

arXiv2024

A strongly degenerate fully nonlinear mean field game with nonlocal diffusion

There are few results on mean field game (MFG) systems where the PDEs are either fully nonlinear or have degenerate diffusions. This paper introduces a problem that combines both difficulties. We prove existence and uniqueness for a strongly degenerate, fully nonlinear MFG system by using the well-posedness theory for fully nonlinear MFGs established in our previous paper. It is the first such application in a degenerate setting. Our MFG involves a controlled pure jump (nonlocal) Lévy diffusion of order less than one, and monotone, smoothing couplings. The key difficulty is obtaining uniqueness for the corresponding degenerate, non-smooth Fokker-Plank equation: since the regularity of the coefficient and the order of the diffusion are interdependent, it holds when the order is sufficiently low. Viscosity solutions and a non-standard doubling of variables argument are used along with a bootstrapping procedure.

math.AP↗

arXiv2024

On fully nonlinear parabolic mean field games with nonlocal and local diffusions

We introduce a class of fully nonlinear mean field games posed in $[0,T]\times\mathbb{R}^d$. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (Lévy) processes. The main results are existence and uniqueness of solutions under general assumptions. These results are applied to non-degenerate equations - including both local second order and nonlocal with fractional Laplacians. Uniqueness holds under monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions.

math.AP↗

arXiv2023

Precise Error Bounds for Numerical Approximations of Fractional HJB Equations

We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. We consider diffusion corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations which are (formally) fractional order and 2nd order methods. It is well-known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) Strongly degenerate problems with Lipschitz solutions, and (ii) weakly non-degenerate problems where we show that solutions have bounded fractional derivatives of order between 1 and 2. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly non-degenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than 1/2.

math.AP↗

arXiv2021

Fractional Poincaré Inequality for Unbounded Domains with Finite Ball Condition: Counter Example

In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Sandeep-Mancini showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain does not allow balls of arbitrarily large radius (finite ball condition). We prove that such a result can not be true in the `nonlocal/fractional' setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all eigenvalues of fractional Dirichlet problems on long cylindrical domains is addressed.

math.AP↗

arXiv2021

On Numerical approximations of fractional and nonlocal Mean Field Games

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

math.AP↗

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↗

arXiv2017

On the rate of convergence for monotone numerical schemes for nonlocal Isaacs' equations

We study monotone numerical schemes for nonlocal Isaacs equations, the dynamic programming equations of stochastic differential games with jump-diffusion state processes. These equations are fully-nonlinear non-convex equations of order less than $2$. In this paper they are also allowed to be degenerate and have non-smooth solutions. The main contribution is a series of new a priori error estimates: The first results for nonlocal Isaacs equations, the first general results for degenerate non-convex equations of order greater than $1$, and the first results in the viscosity solution setting giving the precise dependence on the fractional order of the equation. We also observe a new phenomena, that the rates differ when the nonlocal diffusion coefficient depend on $x$ and $t$, only on $x$, or on neither.

math.AP↗

arXiv2015

On the differentiability of the solutions of non-local Isaacs equations involving $\frac 12$-Laplacian

We derive $C^{1,σ}$-estimate for the solutions of a class of non-local elliptic Bellman-Isaacs equations. These equations are fully nonlinear and are associated with infinite horizon stochastic differential game problems involving jump-diffusions. The non-locality is represented by the presence of fractional order diffusion term and we deal with the particular case of $\frac 12$-Laplacian, where the order $\frac 12$ is known as the critical order in this context. More importantly, these equations are not translation invariant and we prove that the viscosity solutions of such equations are $C^{1,σ}$, making the equations classically solvable.

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↗