# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### monotonicity of Dirichlet energy under different Dirichlet boundary conditions.

Let $\Omega$ be a domain with two smooth boundaries in $\mathbb{R}^n$. Suppose $u_1$ is a harmonic function in $\Omega$ satisfying that $u=0$ in $\partial_1\Omega$ and $u=1$ in $\partial_2\Omega$. ...
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### Smooth real-valued harmonic function can be written as sum of real part and scaled log. [duplicate]

I am working on a previous year's qualifying exam problem, and I'm stuck. Here's the problem and what I know so far: The Question: Write $z=x+iy$. Let $f(x,y)$ be a smooth, real-valued, harmonic ...
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### Solving the Poisson equation

I am an undergraduate student, in this semester I am taking the course of partial differential equations. So reading about Poisson equation by Evan's classic book for pdes, i have some questions: ...
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### Finding the harmonic conjugate of the function

$u(x,y)=e^{x^2-y^2}(e^y \cos(x-2xy)+ e^{-y} \cos(x+2xy))$ Solving the Cauchy-Riemann equations for this is not practical. Alternatively, I could try to express $u(x,y)$ in the form of the real or ...
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### A mathematical statement about Laplace's equation from famous physicist Landau's book

In the book titled Classical Theory of Fields, by famous Physicist Landau, writes, From $\nabla^2\phi(x,y,z)=0$, ...it follows, in particular, that the potential $\phi$ of the electric field can ...
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### Possible to have only zero eigenvalues of the Hessian of a harmonic function that is neither of the form $ax+by+cz+d$ nor a constant?

(Following an earlier post here) I intuit that if we restrict to functions $f(x,y,z)$ that are harmonic (i.e. satisfying $\nabla^2f=0)$ but neither of the form $ax+by+cz+d$ ($a,b,c,d\in\mathbb{R}$) ...