The heat equation with an inhomogeneous boundary condition in the half-line. Part 1: Explicit Solution of the Dirichlet problem

In this series of blog posts, we'll consider the standard heat equation in the half-line $latex \partial_t u(t,y)=\partial_{yy}(t,y), \qquad (t,y)\in (0,\infty)\times (0,\infty),$ together with an inhomogeneous boundary condition at $latex y=0$. Initially, for simplicity we'll assume the initial condition to be homogeneous, i.e. $latex u(0,y)= 0\qquad \forall y>0. $ We will consider three types of … Continue reading The heat equation with an inhomogeneous boundary condition in the half-line. Part 1: Explicit Solution of the Dirichlet problem →

Minimizing the area between the graph of a function and the hyperplane y=0 by shifting.

Today I will present a simple problem from elementary calculus, which I find quite instructive. Let $latex \Omega\subset\mathbb{R}^n$ be a bounded non-empty open set and  let $latex u:\Omega\to \mathbb{R}$ be a continuous function. Can we find a real number $latex t=c$ which minimizes the real-variable function  $latex F(t):= \int_{\Omega}|u-t|$? When I first saw this problem … Continue reading Minimizing the area between the graph of a function and the hyperplane y=0 by shifting. →

Optimal path and speed for a person walking under the rain

Today I will present a topic which starts off as a rather divulgative problem with an accessible solution but then gets rather involved when the analysis is extended beyond the simplest cases. Suppose a person is walking under the rain, from point $latex A$ to point $latex B$, and wants to minimize the rain that … Continue reading Optimal path and speed for a person walking under the rain →

A proof of the Riesz Lemma through the Hahn-Banach theorem, and the reflexive case.

The Riesz Lemma is typically introduced in order to prove a fundamental result on the strong topology of normed spaces. That is, that the closed unit ball of an infinite-dimensional normed space is never compact. Today I was trying to prove it on my own and came up with this proof. The traditional proof does … Continue reading A proof of the Riesz Lemma through the Hahn-Banach theorem, and the reflexive case. →

The diagonalization procedure in analysis, with an application to Sobolev spaces

When discussing convergences of sequences in analysis, it is often the case that one is only able to obtain the convergence of a subsequence. This is already obvious from the definition of sequential compactness - a topological space is sequentially compact when every sequence has a converging subsequence. Here is a few more examples: $latex … Continue reading The diagonalization procedure in analysis, with an application to Sobolev spaces →

Can almost everywhere convergence be represented by a topology?

In this post, I will discuss the following issue: Can almost everywhere convergence on a $latex L^p$ space be represented by a topology? Naively, I was convinced for a long time that any notion of convergence on a set could be given by the convergence in a topology on that set. By a notion of … Continue reading Can almost everywhere convergence be represented by a topology? →