Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

8
votes
2answers
153 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
4
votes
0answers
97 views
+100

Compute the derivative $\partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves (in the sense of distributions) an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic ...
2
votes
1answer
70 views
+200

Examples for increasing homeomorphisms related to $\varphi$-laplacian

Let $\varphi$ be an odd function satisfying $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \mathbb{R}_+:=[0,\infty).$ Then $\varphi'(s)=\frac{s^4+3s^2}{(1+s^2)^2}>0$ for $s \neq 0,$ so that $\varphi:\...
1
vote
1answer
73 views
+100

Propagation of regularity for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider the following boundary value problem for the heat equation:...
6
votes
1answer
45 views
+50

Existence of the $\Omega$ set in “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization”

In the proof of Lemma 4.3 in [1], they claim the following: Let $U$ be a subspace of $\mathbb{R}^{m\times n}$ with dim$(U)=d$ and let $\delta>0$. Then, there exists a set $\Omega\subset\mathbb{R}^{...
2
votes
1answer
76 views
+100

Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities

Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $ \mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \...