1993 Preliminary Examination\ 0.5 ] Differential Equations

Show that solutions of initial value problems for (2) are defined on \((-\infty ,\infty )\).

Let \(\operatorname {trace} A(t)\equiv 0\), \(t\in (-\infty ,\infty )\) and let \(\overline {X}(t)\) be a matrix solution of (2) show that \(\det \overline {X}(t)=\text {constant}\), \(t\in (-\infty ,\infty )\).

Let \(\overline {X}(t)\) be a matrix solution of (2) with \(\det \overline {X}(t_0)\ne 0\), some \(t_0\). Characterize all other matrix solutions of (2) and provide a characterization of all solutions of (2).

Assume that \(\overline {X}(t)\) is a matrix solution of (2) with \(\overline {X}(0)\) nonsingular and \(\overline {X}(t)\) uniformly bounded on \([0,\infty )\). Also assume \[ \liminf _{t\to \infty }\operatorname {Re}\int _{0}^{t}\operatorname {trace} A(s)\,ds >-\infty . \] Prove that \(\overline {X}^{-1}(t)\) is uniformly bounded on \([0,\infty )\) and no solution \(\varphi (t)\) of (2), not identically zero, can satisfy \(\varphi (t)\longrightarrow 0\) as \(t\longrightarrow \infty \).

Assume the conditions of d) and let \(B(t)\) be a continuous matrix function with \[ \int _0^\infty \lVert A(t)-B(t)\rVert \,dt <+\infty . \] Let \(\Psi \) be a solution of \[ \Psi '=B(t)\Psi . \] Show that \(\Psi \) must be uniformly bounded on \([0,\infty )\).

Hint: \(\Psi \) also solves \[ \Psi '=A(t)\Psi +(B(t)-A(t))\Psi . \] Use variation of constants.

Define what is meant by a distribution on \(G\).

Let \(f:G\longrightarrow \mathbb {R}\) be a measurable function such that \(\int _{G_0}\lvert f\rvert \,dx<\infty \) where \(G_0\) is an open subset of \(G\) with \(\overline {G_0}\) compact, i.e. \(f\in L^1_{\operatorname {loc}}(G)\), show that each such \(f\) defines a unique distribution on \(G\). Give an example of a distribution which is not defined by such an \(f\).

If \(T\) is a distribution on \(G\), define \(\partial ^\alpha T\), where \(\alpha \) is a multi-index. Show that if \(f\in C^1(G)\), then \(f\) and its partial derivatives \(\partial f/\partial x_i\), \(1\le i\le n\), define unique distributions \(T\), \(T_i\), \(1\le i\le n\), on \(G\) and \(\partial ^\alpha T=T_i\); \(\alpha =(0,\ldots ,1,0,\ldots ,0)\), where the \(1\) appears in the \(i\)th place.

Let \[ f=x_+^\lambda = \begin {cases} 0, & x\le 0,\\ x^\lambda , & x>0, \end {cases} \] where \(-1<\lambda <0\). Then \(f\in L^1_{\operatorname {loc}}(\mathbb {R})\), hence \(f\) may be thought of as a distribution.

Compute \(\partial f\). Note that \(\partial f/\partial x\notin L^1_{\operatorname {loc}}(\mathbb {R})\).

Let \(U:\mathbb {R}^2\longrightarrow \mathbb {R}\) be given by \[ U(x,t)=f(x+t)+g(x-t), \] where \(f,g\in L^1_{\operatorname {loc}}(\mathbb {R})\). Show that \[ \partial _x^2 U=\partial _t^2 U \] in the sense of distributions.

Evaluate \(\Delta _3(1/r^2)\), \(r^2=x^2+y^2+z^2\) in \(\mathbb {R}^3\), where \(\Delta _3\) is the distributional Laplacian.

Define the Sobolev spaces \(H^m(G)\) and \(H_0^m(G)\), where \(m\) is a nonnegative integer.

Give a characterization of these spaces in terms of \(L^2(G)\) functions and their distributional derivatives.

Let \(G\) be an open set in \(\mathbb {R}^n\) with \[ \sup \{\lvert x_1\rvert :(x_1,x_2,\ldots ,x_n)\in G\}=K<+\infty . \] Then \[ \lVert \varphi \rVert _{L^2(G)}\le 2K\lVert \partial _1\varphi \rVert _{L^2(G)}, \quad \forall \varphi \in H_0^1(G). \tag {6} \] \[ \left (\text {Hint: For }\varphi \in C_0^\infty (G)\text { compute } \frac {\partial }{\partial x_1}(x_1\lvert \varphi (x)\rvert ^2)\right . \] \[ \left .\text {and use the divergence theorem.}\right ) \] Use (6) to obtain Poincaré’s inequality for functions \(\varphi \in H_0^1(G)\), whenever \(G\) is a bounded domain.

\((Ax,x)\) is real for all \(x\in D(A)\), \(((\cdot ,\cdot )\) is the inner product of \(H)\).

All eigenvalues of \(A\) are real.

Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Let \(\lambda \in \mathbb {C}\) with \(\operatorname {Im}\lambda \ne 0\). Show that \(\lambda -A\) is invertible.

Let \(G=(0,1)\), \(V=H_0^1(G)\), and \(H=L^2(G)\). Define \[ a(u,v)=\int _0^1 \partial u\,\partial \overline {v}\,dx. \] Use what you know about coercive and elliptic sesquilinear forms and the spectral theorem for operators defined by them to find a basis for \(L^2(G)\). Be sure to present a fairly complete discussion.

Define the concepts “strongly continuous semigroup on \(H\)” and “infinitesimal generator” of the semigroup.

Let \(\{T(t):t\ge 0\}\) be a strongly continuous semigroup on \(H\) with infinitesimal generator \(A\). Show that \[ T(t):D(A)\longrightarrow D(A) \] and that the map \[ t\longmapsto T(t)x,\qquad t\in [0,\infty ), \] is differentiable and \[ \frac {d}{dt}T(t)x=AT(t)x=T(t)Ax,\quad x\in D(A). \]

State necessary and sufficient conditions in order that a linear operator be the generator of a strongly continuous contraction semigroup on \(H\).

Use the above to establish the existence and uniqueness of solutions to the initial value problem \[ U_t=U_{xx}+f(t,x),\quad t>0,\quad x\in (0,1) \] \[ U(0,x)=\varphi (x) \] \[ U(t,0)=U(t,1)=0; \] be sure to provide the appropriate function spaces for \(U\), \(f\), \(\varphi \).