Search arXiv⌕ Search

arXiv subjects

Tilak Bhattacharya

Publications and source records attributed to Tilak Bhattacharya.

10 recordsLinked to original sources
arXiv2022

A Strong Minimum principle and Large Time Asymptotics for viscosity solutions to a class of doubly nonlinear possibly degenerate parabolic equations

We study a version of the strong minimum principle, and large time asymptotics of positive viscosity solutions to classes of doubly nonlinear parabolic equations of the form $$ H(Du,D^2u)-u^{k-1}u_t=0,\;\;k\geq 1,\quad\mbox{in $Ω\times [0,T)$},$$ where $Ω\subset \mathbb{R}^n$ is a bounded domain and $0<T\leq \infty$. The spatial operator $H$ is homogeneous with power $k$.

math.AP↗
arXiv2017

On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations

In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+χ(t)|Du|^Γ-f(u)u_t=$ in $Ω\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions, $Γ>0$ and depends on the degree of homogeneity of $H$, $f>0$, increasing and meets a concavity condition. We also consider the case $f\equiv 1$ and prove existence of solutions without sign restrictions.

math.AP↗
arXiv2016

On the viscosity solutions to some nonlinear elliptic equations

We consider viscosity solutions of a class of nonlinear degenerate elliptic equations on bounded domains. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of a first eigenvalue and a first positive eigenfunction.

math.AP↗
arXiv2015

On the viscosity solutions to Trudinger's equation

We study the existence of positive viscosity solutions to Trudinger's equation for cylindrical domains $Ω\times[0, T)$, where $Ω\subset \mathbb{R}^n,\;n\ge 2,$ is a bounded domain, $T>0$ and $2\leq p<\infty$. We show existence for general domains $Ω,$ when $n<p<\infty$. For $2\leq p\leq n$, we prove existence for domains $Ω$ that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron's method.

math.AP↗
arXiv2013

An Eigenvalue problem for the Infinity-Laplacian

We study an eigenvalue problem for the infinity-Laplacian on bounded domains. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The work also contains existence results when the parameter, in the equation, is less than the first eigenvalue. A comparison principle applicable to these problems is also proven. Some additional results are shown, in particular, that on star- shaped domains and on C^2 domains higher eigenfunctions change sign. When the domain is a ball, we prove that the first eigenfunction has one sign, radial principal eigenfunction exist and are unique up to scalar multiplication, and that there are infinitely many eigenvalues.

math.AP↗
arXiv2011

Inhomogeneous Dirichlet problems involving the infinity-Laplacian

Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $Δ_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of Perron's method and a priori estimates to give general sufficient conditions on the right hand side $f$ that would ensure existence of viscosity solutions to the Dirichlet problem. Examples show that these sufficient conditions may not be relaxed. We also identify a class of inhomogeneous terms for which the corresponding Dirichlet problem has no solution in any domain with large in-radius. Several results, which are of independent interest, are developed to build towards the main results. The existence theorems provide substantial improvement of previous results, including our earlier results \cite{BMO} on this topic.

math.AP↗