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Special Session 77: Recent developments in variational problems and geometric analysis

Some new results on elliptic equations involving Logarithmic Laplacian

Rakesh Arora

Indian Institute of Technology (IIT-BHU)
India

Co-Author(s): Jacques Giacomoni and Arshi Vaishnavi

Abstract:

This talk presents new existence, non-existence and uniqueness results for the following problem

$L_{\Delta} u = f(x,u) \ \text{in} \ \Omega, \quad u = 0 \ \text{in} \ \mathbb{R}^N \setminus \Omega$ where

$L_\Delta$ is the Logarithmic Laplace operator and

$f$ satisfies sub-critical, critical and super-critical nonlinearities growth conditions. Such type of problems are connected to the model problems in population dynamics, optimal control, approximation of fractional harmonic maps, and fractional image denoising.

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