Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Uniqueness Results for Poisson Equation

In course

MATH-305: Introduction to partial differential equations

This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs. ...

Description

This lecture covers uniqueness results for the Poisson equation, discussing the existence and uniqueness of solutions with different boundary conditions. The presentation includes proofs of the uniqueness of solutions and explores the concept of harmonic functions. The lecture also delves into the fundamental solutions of the Laplace equation and the bounds on the solutions, emphasizing both sub and super harmonic properties.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Fabio Nobile

Nicola De Nitti

Official source

Related concepts (32)

Laplace's equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where is the Laplace operator, is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

Dirichlet boundary condition

In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of \mathbb R^n, that satisfies Laplace's equation, that is, everywhere on U. This is usually written as or The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics.

Robin boundary condition

In mathematics, the Robin boundary condition (ˈrɒbɪn; properly ʁɔbɛ̃), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.

Related lectures (50)

Groundwater Flow: Laplace Equation and Boundary Conditions

Explores the Laplace equation, harmonic potential, and boundary conditions in groundwater flow, with practical examples.

Electrostatics and Green's Functions: Mathematical Methods

Discusses electrostatics, Green's functions, and the application of complex analysis in deriving potentials.

Numerical Groundwater Flow Resolution

Explains the numerical resolution of groundwater flows using finite differences and demonstrates the method with Excel.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Recent Advances in the Dimer Model and Its Applications