PDE: If u is a solution to a certain bound problem, question about laplacian u

  • Context: Graduate 
  • Thread starter Thread starter calvino
  • Start date Start date
  • Tags Tags
    Bound Laplacian Pde
Click For Summary

Discussion Overview

The discussion revolves around the properties of the Laplacian operator applied to a function \( u \) that is a solution to a boundary problem, specifically questioning whether the Laplacian of \( u \) is always zero and exploring related characteristics of solutions to partial differential equations (PDEs).

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions why the Laplacian of \( u \) equals zero when \( u \) is a solution to a boundary problem and whether this is a universal condition.
  • Another participant reflects on their initial misunderstanding, suggesting that the behavior of the Laplacian is contingent on specific stipulations related to the problem.
  • A third participant explains that if \( Lu=0 \) (where \( L \) is the Laplacian operator), then \( u \) is termed a harmonic function, and discusses the implications of the Laplacian being a linear operator that adheres to the superposition principle, though they acknowledge the informality of their explanation.
  • A later reply requests clarification on the initial question posed by the first participant.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the Laplacian of \( u \) is zero, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants have not fully defined the specific boundary conditions or types of PDEs being discussed, which may influence the behavior of the Laplacian.

calvino
Messages
108
Reaction score
0
Why does the laplacian of u=0 when u is a solution to a certain boundary problem? Is this always the case?
 
Physics news on Phys.org
after working it out, I realized I was on a totally wrong track. It's simply dependent on the stipulations on the laplacian. Perhaps one can help me with one more thing. Are there any special facts about the laplacian of a solution to a pde problem?
 
Lu=0 is the laplace equation (L is the laplacian operator), which is itself a PDE. A Function u which satisfies the laplace equation is called an harmonic function. If the solution of a different PDE satifies also the laplace equation (it's laplacian is zero) that solution is itself an harmonic function or a sum of them or may be expresed as a series of harmonic functions, because the laplacian is a linear operator and it obbeys the superposition principle. Of course this is a very informal explanation, but i think that is what you are asking for,, maybe
 
Calvino, spesify your question?
 

Similar threads

Undergrad Stuck on solving a non homogenous (linear) PDE
  • · Replies 36 ·
2
Replies
36
Views
7K
Graduate Determining whether the Dirichlet problem is nonhomogenous
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Heat conduction equation in cylindrical coordinates
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Generic Solution of a Coupled System of 2nd Order PDEs
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Poisson's inhomogeneous PDE and its solutions
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Heat Equation: Solve with Non-Homogeneous Boundary Conditions
  • · Replies 4 ·
Replies
4
Views
5K
Undergrad Graph Representation Learning: Question about eigenvector of Laplacian
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Neumann Boundary Value Problem in a Half Plane
  • · Replies 1 ·
Replies
1
Views
2K
Graduate How to solve simple 2D space-time PDE numerically
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Finding the time for the first shock for a quasilinear first order PDE
  • · Replies 4 ·
Replies
4
Views
4K