APS
Speaker: Jayant Sharma, IIT Bombay
Host: Manas Rachh
Title: Analysis of Laplace and Helmholtz Equations in Unbounded Domains: Solvability and Radiation Conditions
Time, day and date: 3:00:00 PM - 4:00:00 PM, Tuesday, September 22
Venue: Ramanujan Hall
Abstract: Solving elliptic partial differential equations (PDEs) in unbounded domains introduces significant challenges regarding the existence and uniqueness of solutions. Without appropriate asymptotic conditions at infinity, solutions may fail to exist or may exhibit arbitrary behaviour, leading to a loss of uniqueness.
This seminar presents an analysis of the Laplace and Helmholtz equations within unbounded domains.
We begin by solving the exterior problem for a unit circle using the separation of variables to establish foundational intuition. These concepts are then generalized to an unbounded domain in R²/R³, followed by a rigorous proof of the uniqueness of the solution for the Laplace equation. Furthermore, we will see solutions to the Helmholtz equation are analytic and this property will be used to prove Rellich's lemma which gives the uniqueness of the solution for the Helmholtz equation.