Almost six years ago, Michael Hardy raised the issue of the "partitions" tag being used for some very different concepts, and subsequently edited its tag wiki excerpt to at least be clear about what the various concepts covered were. Two months ago the tag wiki was changed to be solely about in...

If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$, then find $B^{16}$ My Method: Given $$AB^2=BA \tag{1}$$ Post multiplying with $B^2$ we get $$AB^4=BAB^2=B^2A$$ Hence $$AB^4=B^2A$$ Pre Multiplying with $A$ and using $(1)$ we get $$A^2B^4=(AB^2)A=BA^2$$ hence $$A^2B^4=BA^2 \t...

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...