Solving a PDE with Boundary Conditions: A Minimization Problem

[yakamozyu]

Homework Statement

Let Ω$\subset$R² be a region with boundary $\Gamma$=$\Gamma$₁$\bigcup$$\Gamma$₂. On Ω we must solve the PDE

$$-{div}(\frac{h^{3}}{12\mu}{grad} p+\frac{h}{2}{u})+kp=f$$


with h and f functions of the spatial coordinates, \mu and k given constants, u a given constant velocity vector and the pressure p unknown. The boundary conditions fot this PDE are:

p = p₀ on $\Gamma$₁

(n, \frac{h^{3}}{12\mu}{grad} p+\frac{h}{2}{u} ) = 0 on $\Gamma$₂

n is the outward normal and p₀ a given pressure.

Derive the minimization problem corresponding to the PDE, with boundary conditions.






Can anyone give me some hints about this problem? I don't know where to start.
Thanks in advance! :)

 

[anathema]

To start, we can rewrite the PDE in a more standard form as follows:

-∇⋅(h^3/12μ∇p + (hu)/2) + kp = f

This is a second-order elliptic PDE with mixed boundary conditions. To derive the minimization problem, we need to use the principle of minimum potential energy. This principle states that the solution to a boundary value problem for a linear elastic system is the one that minimizes the total potential energy of the system.

In this case, the total potential energy can be expressed as follows:

Π = ∫Ω{h^3/24μ(∇p)^2 + hpu + kpp} dΩ - ∫Γ2p0(∇p⋅n) dΓ

To minimize Π, we differentiate with respect to p and set it equal to 0:

δΠ/δp = ∫Ω{h^3/12μ∇p⋅∇q + hq + kq} dΩ - ∫Γ2p0(∇q⋅n) dΓ = 0

where q is a test function. This gives us the weak form of the PDE. We can then apply the boundary conditions to obtain:

p = p0 on Γ1

∫Γ2(h^3/12μ∇p + hu)⋅n dΓ = 0

This is the minimization problem corresponding to the PDE, with boundary conditions. I hope this helps!