Articles of poissons equation

Prove the uniqueness of poisson equation with robin boundary condition

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 […]

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad x\in B_r, \\ u(x) = g(x), & \qquad x\in \partial B_r, \end{align} for some $f \in C^1(\overline{B}_r)$ and $g \in C(\partial B_r)$ I have to show […]

necessary and sufficient condition for the Poisson's equation to admit a solution$?$

$$−\Delta u = f \text{ in } \Omega$$ $$ \frac{\partial u}{\partial n}= g \text{ on } \partial\Omega$$ where $\Omega\subset\mathbb R^n$ is a bounded domain with boundary $\partial\Omega$, $\Delta$ is the Laplace operator, $f$ and $g $ are given smooth functions and $ \frac{\partial u}{\partial n}$ denotes the outer normal derivative of $u$. How to find […]

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |{f(x’,y’,z’)}|dx’dy’dz’$ is finite. $ r’=\sqrt{|x-x’|^2+|y-y’|^2+|z-z’|^2}$ let: $$g(x,y,z)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac {f(x’,y’,z’)}{4\pi r’}dx’dy’dz’,\ \ \text{s.t.}\ \ \nabla^2g= \frac {\partial^2g}{\partial x^2}+\frac {\partial^2g}{\partial y^2}+\frac {\partial^2g}{\partial z^2}$$ How to show that $\nabla^2g=-f$ almost everywhere? This is […]