Maximum Principles

Formulate and prove the maximum principle for Laplace’s equation \(u_{xx}+u_{yy}=0\).

Formulate and prove the maximum principle for the heat/diffusion equation \(u_t=u_{xx}\).

1. Calculate the Fourier transform of a Gaussian \(f(x)=e^{-x^2}\).
2. Formulate and prove Heisenberg’s uncertainty principle.

Formulate and derive the uncertainty principle. Is its inequality optimal? Explain.

Formulate and derive the uncertainty principle. Is its inequality optimal?

Consider the second-order equation \[ \nabla ^2u-cu=f\qquad \text {in }D\subset \mathbb {R}^n \] with \(c\) constant, \(u,f\in C^2(D)\) and \[ u=g\qquad \text {on }\partial D, \] where \(D\) is bounded and \(\partial D\) is smooth.

1. Assuming a solution \(u\) exists, show that it is unique when \(c>0\). Establish this using two different methods: (i) an integral method and (ii) a Maximum Principle.
2. Suppose that \(n=2\), \(c<0\), \(f=g=0\), and \(D\) is the region \(r^2=x^2+y^2<1\). Show that for \(a_0\) constant, \[ u=a_0J_0(r\sqrt {-c}), \] is a solution as long as \(\sqrt {-c}\) is a zero of the Bessel function \(J_0(x)\), which satisfies \[ \frac {d^2J_0}{dx^2}+\frac 1x\frac {dJ_0}{dx}+J_0=0. \]
3. Suppose that \(n=3\), \(c<0\), \(f=g=0\), and \(D\) is the region \(1<x^2+y^2+z^2<4\). Show that there are non-trivial solutions if \(c=-n^2\pi ^2\), \(n=1,2,3\ldots \).

Let \(U\subset \mathbb {R}^n\) be an open bounded connected set with smooth boundary \(\partial U\). Consider the initial boundary-value problem for the heat equation: \[ \begin {cases} u_t-\Delta u=0, & \text {in }U\times (0,T],\\ u=0, & \text {on }\partial U\times [0,T],\\ u=g, & \text {on }U\times \{t=0\}. \end {cases} \]

1. Use the mean value property for the heat equation \[ u(x,t)=\frac {1}{4r^n}\iint _{E(x,t;r)} u(y,s)\frac {\lvert x-y\rvert ^2}{(t-s)^2}\dd y\dd s, \] where \[ E(x,t;r)=\{(y,s)\in \mathbb {R}^{n+1}:s\le t,\ \phi (x-y,t-s)\ge 1/r^n\} \] and \(\phi (x,t)\) is the fundamental solution of the heat equation, to prove the strong maximum principle for the heat equation.
2. Show that the strong maximum principle implies the principle of infinite propagation speed for the problem above.

Assume that \(u\in C^2(\mathbb {R})\), and \(f\in C_0^2(\mathbb {R})\).

1. Assume \(u\) satisfies \(u''=0\). Without using an explicit representation for the solution, show that the mean value property \[ u(x)=\frac 12\bigl (u(x+r)+u(x-r)\bigr )=\frac {1}{2r}\int _{x-r}^{x+r}u(t)\dd t, \] for each interval \((x-r,x+r)\), is satisfied, and conversely show that the mean value property implies \(u''=0\).
2. Assuming \(u''=0\) in the bounded interval \((a,b)\), use part (a) to prove the strong maximum principle for \(u\).
3. Find a radially symmetric fundamental solution \(\Phi (x)\) for the problem \(u''=0\), and use it to prove that \[ u(x)=\int _{-\infty }^{\infty }\Phi (x-y)f(y)\dd y \] satisfies \(-u''=f\) in \(\mathbb {R}\).

Let \(U\subset \mathbb {R}^n\) be open and bounded, with smooth boundary. For \(T>0\), denote \(U_T=U\times (0,T]\). Let \(\Gamma _T=\overline {U_T}-U_T\) (closure taken in \(\mathbb {R}^n\times \mathbb {R}\)). Prove that functions \(u(x,t)\) satisfying \(u\in C^{2,1}(U_T)\cap C(\overline {U_T})\) and \[ \begin {aligned} \Delta u-u &\ge u_t, && \text {in } U_T,\\ u &\ge 0, && \text {in } \overline {U_T}, \end {aligned} \] also satisfy the maximum principle \[ \max _{\overline {U_T}} u=\max _{\Gamma _T} u. \]

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain, \(0<T<\infty \) be a constant, \(c(x)\in C^\infty (\overline {\Omega })\) such that \(c(x)\le 0\) for all \(x\) and \(\psi (x,t)\in C^\infty (\overline {\Omega }\times [0,T])\). Let \(u\in C^{2,1}(\overline {\Omega }\times [0,T])\) be a solution to \[ \begin {cases} \dfrac {\partial u}{\partial t}=\Delta u+c(x)u, & \text {if }(x,t)\in \Omega \times [0,T],\\ u(x,t)=\psi (x,t), & \text {if }(x,t)\in \Omega \times \{0\}\text { or if }(x,t)\in \partial \Omega \times [0,T]. \end {cases} \qquad (\heartsuit ) \]

1. Prove the Maximum Principle: If \(\psi (x,t)\ge 0\) then the solution satisfies \(u(x,t)\ge 0\).
2. Show that such solutions of \((\heartsuit )\) are unique.

1. Let \(\Omega \subset \mathbb {R}^n\) be open and let \(u\in C(\Omega )\). Define what it means to say that \(u\) is subharmonic in \(\Omega \).
2. Let \(\Omega \) be open, bounded and connected and let \(u\in C^2(\Omega )\cap C^0(\bar {\Omega })\) be subharmonic. State the weak form of the maximum principle for \(u\).
3. Consider the Dirichlet problem \[ \begin {aligned} \Delta u &= f && \text {in }\Omega ,\\ u &= 0 && \text {on }\partial \Omega \end {aligned} \tag {9.1} \] where \(f\in C^0(\Omega )\) and \(\Omega \) is as in (b). Describe in as much detail as you can the major steps in proving the existence of a classical solution of (9.1) via Perron’s Method.

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.