Written Qualifying Examination in DIFFERENTIAL EQUATIONS September 13, 1994

Assume that the eigenvalues of a real \(n\times n\) matrix \(A\) have negative real part strictly less than \(-5\). Prove that there exists a positive constant \(M\) such that for all \(x\in \mathbb {R}^n\) and \(t\ge 0\) \[ \left \lVert e^{At}x\right \rVert \le M\lVert x\rVert e^{-5t}. \]

Consider the initial value problem \[ x'=f(t,x),\qquad x(t_0)=x_0 \] on the set \(X:\lvert t-t_0\rvert \le a\), \(\lvert x-x_0\rvert \le b\). Let \(M=\max \lvert f(t,x)\rvert \) on \(X\), \(h=\min (a,b/M)\). Assume that \(f\) is continuous on \(X\), \(L\ge 0\) and \(f\) satisfies the Lipschitz condition \(\lvert f(t,x)-f(t,y)\rvert \le L\lvert x-y\rvert \) for \((t,x)\in X\), \((t,y)\in X\).

Prove the Picard–Lindelöf existence and uniqueness theorem: There exists a unique solution \(x(t)\) defined on the interval \(\lvert t-t_0\rvert \le h\).

Let \(P\) and \(Q\) be infinitely differentiable. Prove that \(x'=P(x,y)\), \(y'=Q(x,y)\) is a Hamiltonian system iff \(x'=Q(x,y)\), \(y'=-P(x,y)\) is a gradient system.

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 \(\omega \) limit set of a trajectory \(\Gamma \) is closed and if it is also bounded, then it is nonvoid, compact and connected.

Let \(r^2=x^2+y^2\), \(u=r^4-3r^2+1\) and consider the planar system \(x'=-y+xu\), \(y'=x+yu\). Prove by means of the Poincaré–Bendixson theorem that there are two periodic orbits, one in \(0<r<1\) and another in \(1<r<3\).

Prove:

Let \(f\) be of class \(C^1\) in a simply connected region \(E\) in \(\mathbb {R}^2\). Assume the divergence of \(f\) is not identically zero and either nonnegative in \(E\) or non-positive in \(E\). Then \(x'=f(x)\) has no closed orbit which is entirely in \(E\).

Solve the wave equation \[ u_{tt}=c^2u_{xx}+e^{2x},\qquad u(x,0)=0,\qquad u_t(x,0)=0 \] on \(-\infty <x<\infty \), \(0\le t<\infty \).

Suppose \(u\) is of class \(C^2\) and \(\Delta u\ge 0\) in a bounded domain \(\Omega \) with \(C^2\) boundary. State and prove a maximum principle.

Apply the method of odd extensions on the heat kernel formula for \((-\infty ,\infty )\) to derive a formula for the solution of the diffusion equation on \((0,\infty )\) with Dirichlet boundary conditions.

State and prove the Classical Fourier pointwise convergence theorem for the representation of a function by its Fourier series.

Prove by means of the Poisson Formula that harmonic functions are of class \(C^\infty \).

Define the Sobolev space \(H^k(\Omega )\) and give an example of a set \(\Omega \) and a function \(f\in H^1(\Omega )\) which does not agree a.e. with any \(C^1\) function.

Let \(\mathcal {H}\) be a Hilbert space with inner product \((x,y)\) and let the set \(\{e_n\}_{n=1}^{\infty }\) be orthonormal in \(\mathcal {H}\). Complete the following:

1. Prove that the formal series \[ g=\sum _{n=1}^{\infty }(f,e_n)e_n \] converges for each \(f\in \mathcal {H}\) to an element \(g\in \mathcal {H}\).
2. Let the mapping \(\mathcal {P}\) be defined on \(\mathcal {H}\) by \(\mathcal {P}f=g\), \(g=\sum _{n=1}^{\infty }(f,e_n)e_n\). Prove that \(\mathcal {P}\) is an orthogonal projection on \(\mathcal {H}\).

Define a regular Sturm-Liouville operator by the equation \[ \mathcal {L} u=-x^2(x^2u')' \] on \(1/2\le x\le 1\) using boundary conditions \(u(1/2)=u(1)=0\). Complete the following:

1. Explain how \(\mathcal {L}\) and the boundary conditions can be used to construct a selfadjoint operator \(S\) in a suitable Hilbert space \(\mathcal {H}\). Do not give proofs.
2. Calculate the eigenvalues and eigenfunctions of the selfadjoint operator \(S\), using the solution basis \(u_1(x)=\sin (\lambda /x)\), \(u_2(x)=\cos (\lambda /x)\) of the differential equation \[ x^2(x^2u')'+\lambda u=0. \] Do not derive the basis.
3. Verify by direct integration that the eigenfunctions are orthogonal in the Hilbert space \(\mathcal {H}\).

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\). Complete the following:

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\xi =re^{i\theta }\) and \(\sqrt {\xi }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \). \[ G(x,y,\xi )= \begin {cases} e^{i\sqrt {\xi }y}\sin (\sqrt {\xi }x)/\sqrt {\xi } & \text {for }0\le x\le y<\infty ,\\ e^{i\sqrt {\xi }x}\sin (\sqrt {\xi }y)/\sqrt {\xi } & \text {for }0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

Let \(T\) be a linear operator in a Hilbert space \(\mathcal {H}\) with domain \(D(T)\) dense in \(\mathcal {H}\). Complete the following:

1. Define the adjoint operator \(T^*\).
2. Let \(G(T)\) be the graph of \(T\) in \(\mathcal {H}\oplus \mathcal {H}\). Define the operator \(V\) on \(\mathcal {H}\oplus \mathcal {H}\) by \(V(f,g)=(g,-f)\). Prove that the graph of \(T^*\) is the orthogonal complement in \(\mathcal {H}\oplus \mathcal {H}\) of \(V(G(T))\), that is, \[ G(T^*)=(V(G(T)))^\perp . \]
3. Let \(\overline {G(T)}\) be the closure in \(\mathcal {H}\oplus \mathcal {H}\) of the graph of \(T\). Prove that \[ \mathcal {H}\oplus \mathcal {H}=V(\overline {G(T)})\oplus G(T^*). \]