I'm having trouble solving the following PDE problem. We're in the open unit ball in the plane, centered at the origin, $$B=\{(x,y)\in \mathbb{R}^{2},\ \ x^2+y^2<1\}$$ The following boundary problem is given $$\Delta u(x,y)=0,\ \ \text{in}\ B,\ \ \ u(x,y)=\sin(x)\ \ \text{on}\ \partial B$$ ...

Let $u:\Omega\rightarrow \mathbb{R}$ be an harmonic function (this is a smooth function) such that $$\Delta u =0 \quad \mathrm{ in }\quad \Omega,$$ where $\Omega\subseteq \mathbb{R}^{2}$ is an open set. Suppose that $0\in \Omega$ and $\rho>0$ such that $\mathcal{B}_{\rho}(0)\subset \Omega$, whe...

I'm having a bit of a problem proving the equality: $$u(x) = \frac{1}{\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{n}{\omega_n r^n}\int_{B (x,r)} u\, dV$$ Which is the mean value theorem for Harmonic functions, where $\omega_n$ is the area of $S^n$ and $B(x,r)$ is the ball in $\...

A well-known feature of harmonic functions on (domains of) $\mathbb{R}^n$ is the mean-value property: that is, if $\Delta u = 0$, then $$ u(x_0) = \frac{1}{\text{Vol}(\partial B_r(x_0))}\int_{\partial B_r(x_0)}{u\,dS} = \frac{1}{\text{Vol}(B_r(x_0))}\int_{B_r(x_0)}{u\,dV}. $$ Is the same true on ...

Theorem If $u\in C(\overline{B_R(x_0)})$ and is harmonic in $B_R(x_0)$, then $$|D^mu(x_0)|\leq\frac{n^m\exp(m-1)m!}{R^m}\max_\limits{\overline{B_R~(x_0)}}|u|$$ We can prove the theorem by induction, but I am stuck at the first step of the proof. When $m=1$, we have $$\triangledown u(x_0)=\frac{n...

Let $A \subset \mathbb{R}^2$ be open and connected and let $u \in C^2(A)$ be harmonic. Then $u$ satisfies $$u(x)=\frac{1}{2\pi}\int_0^{2\pi}u(x+r\hat n(\theta)) \, d\theta$$ I'm given the following proof: Let $x \in A$ and take $r>0$ so that $B_r(x) \subset A$ . Since $u$ is harmonic it satisfi...

A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is the same.) The proof involves distributions, which I am not very familiar with, so there is a step ...

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then $$u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{\partial B(x,r)}u \, dS=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{B(x,r)} u \, d...

A search result for Mean Value Theorem gives us 2715 results, and results on the page are like ones I think we can include in the tag. The theorem is an important result in calculus, and questions relating to its applications, proofs. I think it would be useful if could have the tag, as it can gr...

Let $A$ be a subset of $\mathbb{R}^n$ then the rearrangement of $A$ denoted by $A^*$ is the ball $B(0,r)$ having the same volume as $A$ i.e if $|A| =|B(0,r)|$ with respect to Lebesgue measure then $$A^*= B(0,r)$$ Let $f$ be a function from $\mathbb{R}^n$ to $\mathbb{R}$. Then its symmetric de...

I'm studying section 3.3 of Analysis by Lieb and Loss, about symmetric-decreasing rearrangement of functions. Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing re...

I found this statement about rearrangement from analysis Lieb and Loss in chapter 3. Suppose f, g are nonnegative functions in $L^2(\Bbb{R^n})$, then $||f^*-g^*||_2 \le||f-g||_2$ Where $f^*$ is the symmetric- decreasing rearrangement of $f$. $f^*(x):=\int_0^{\infty} \chi_{\{|f|>t\}^*}(x)dt$. ...

Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it. As you can see in Leoni (or in Lieb & Loss), the decreasing and the increasing rearrangement of a measurable function $u:\Omega \to [0,\infty[$ ($\Omega \subseteq \mathbb{R}^N...

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can define the spherically symmetric and decreasing rearrangement $f^* \, \colon \mathbb{R}^d \rightarrow...

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement $$\alpha_k=\inf\{\alpha>0|F(\alpha)<2^k\}$$ I have to prove the following $$\sum_{k:\alpha_k>\alpha}2^k\leq 2 F(\alpha)$$...

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: u(x)\geq t \} > \omega_N |x|^N\} $$ where $\mu$ is the Lebesgue measure in $\mathbb{R}^N$, $w_N=\...

If $f(x)=|\arctan(x)|$, how does one constructively prove that its decreasing rearrangement is given by the constant function $f^\star(y)=\pi/2$ defined on $[0,\infty)$? The decreasing rearrangement of a function $f$ is defined as $$f^\star(x)=\inf\{u:|\{t:f(t)>y\}|\leq x\}$$

Two functions $f(x)$ and $g(x)$ are called equimeasurable if $m(\{x:f(x)>t\})=m(\{x:g(x)>t\})$. Nondecreasing rearrangement of a function $f(x)$ is defined as $$f^*(\tau)=\inf\{t\in \mathbb{R}^1:m(\{x:f(x)>t\}\leq\tau\}.$$ Prove that $f^*(\tau)$ and $f(x)$ are equimeasurable.

Pluralize division-ring.

I put this in a comment, and @Martin Sleziak suggested that I make it an answer, it here it is again (with better formatting): There is also such a thing as a Lebesgue–Stieltjes integral[^1], and one might want to ask in general about Stieltjesization (if I may say that) about any notion of inte...

I suggest not to have this tag. The mean value property is probably the most characteristic property of harmonic function. Indeed a $C^2$ function is harmonic iff it satisfies the mean value property. Thus if a question is related to mean value property, almost sure the OP will tag "harmonic fu...

In the general situation of $f:S\to \mathbb R^m$ where $S\subset \mathbb R^n$. There is a form of the mean value theorem: $a\cdot (f(y)-f(x))=a\cdot (f'(z)(y-x))$ which requires a vector $a$ and dot products. In Tom Apostol's Mathematical Analysis (Second Edition), page No. 355, I found that aft...

Can someone show me the proof of this form of the mean value theorem for vector valued functions? Let $f:R^n \rightarrow R^n$ be a function of class $C^1$ and $a,b\in R^n$, than there exists some $d\in R^n : a<d<b$ such that $(b-a)\cdot (f(b)-f(a))=(b-a)\cdot (f'(d)(b-a)) $ where $\cdot$ is a ...

Given that $f$ is differentiable on $\mathbb{R}$, I know that "between the zeroes of $f$, there is a zero of $f'$. But given that $f'$ has $k$ roots, then is it true that $f$ has at most $k+1$ roots.

Using transformation formula, we get $\tan\left(\frac{\alpha+\beta}2\right)$. Don't know how to proceed further. Thanks in advance.