Posts Tagged ‘harmonic’

Sleek proof that the harmonic series diverges

April 14, 2012 7 comments

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]}.

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Problem regarding subharmonic functions

March 15, 2012 Leave a comment

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.

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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

November 23, 2010 Leave a comment

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

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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

April 30, 2010 Leave a comment

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.
Miklos Schweitzer 2001
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