Homework for Part 5
Week 9 : Solve exercises 2, 4, 5, 6.
Week 10 : Solve exercises 7, 8, 9, 10, 11.
Exercise 1 • From Chipot, 1.3
Describe a theory of elasticity in dimension 3.
Exercise 2 • From Chipot, 1.3
Let \(f : \Omega \to \mathbb{R}\) be a continuous function. Let \(B\) be a ball in \(\Omega.\) Show that \[\int_B f dv = 0 \quad \forall B \subset \Omega\] implies that \(f \equiv 0\) in \(\Omega.\)
Exercise 3 • From Chipot, 1.3
In the case of diffusion of population, what kind of dependence in \(u\) would be reasonable to consider for \(a=a(u)\) ?
Exercise 4 • From Chipot, 1.3
Give various examples of parial differential equations with no solution.
Exercise 5 • From Chipot, 1.3
If \(\varphi\) denotes a mapping from \(\Omega\) into itself, what kind of diffusion properties could be modelled though a coefficient of diffusion of the type: \[a=a(u(\varphi(x))) \; ?\]
Exercise 6
Show that the surface of the plane, with unit normal \(\nu\) above the rectangle of sizes \(\Delta h_1,\) \(\Delta h_2\) is given by \[\frac{\Delta h_1 \Delta h_2}{\nu_3}, \quad \nu = (\nu_1, \nu_2, \nu_3), \nu_3 \neq 0.\]
Show then that the element of surface of a surface defined by \(x_3 = \varphi(x_1,x_2)\) is given by \[da \approx \sqrt{1+\vert \nabla\varphi\vert^2} dx_1 dx_2\]
Exercise 7
Show that the functions \(r^n \cos n\theta\) , \(r^n\sin n\theta\) are harmonic in \(\mathbb{R}^2.\)
Hint: \(r^n\cos n\theta + i r^n \sin n\theta = (x_1 + i x_2)^n\) with \(x_1 = r\cos \theta, x_2 = r \sin \theta.\)
Exercise 8
Solves the problem \(-u'' = 1\) with \(u(\pm 1)=0.\)
Exercise 9
Let \(\Omega = \mathbb{R}\times (-\pi,\pi).\)
Show that \(u = e^{x_1} \sin x_2\) is solution to \[- \Delta u =0 \quad \text{in} \; \Omega, \quad u=0 \; \text{on} \; \partial\Omega\] Does this proble have another solution? In the lemma, where \(\Omega\) bounded is used?
Exercise 10
Let \(\Omega\) be a bounded open set of \(\mathbb{R}^n.\)
Show that the problem \[\Delta u = u^3 \quad \text{in} \;\Omega, \quad u=0 \; \text{on} \; \partial\Omega\] has no other solution than \(u=0.\)
Exercise 11
Let \(O\) an orthogonal \(n\times n\) matrix, i.e. \(O\) is such that \(O\cdot O^T = O^T \cdot O = \mathrm{Id}.\) Show that the Laplace equation is invariant through such a transformation, i.e. if \[\Delta u = 0 \quad \text{in} \; \mathbb{R}^n, \quad v(x) = u(Ox) \; \text{then} \quad \Delta v =0 \quad \text{in} \; \mathbb{R}^n.\]