Tagged Questions

18 views

How to use second derivative test?

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
32 views

Laplace's equation boundary conditions

I am supposed to determine a solution from the following boundary conditions in a rectangular area $V$. $f(0,y)=f(x,a) = f(x,0)=0$ and $f(b,y) = C\ sin(\frac{\pi}{a}y)$, for $a,b \in \mathbb{R}$. I ...
36 views

Regularity of weakly harmonic map

Suppose $(M,g)$ is a smooth $n$-dimensional manifold with $C^k$-metric $g$ and let $U\subset M$ be an open subset. Does anyone have a reference for a statement about the regularity of a map ...
39 views

Mean Value Property and Harmonic functions: a simple exercise

Let $f:[a,b]\to\mathbb{R}$ such that for every $h>0$ such that $$(x-h,x+h)\subset[a,b]$$ we have $$f(x)=\frac{1}{2}(f(x-h)+f(x+h)).$$ How can I conclude that $f$ is harmonic in $[a,b]$? My idea ...
28 views

example of harmonic function on sphere

Can anyone give me an example of a harmonic function on the sphere $S^{2}=\{(x,y,z):x^2+y^2+z^2=1,x,y,z\in{\mathbb{R}}\}$, which equals $1$ on the northern hemisphere and $-1$ on the southeren ...
35 views