# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

1,385 questions
0answers
14 views

### Gradient estimate and divergence theorem in an open ball (PDE)

I need to prove the estimate $|\partial_{x_i}u(x_0)| \leq \frac{N}{R}u(x_0)$ where $u \in C(B(x_0,R))$ and $u$ is nonnegative and harmonic in $B(x_0,R)$, and $i=1, \ldots,N$ I found this ...
0answers
12 views

### Perron's Method with Measurable Boundary Data

Perron's method is a standard construction to solve the Dirichlet Problem with continuous boundary data on domains with sufficient regularity. My question is about Boundary data that is no longer ...
0answers
9 views

### converting a function to a harmonic one in dirichlet boundary condition

I am trying to convert to a harmonic one in a manner that the harmonic function to be equal to the original function at $x=0$. is there any method to use for this purpose? thanks
1answer
71 views

### Convergence of harmonic functions in $L^1$ implies uniform convergence on compact sets

Resorting to an analog of what's done here, I'm trying to prove the following statement: Let $u_m: \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions and suppose there exists a ...
0answers
53 views

### Understanding proof: if $u: \Omega \subset \mathbb{R}^N \to \mathbb{R}$ is harmonic and $\nabla u \in L^2(\mathbb{R}^N)$, then $u$ is constant.

I'm trying to solve the following question: If $u$ is harmonic in $\mathbb{R}^N$ e $\nabla u \in L^2(\mathbb{R}^N)$, then $u$ is constant. I've found this solution, which I posted below, but I ...
1answer
37 views

2answers
46 views

### Is a harmonic function in $\mathbb{R}^2$ which is $o(\ln |x|)$ a constant?

Let $f : \mathbb{R}^{ 2 } \rightarrow \mathbb{R}$ be a harmonic function. Suppose $$\lim _ { | x | \rightarrow \infty } \frac { | f ( x ) | } { \ln | x | } = 0$$ Prove or disprove that $f$ is a ...
0answers
35 views

### Regularity of harmonic functions

I have a question on a fundamental property of harmonic functions. Let $\Omega \subset \mathbb{R}^d$ be a domain. We define $H^{1}(\Omega)$ by \begin{align*} H^{1}(\Omega)=\{u \in L^{2}(\Omega,m) \...
0answers
11 views

### separability of the space of harmonic functions on a domain

I tried hard to find a reference for this but I could not; please help me. Let D be an open set in $R^d$ (connected, if necessary), and let H be the set of harmonic functions on D. We may define a ...
0answers
41 views

0answers
22 views

### Solutions to $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ that only depend on r

Find all the solutions of $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ in three dimensions that depend only on $r = 􏰃x^2 + y^2 + z^2$, the radial variable in polar coordinates. Use the following ...
1answer
13 views

### Maximum Principle for estimate norm

I have two functions. A harmonic funcion $u$ in unit ball $B_{1}\subset \mathbb{R}^{n}$ and a function $h$ defined on $B_{1}$ such that $\Delta h=u$ in $B_{1}$ and $h=u$ in $\partial B_{1}$, the ...
1answer
54 views