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Posts Tagged ‘harmonic’

Sleek proof that the harmonic series diverges

Suppose that the harmonic series does converge. Then the sequence of integrable functions functions $f_n : \Bbb{R}_+ \to \Bbb{R}_+,\ f_n=\frac{1}{n} \chi_{[0,n]}$ is bounded above by an integrable function $g: \Bbb{R}_+ \to \Bbb{R}_+,\displaystyle \ g=\sum_{i=1}^\infty \frac{1}{i} \chi_{[i-1,i]}$.

Categories: Analysis, Measure Theory

Problem regarding subharmonic functions

Suppose that a function $f: \Bbb{R}^2 \to \Bbb{R}$ of class $C^2$ satisfies the inequality $f_{xx}+f_{yy} \geq 0$ at every point of $\Bbb{R}^2$. Suppose also that all its critical points are non-degenerate, i.e. the matrix of second order derivatives at the critical point has non-zero determinant. Prove that $f$ cannot have local maxima.

Weakly harmonic implies harmonic

Suppose $u \in C^0(\Omega)$ is weakly harmonic on an open set $\Omega$, i.e. the relation $\displaystyle \int_\Omega u \Delta \phi dx =0$ holds for all $\phi \in C_0^2(\Omega)$. Show that $u$ is harmonic in $\Omega$.

PHD Iowa (6202)

See this blog post for a proof.

Divergent integral

Let $u$ satisfy $u \in \mathcal{C}^2(\Bbb{R}^n),\ \Delta u=0$ on $\Bbb{R}^n$. Show that the integral $\displaystyle \int_{\Bbb{R}^n}u^2 dx$ is convergent if and only if $u \equiv 0$.
PHD 6201

Harmonic functions and holomorphic functions

May 3, 2010 1 comment

Prove that a function $u:\mathbb{R}^2 \to \mathbb{R}$ is harmonic if and only if there exists an entire function such that $u=\text{Re} f$. ( if and only if it is the real part of an entire function)

Harmonic function

Let $e_1,e_2,...,e_n$ be semilines on the plane starting from a common point. Prove that if there doesn’t exist any function $u \not\equiv 0$ harmonic on the whole plane that vanishes on the set $e_1\cup e_2\cup...\cup e_n$, then there exists a pair $(i,j)$ such that there is no function $u\not\equiv 0$ harmonic on the whole plane such that $u$ vanishes on $e_i \cup e_j$.