Solving Uniqueness Questions for Laplace Problem

[miner3]

I am struggling with proving uniqueness and have multiple HW questions, which basically just have different boundary conditions. Something just isn't clicking for me when doing uniqueness analyses. The approach we were taught is the following:
del^2(phi) = 0 in a domain
del(phi) * n = d(phi)/d(n) = f(x,y,z) on the surface (d's in this case are partials and n is the normal unit vector)

So, del^2(phi1) = 0 and del^2(phi2) = 0
We can also sub phi1 and phi2 into the second equation.

Then let K = phi1 - phi2 and d(K)/d(n) = 0

We then were taught to apply Green's Theorem and solve so that eventually we get:
The integral over the volume of (del(K) . del(K) dv = integral of K * d(K)/d(n) ds = 0

However, I know this doesn't hold for every situation (particularly if del^2(phi) is not equal to zero. Something in this process is missing for me.

In question one we are asked to show unqiueness for the Dirichlet condition (Laplace problem). Problem 2 gives conditions such as:
del^2(u) = Pu in volume/domain
P = P(x,y,z)
B.C.s = d(u)/d(n) = f on surface

If anyone can provide some information, link for something to read, or reference of something to read that might help it would be greatly appreciated!

 

[nilesh_pat]

Thank you! The approach outlined above is called the Method of Separation of Variables and it is used to solve Laplace equations with various boundary conditions. To prove uniqueness for a Dirichlet problem, one needs to use the Maximum Principle, which states that the maximum value of a harmonic function in a domain is attained on the boundary. This can be used to prove that when two solutions satisfy the same boundary conditions, they must be equal.For the second question, a slightly different approach is necessary. Here, you need to use the weak formulation of the equation. This means transforming the partial differential equation into an integral equation, which can then be used to prove uniqueness. The weak formulation involves multiplying the equation (del^2(u) = Pu) by a test function v and integrating over the domain. The result of this integration is an equation of the form:int_V (del(u) * del(v) dV) - int_V (Pu * v dV) = 0Note that this equation is satisfied for all v, so it can be used to prove uniqueness. The idea is to choose two solutions u1 and u2 such that they satisfy the same boundary conditions and plug them into the equation. Then, take the difference between the two equations and integrate over the domain. Since the left-hand side of the equation will be 0, the result is that the difference between u1 and u2 must also be 0. Thus, u1 and u2 are the same solution.

 

[piercas]

I understand your struggle with proving uniqueness for Laplace problems. It can be a complex and challenging task, but with the right approach and understanding, it can be achieved.

First, let's start by understanding the Laplace problem. It is a type of partial differential equation that describes the steady-state distribution of a scalar field, such as temperature, in a given domain. The uniqueness of the solution to this problem is crucial because it ensures that the solution is well-defined and independent of the specific boundary conditions.

The approach you were taught is known as the method of Green's functions. This method relies on the use of Green's theorem, which relates the surface integral of a vector field to the volume integral of its divergence. This theorem is a powerful tool in solving partial differential equations, but as you mentioned, it may not hold for every situation.

One possible reason for this is that the Laplace problem is a linear equation, meaning that the sum of any two solutions is also a solution. This property can lead to multiple solutions that satisfy the same boundary conditions, making it challenging to prove uniqueness.

To overcome this, one can use the maximum principle. This principle states that the maximum or minimum of a harmonic function occurs on the boundary of the domain. In other words, if a solution satisfies the boundary conditions and is also a maximum or minimum in the interior of the domain, then it must be the only solution.

Another approach is to use the method of separation of variables. This method involves expressing the solution as a product of functions of each variable, which are then substituted into the Laplace equation. The resulting equations can then be solved using boundary conditions to determine the unique solution.

Additionally, it may be helpful to read about the properties of harmonic functions and the Dirichlet and Neumann boundary conditions. These concepts are fundamental to understanding and solving Laplace problems.

In conclusion, proving uniqueness for Laplace problems can be challenging, but with a solid understanding of the underlying concepts and the use of appropriate methods, it can be achieved. I hope this information helps, and I wish you success in your future problem-solving endeavors.