# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Bound by limit plane of harmonic function on half space vanishing at infinity

The problem is Suppose $\phi\in\mathcal{S}(\mathbb{R}^2)$, $u$ on $\mathbb{R}^3_-=\mathbb{R}^2\times(-\infty,0)$ satisfies \begin{cases} \Delta u(x,y,z)=0\\ \lim\limits_{z\to 0^-} u(x,y,z)=\phi(x,y)\\ ...
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### Why the harmonic lifting of $v_n$ still holds $V_n(y)\to u(y)$

In the book "Elliptic Partial Differential Equations of Second Order", the theorem 2.12 says the function $f(x)=\sup_{v\in S_\varphi}v(x)$ is harmonic in $\Omega$. In the proof, it says ...
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### Hypothesis on a parameter to ensure the unique solvability of a modified Laplace equation

Let $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain, $n$ the normal vector on its boundary and $q\in L^{\infty}(\Omega)$. I want to find the minimal hypothesis on $q$ such that the following ...
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### Understanding construction of fundamental solution to laplace's equation

We take a look at Laplace's equation $\Delta u=0, u:\mathbb{R}^n\rightarrow\mathbb{R}$ and want to look for explicit solutions, firstly, since Laplace's equation is invariant under rotations, for ...
1 vote
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### Question on Theorem 2.2.13 in Partial Differential Equations by Evans

I'm working through the section on Laplace's Equation in Partial Differential Equations by Evans. I'm having trouble following a step in Evans's proof of the symmetry of Green's function (Theorem 2.2....
• 198
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### How to solve the two-dimensional Laplace equation in this unbounded and multiply-connected domain?

I have the Laplace equation $$\nabla^2 \phi(x,y) = 0$$ where $\phi$ is a function defined on the domain $\mathbb{R^2}\setminus(C_1 \cup C_2)$. $C_1$ and $C_2$ are two circles of radius $r = 0.5$ and ...
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### Solve $\nabla^2 u(x,y) = 1-y$ + B.C.

I am trying to solve: $$\nabla^2 u(x,y) = \partial_{xx} u(x,y) + \partial_{yy} u(x,y) = 1-y$$ over the domain $[-1,1]\times [0,1]$, with the B.C.: $$u(x,0) = 0 = u(\pm 1, y).$$ Is there any closed-...
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### Question about Theorem 1 in Chapter 2 of Evans's Partial Differential Equations

I am working through Partial Differential Equations by Evans and I am struggling to understand a step of his proof of theorem 1 in chapter 2. He presents the following argument: The function $\phi$ ...
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### poisson's formula for half-spce, evans page 38

I have questions about the proof of Theorem 14 (poisson's formula for half-space) in Page 38. Let $K(x,y)$ be the Poisson's kernel for $\mathbb R^n_+$: K(x,y)=\frac{2x_n}{na(n)}\frac{1}{|x-y|^n}\...
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$K$ is a baseline kernel function that is nonnegative, symmetric and supported on [-1,1]; $h$ is a bandwidth in local smoothing .Let $p_j$ denote the marginal density of $X_j$.We define the estimator ...