Differential Equations Preliminary Examination

State the fundamental existence uniqueness theorem for solutions of (2.2) and sketch a proof.

State a general principle about the existence intervals for solutions of (2.2) and the behavior of solutions at the endpoints of these intervals.

Use the above and a Gronwall inequality argument (note: \(\lvert f(t,x)\rvert \le \lvert f(t,x)-f(t,0)\rvert +\lvert f(t,0)\rvert \)) to show that solutions of (2.2) are defined on all of \(\mathbb {R}\).

Define what is meant by a fundamental matrix solution \(\Phi (t)\) of (2.3).

Use general existence uniqueness principles to give the form of all fundamental matrix solutions of (2.3).

Let \(\Psi (t)\) be a fundamental matrix solution of (2.4). Prove that \[ \Psi ^T(t)\Phi (t) \] is a nonsingular constant matrix.

Derive the variation of constants formula (using the fundamental solution matrix \(\Phi \)) for solutions of \[ x'=A(t)x+h(t). \tag {2.5} \]

Let \(\Phi (0)=\text {identity}=\Psi (0)\). Give the variation of constants formula derived above in terms of \(\Phi \) and \(\Psi \). Do you see any computational advantages in this form of the variation of constants formula.

Let \(G\subset \mathbb {R}^N\) be an open set. Define what is meant by a distribution on \(G\). Let \(f\in L^2(G)\) be a real function. Show that \(f\) may be identified with a distribution \(T_f\).

Let \(T\) be a distribution on \(G\) and let \(\alpha =(\alpha _1,\ldots ,\alpha _N)\) be a multiindex. Define the concept of the \(\alpha \)-th distributional derivative of \(T\).

Let \[ r(t)= \begin {cases} 0, & t\le -1,\\ 1, & -1<t<1,\\ 0, & t\ge 1. \end {cases} \] Think of \(r\) as a distribution on \(\mathbb {R}\) and compute all distributional derivatives of \(r\).

Provide the higher dimensional analogue of the previous exercise, i.e., let \[ r:\mathbb {R}^N\to \mathbb {R} \] be defined by \[ r(x)=1,\quad \lvert x\rvert <1, \qquad r(x)=0,\quad \lvert x\rvert \ge 1. \] Compute the distributional gradient \(\nabla r\) of \(r\).

Let \(G\) be an open set. Define the Sobolev spaces \(H^m(G)\), \(m=1,2,\ldots \), and \(H^m_0(G)\), \(m=1,2,\ldots \). Indicate why these spaces are Hilbert spaces.

Let \(G\) be a bounded open set. State and prove Poincaré’s inequality for functions \(u\in H^1_0(G)\) and thus verify that \(H^1_0(G)\) is also a Hilbert space with respect to the inner product \[ (u,v)_{H^1_0(G)}=\int _G \nabla u\cdot \overline {\nabla v}\,dx. \]

Let \(G\) be an open subset of \(\mathbb {R}^N\) and let \(f\in L^2(G)\). Show that \(f\) defines a continuous linear functional on \(H^1(G)\).

State the Lax-Milgram theorem for subspaces \(V\subset H^1(G)\).

Show that in the sense of distributions there exists a unique solution \(u\in H^1_0(G)\) of the equation \[ -\Delta u+u=f. \]

Characterize the orthogonal complement of \(H^1_0(0,\infty )\) in the space \(H^1(0,\infty )\).

Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0,\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \] Solve this problem explicitly.

Also state a result about the initial value problem \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]