Partial Differential Equations 7
Proof of Theorem of solution of wave equation in the case $n = 1$
Solution of Brezis Problem 8.24 (1) and (2)
Solution of Evans PDE Problem 13
Collected Lectures on Partial Differential Equations (PDE)
📝 Notes on Partial Differential Equations - John K. Hunter (University of California at Davis) 📝 Partial Differential Equations: Lecture Notes - Erich Miersemann (Leipzig University) 📝 Linear Methods of Applied Mathematics - E. Harrell, J. Herod (Georgia Tech)
A lemma of J. L. Lions
This post explores J. L. Lions’ lemma about Banach spaces with compact injection, including applications to functional analysis. Lemma statement: Let $X$, $Y$, and $Z$ be three Banach spaces with norms $|| \cdot ||_X$, $|| \cdot ||_Y$, and $|| \cdot ||_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection. Prove that $$ \forall \varepsilon > 0, \exists C_\varepsilon > 0 \text{ satisfying } || u ||_Y \leq \varepsilon || u ||_X + C _{\varepsilon}|| u ||_Z,\quad \forall u \in X $$
Riesz Representation Theorem
1. Riesz Representation Theorem # Let $H$ be a Hilbert space over $\mathbb{R}$ or $\mathbb{C}$, and $T$ be a bounded linear functional on $H$ (a bounded operator from $H$ to the field $\mathbb{R}$ or $\mathbb{C}$, where $H$ is defined over that field). The following is known as the Riesz Representation Theorem: Theorem 1: If $T$ is a bounded linear functional on the Hilbert space $H$, then there exists $g \in H$ such that for every $f \in H$, we have: $$ T(f) = \langle f, g \rangle. $$