Differential Equations Preliminary Examination

Assume that the eigenvalues of a real \(n\times n\) matrix \(A\) have negative real part. Select with a check-mark ☑ and solve either A-2.I or A-2.II: ☐ **A-2.I.** **(a)** Prove that positive constants \(M\) and \(\alpha\) exist such that for all \(x\in\mathbb{R}^n\) and \(t\ge0\) \[ \lVert e^{At}x\rVert\le M\lVert x\rVert e^{-\alpha t}. \] **(b)** Prove that the zero solution of \(u'=Au\) is asymptotically stable. ☐ **A-2.II.** Prove that for \(h(t)\) continuous and \(T\)-periodic, the equation \(u'=Au+h(t)\) has a unique \(T\)-periodic solution \(u(t)\).

Let the system \(x'=f(x)\) define a \(C^1\) flow \(\phi_t\) on the open set \(E\) contained in \(\mathbb{R}^n\). Prove that the positive limit set \(\Gamma^+(v)\) of a trajectory \(x(t)\) with \(x(0)=v\) is closed. Then, select with a check-mark ☑ and solve either A-3.I or A-3.II: ☐ **A-3.I.** Consider the autonomous planar dynamical system \[ x'=6x-2xy-6x^2, \qquad y'=-7y+2xy-y^2. \] **(a)** Compute the four rest points (=equilibrium or stationary points) of the system and the linearization about each rest point. **(b)** Make a table in which each row contains a rest point, the classification stable or unstable, and the geometric classification node, spiral, center or hyperbolic point. **(c)** Sketch the phase diagram showing the rest points and the local behavior of solution curves (rough and brief!). ☐ **A-3.II.** Let \(r^2=x^2+y^2\), \(w=(r^2-1)(r^2-4)\) and consider the planar system \(x'=-y+xw\), \(y'=x+yw\) (\(r'=rw\), \(\theta'=1\) in polar coordinates). Apply the Poincar\'e-Bendixson theorem to prove that \(r=1\) and \(r=2\) are limit cycles (a periodic orbit \(\gamma\) with \(\gamma=\Gamma^+(v)\) or \(\gamma=\Gamma^-(v)\) for nearby \(v\)).

Select with a check-mark ☑ and solve either A-4.I or A-4.II: ☐ **A-4.I.** Assume \(f:D\to\mathbb{R}^N\) is continuous and \(f\) is bounded by a constant \(m\) on a subdomain \(D_0\subset D\). Let \(u(t)\) be a solution of \(u'=f(t,u)\) with \((t,u(t))\in D_0\) on \(a0>f(a)\), then \(d(f,(a,b),0)=1\). **(b)** If \(f(b)<00\), then \(d(f,(a,b),0)=0\).

Consider the Sturm-Liouville problem \[ x^2(x^2u')'+\lambda u=0 \] on \(1/2\le x\le 1\) with boundary conditions \(u(1/2)=u(1)=0\).

1. State without proof the main theorem on eigenfunction expansions which applies to this example.
2. Use the change of variables \(w(t)=u(1/t)\) to transform the differential equation into \(d^2w/dt^2+\lambda w=0\). Then calculate the eigenvalues \(\lambda _n\) and eigenfunctions \(u_n\), by citing without proof a result for the Sturm-Liouville problem \(y''+\lambda y=0\), \(y(a)=0=y(b)\).
3. Sturm oscillation theory and the Prüfer transformation are used in the general theory to produce the candidate eigenvalues and eigenfunctions. Sketch briefly how this is accomplished, without proofs.

Select with a check-mark ☑ and solve either B-2.I or B-2.II: ☐ **B-2.I:** Define the Sobolev space \(H^m(\Omega)\) for open \(\Omega\subset\mathbb{R}^n\). Then: **(a)** Prove that \(H^m(\Omega)\) is a Hilbert space. **(b)** Give an example of a sequence which shows that the subspace \(C([0,1])\) in \(L^1(0,1)\) is not complete in the \(L^1\)-norm. **(c)** Compute the distributional derivatives \(\partial f\) and \(\partial^2f\) for \(f(x)=\lvert x\rvert\) in \(H^2(-\infty,\infty)\). Assume results for the Heaviside unit step and Delta. ☐ **B-2.II:** Define what it means for \(H\) to be a Hilbert space. Then: **(a)** Explain the meaning of the formula \(H=M\oplus M^\perp\) and give conditions on \(M\) for when it is true (do not give proofs). **(b)** State the Riesz representation theorem and use (a) to prove it.

Select with a check-mark ☑ and solve either B-3.I or B-3.II: ☐ **B-3.I:** Let \(u,h\) denote elements of some Sobolev space and consider the distributional differential equation \(-u''+u=h\) with Dirichlet boundary conditions \(u(0)=u(1)=0\). **(a)** Formulate an abstract boundary value problem \(a(u,v)=(h,v)\), by defining the sequilinear form \(a(u,v)\), the Hilbert space \(H\) and the inner product \((\cdot,\cdot)\). **(b)** Discuss in detail how the Riesz theorem applies to solve the abstract boundary value problem. **(c)** Is enough known about the Hilbert space solution \(u\) for it to be a solution of the distributional differential equation? Explain. ☐ **B-3.II:** Let \(\phi\in H^1_0(G)\) and denote by \(\lVert \cdot\rVert\) the usual norm in \(L^2(G)\), \(G\) open in \(\mathbb{R}^n\). Assume \(\lvert x_1\rvert\le K\) for all \(x\in G\). **(a)** Prove the Poincar\'e inequality \[ \lVert \phi\rVert\le2K\lVert \partial_{x_1}\phi\rVert. \] **(b)** Explain, without proof, how to use generalizations of the Poincar\'e inequality to solve the abstract boundary value problem for distributional differential equations of the form \(\Delta u=f\).

Sobolev proved an imbedding inequality of the form \(\lVert f\rVert_B\le M\lVert f\rVert_A\), where \(A=H^m(G)\) and \(B\) is the set of functions \(u\) such that \(D^\alpha u\) is uniformly continuous on \(G\) for \(\lvert\alpha\rvert\le k\). Give, without proof, the conditions on the open set \(G\subset\mathbb{R}^n\) and the integers \(m\) and \(k\). Select with a check-mark ☑ and solve either B-4.I or B-4.II: ☐ **B-4.I:** Under Sobolev's conditions on \(G\), \(m\) and \(k\), each \(f\in H^m(G)\) satisfies \(\partial^\alpha f=D^\alpha g\) a.e. for some \(k\)-times continuously differentiable function \(g\), \(\lvert\alpha\rvert\le k\). **(a)** Prove this, assuming the imbedding inequality above. **(b)** Determine for \(n=2\) the least \(m\) such that an element in \(H^m(G)\) has 4 continuous derivatives (\(G\) as above). ☐ **B-4.II:** Regularity theory implies that certain abstract boundary value problems \(a(u,v)=(F,v)\) can be solved for \(u\in H^{2+s}(G)\) provided \(F\in H^s(G)\). Consider the Dirichlet problem for \(\Delta u=F\), \(x\in G\). **(a)** Assume \(G\) is a disk in \(\mathbb{R}^2\). Give without proof a smoothness condition on \(F\) for the existence of \(C^2\) solutions \(u\). **(b)** Assume \(G\) is bounded and open in \(\mathbb{R}^2\). Give without proof conditions on \(G\) and \(\partial G\), and a smoothness condition on \(F\), for the existence of \(C^2\) solutions \(u\).