For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).
Questions tagged [elliptic-equations]
682 questions
3
votes
1 answer
PDE: How to show that this function is the zero function?
Let $\Omega \subset R^n$ be a bounded domain with smooth boundary. Let $u \in C^{2}(\Omega) \cap C(\overline{\Omega})$ a function that satisfies $\Delta u = u^3$ in $\Omega$ and $u = 0$ on $\partial \Omega $. Prove that $u$ is the null function.
I…
math student
- 4,566
2
votes
0 answers
Why does this elliptic quasilinear problem have a weak solution?
Let $\Omega \subset R^n$ a bounded domain with smooth boundary and $\beta \in (0,1)$ fixed. Let $\overline{U}$ the weak solution of the problem
$$
\Delta \overline{U} = 1 \text{ in } \Omega \\
\overline{U} = 0 \text{ in } \partial \Omega .
$$
It can…
math student
- 4,566
2
votes
1 answer
Can someone help with finding the area of a super ellipse?
I have super ellipse x^4+y^4=9592^4 inside a square with edges equal to 9592*2.
I want to find out what the area is between the square and the super ellipse but the super ellipse math for area is a little beyond my abilities.
In reality I am…
Joe
- 534
1
vote
1 answer
How to draw this equation of point (U, V)
$ U = 9 - 3 \cos(2t) + 8.5 \sin(2 t) $
$ V = 5.5 - 0.5 \cos(2t) + 6 \sin(2t) $
1)How to know the point (U, V) will form an ellipse?
2)How to draw it?
3)Is there any tool which can display this path?
Attempt:
I read the wiki:…
Francis Bacon
- 211
1
vote
0 answers
Well-posedness of the Poisson equation with mixed directional derivative/Dirichlet boundary conditions
The Problem
Let $\Omega \subseteq \mathbb{R}^d$ with regular enough boundary $\partial \Omega = \Gamma_1 + \Gamma_2$.
Let $K \in \mathbb{R}^{d \times d}$ a symmetric positive-definite matrix.
Consider the following Poisson problem with directional…
FredV
- 163
1
vote
0 answers
A question about weak solutions of elliptic equations
Let $B_r:= \{ x \in \mathbb{R}^N; |x| < r\}$. Consider $u \in H^{1}_{0}(B_1)$ and $f \in L^2({B_1})$ such that
$$\int_{B_{\frac{1}{2}}} \nabla u \nabla v dx = \int_{B_{\frac{1}{2}}} fv dx, \forall v \in H^{1}_{0}(B_{\frac{1}{2}})$$
and
$$\int_{A}…
math student
- 4,566
1
vote
0 answers
a Problem about Harmonic Function
Let $\Omega\subset\mathbb R^n$, $u\in W^{1,p}(\Omega)$, $p>1$ satisfies
$$\int_{\Omega}|\triangledown u|^{p-2}\triangledown u\triangledown\phi=0~~~,\forall\phi\in C_0^{\infty}(\Omega)$$
Then we call $u$ harmonic function.
When $p=n$, prove $u\in…
gaoxinge
- 4,434
1
vote
0 answers
Neumann-like boundary conditions for Poisson equation
Consider following problem for Poisson equation in some bounded domain $G$ with piecewise smooth Lipschitz continuious boundary $\partial G$:
$$
-\Delta u = \rho \text{ in } G\\
(\boldsymbol{\nu} \nabla) u = \sigma \text{ on } \partial G
$$
where…
uranix
- 7,503
0
votes
1 answer
Weak maximum principle for eliptic PDEs proof question
I have been given a proof for the weak maximum principle and I do not understand the logic in one of the steps.
Let $a_{ij}$ be syymetric then: $$Lv=\sum_{i,j=1}^N a_{ij}(x)\partial_{x_i}\partial_{x_j}+\sum_{i=1}^Nb_i(x)\partial_{x_i}v $$
Then in…
Tim Coulter
- 11
0
votes
2 answers
find the locus of the vertices of equilateral triangle circumscribing the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
My try : I am confused in this question , I have only tried questions who says to find the locus of a point but here we have to find locus of three points and there are only two relation , seems pretty odd and new to me. Need your help in this.
Shinobi
- 351
- 4
- 14
0
votes
0 answers
How to get the equation of an ellipse given the center, directrix and length of latus rectum?
Find the equation of the ellipse with centre $(5, -2)$, $x+3=0$ as the directrix, and length of the latus rectum equal to $6$.
