DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN DIFFERENTIAL EQUATIONS Autumn 1997

Show that \(\dfrac {\partial \varphi }{\partial \xi }(t;\xi )\) exists for all \(t\ge 0\), \(\xi \in \mathbb {R}\).

Let \(f(t,y)=\dfrac {y-3}{1+y^2+t^2}\). Find \(\dfrac {\partial \varphi }{\partial \xi }(t;3)\).

Define: \(z\) is a uniformly Liapunov stable solution of (2).

Suppose \(f(t,x)=Ax\) where \(A\) is a constant matrix. State the precise conditions on \(A\) under which \(z(t)\) is uniformly Liapunov stable in equation (2).

Assume that \(A\) is a real matrix whose eigenvalues have negative real part \((\Reals \,\sigma (A)<0)\) and \(B(t)\) is continuous for all \(t\in \mathbb {R}\). Show that there is an \(\epsilon >0\) depending on \(A\) so that if \(\norm {B}\le \epsilon \) then the zero solution of (2) with \(f(x,t)=Ax+B(t)x\) is uniformly Liapunov stable.

Define what is meant by a weak solution of (3).

Using the method of characteristics, solve the initial value problem (3) assuming \(f(x,t)=x\) and \(\varphi (s)=s\).

Assume \(f(x,t)=0\). Show by example that if \[ \varphi (x)= \begin {cases} \alpha , & \text {if }x<0,\\ \beta , & \text {if }x\ge 0. \end {cases} \] where \(\alpha <\beta \) are constants, then the IVP (3) may have more that one weak solution. (Check that your examples are weak solutions.)

Under what additional condition are the weak solutions unique?

Show that there is a constant \(c<\infty \) depending only on \(n\) and \(r\) so that \[ \sup \{\lvert Du(x)\rvert :x\in \overline {B_r(0)}\}\le c\sup \{\lvert u(x)\rvert :x\in \partial B_{2r}(0)\}. \] for all harmonic functions \(u\) on \(B_{2r}(0)\).

Show that there is a sequence of functions \(u_i\in S_\varphi \) so that \(u_i\to w_\varphi \) uniformly in \(\overline {B_r(\xi )}\).

Show that \(w_\varphi \) is harmonic in \(B_r(\xi )\).

Show that \(F\) is a distribution on \(\mathbb {R}\).

Let \(\zeta \in C_0(\mathbb {R})\) be continuous with compact support such that the integral \[ \int _{-\infty }^{\infty }\zeta (x)\,dx=1. \] Show that as distributions, the convolutions \(\zeta _\epsilon *F\to F\) as \(\epsilon \to 0\), where \[ \zeta _\epsilon (x):=\frac {1}{\epsilon }\zeta \left (\frac {x}{\epsilon }\right ). \]

Find \(\dfrac {d^k}{dx^k}F\) for all integers \(k>0\).

State the Poincaré inequality for functions \(u\in H^1_0(\Omega )\).

Find a condition relating \(\Omega \) and \(V\) which implies there exists a unique weak solution \(u\in H^1_0(\Omega )\) for problem (6). Prove the existence and uniqueness under your condition.

Define the Sobolev Space \(H^k(\T ^n)=H^{k,2}(\T ^n)\) for \(k\) a nonnegative integer. Explain how to make sense of \(H^s(\T ^n)\) for \(s\in \mathbb {R}\).

Show that for all nonnegative integers \(k\) there is a constant \(c(k,n)<\infty \) so that for all \(u\in H^{k+2}(\mathbb {T}^n)\) we have \[ \norm {u}_{H^{k+2}(\mathbb {T}^n)}\le c\left (\norm {Lu}_{H^k(\mathbb {T}^n)}+\norm {u}_{H^0(\mathbb {T}^n)}\right ). \]

Assume that \(u\in H^1(\mathbb {T}^n)\) is a weak solution to \(Lu=f(x)\) on \(\mathbb {T}^n\), where \(f\in C^\infty (\mathbb {T}^n)\). Show that \(u\in C^\infty (\mathbb {T}^n)\).