Let $\Omega$ be a domain of $\mathbb{R}^n$, and $u:\Omega\to\mathbb{R}$ a continuous function. We call $u$ subharmonic if for any ball $B\subset\subset\Omega$ and any $h:\overline B\to\mathbb{R}$ which is continuous on $\overline B$, harmonic in $B$, and satisfies $u|_{\partial B}\leq h|_{\partial B}$, then $u\leq h$ on the whole $\overline B$. No regularity assumed for $u$, except for […]

Suppose that $D$ is a domain in $\mathbb{R}^n$ (that is, an open, bounded and connected subset), and that $u$ is an harmonic function on $D$. Let $x_0$ be a point at the boundary of $D$. Question (vague version): Can we say something about the limit $$\lim_{x \to x_0} u(x)?$$ I guess that the short answer […]

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem: $$ \Delta \omega = 0 \text{ in } M \\ \omega = f \text{ in } \partial M $$ I would like to find some bounds on […]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ intersect.How do I show that the level curves intersect at right angles (by calculating)? 2 What is the conceptual explanation behind the right angle intersection? What I […]

The general solution of the Laplace equation in spherical coordinates is (independant of $\phi$): $$V(r,\theta ) = \sum ^{\infty} _{l=0} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta ))$$ An example (Griffiths EM, 3rd edition, example 3.6): The potential $V_0 (\theta )$ is specified on the surface of a hollow sphere of radius $R$. […]

Does there exist a smooth harmonic map $f:\mathbb{R}^2 \to \mathbb{R}^2$ such that: $\det(df)$ is constant. $f$ is not affine. Is there such a map with $\det(df) \neq 0$? ($f$ is harmonic if each of its two components is a harmonic function).

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial n}=\triangledown u\cdot n$ ($n$ is normal vector). My thoughts: Suppose there are $u_1$ and $u_2$, satisfis the above equations. Let $w=u_1-u_2$, then we have $\Delta […]

Usually second-order linear PDE’s are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE’s: Elliptic – Laplace’s equation $\nabla^2 u = 0$. Parabolic – the heat equation $u_t = \nabla^2 u$. Hyperbolic – the wave equation $u_{tt} […]

A map $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ preserves harmonic functions if $f\circ\varphi$ is harmonic for every harmonic function $f:\mathbb{R}^n\to\mathbb{R}$. It is known that these maps are, in fact, compositions of isometries and dilations. I am looking for a concrete reference of this result, because I only have the reference included in Composition of a harmonic function with a holomorphic […]

Show that the time-independent Schrödinger equation for a simple harmonic oscillating potential $$-\frac{\hbar^2}{2m}\frac{d^2 u}{dx^2}+\frac12 m\,\omega_0^2 x^2u=E\,u$$ can be written as $$\frac{d^2u}{dy^2}+(\alpha – y^2)u=0\tag{1}$$ where $$y=\sqrt{\frac{m\,\omega_0}{\hbar}}x$$ and $$\alpha=\frac{2E}{\hbar\,\omega_0}$$ So by my logic $$\frac{dy}{dx}=\sqrt{\frac{m\,\omega_0}{\hbar}}$$ and $$\frac{d^2y}{dx^2}=0$$ clearly something has gone wrong or I am going about this the wrong way. My lecturer mentioned in the lecture that: […]

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