# Questions tagged [elliptic-equations]

For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).

531 questions
Filter by
Sorted by
Tagged with
12 views

### Calderon-Zygmund in solving $\Delta u + (d f, \theta)_{\omega}=e^{u}$ when we have the $L_{\infty}$ estimation of $u$ on complex compact manifold

I'm reading the paper On Chern-Yamabe problem in this paper, $\Delta^{C h} f=\Delta_{d} f+(d f, \theta)_{\omega}$, after getting the uniform $L_{\infty}$ bound of the constructed sequence $f_{t_{n}}$, ...
25 views

### Calculate an ellipsoid given two points and a arc length.

I have a problem and I would like some insight into if people think it is possible. If they think it is possible do they also have an idea of how to solve it. Given two points $A$, $B$ in the ...
• 13
31 views

• 23
34 views

### On the elliptic regularity theory of a Schrödinger equation

Let $u\not\equiv0$ be a non-negative solution of the following equation: $$-\Delta u+u=|u|^{p-2}u,\quad u\in H^1(\mathbb{R}^N),$$ where $N\geq3$ and $2<p<2^*=2N/(N-2)$. My goal is to prove ...
• 319
20 views

### Choose Green's function of Possion equation on half-space

Background: Consider the Dirichlet problem (A) defined in $\mathbb{R}^3$ $$-\Delta u=0~~~~\text{in \mathbb{R}_+^3}=\{x=(x_1,x_2,x_3):x_3>0\},$$ $$u=g~~~~\text{in \partial \mathbb{R}_+^3 }.$$ We ...
• 203
54 views

### Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

This is intended to be an example of an elliptic integral but I'm not sure how to go about showing it. I'm not sure which identities to start with, any tips would be greatly appreciated!
• 23
53 views

### A sequence of solutions of elliptic equations converges uniformly on compact subsets

$\textbf{Theorem 3.}$ $\Omega$ is a bounded domain in $\mathbb{R^2}$. Suppose $(u_n)$ is a sequence of solution of $$-\Delta u_n=V_n(x)e^{u_n}\quad\text{in}\quad \Omega$$ for some $1<p\leq \infty$....
31 views

22 views

### Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I am reading a paper of Brezis and Oswald about existence and uniqueness of positive solutions to sublinear elliptic equations: - \Delta u = f(x, u), \ u \geq 0, \ u \not\equiv 0 \...
24 views

### Constant solution for an elliptic equation

Let $U$ be a open and connected subset of $\mathbb{R}^n$ with regular boundary. Consider the fallowing elliptic problem $$\Delta u + c(x) u = u^3, U\\ \hspace{2cm}u = 0, \partial U,$$ where $c$ may ...
• 301
1 vote
64 views

### Reference for $L^p$ estimates

My PDE professor showed the following result: Let $f \in L^{p}(\Omega)$, for $1 < p < \infty$. Also consider $u \in L^{1}_{loc}(\Omega)$ a solution of \begin{align} -\Delta u + a(x)u&= f, \...
• 301
29 views

### Estimate the error caused by domain perturbation for Dirichlet Laplacian equation

$\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, consider the Dirichlet laplacian equation on $\Omega$. \begin{cases} -\Delta u=f & \mbox{in }\Omega\\ u|_{\partial \Omega}=0 \end{cases} ...
• 322
20 views

### Question about the use of the Min-Max method in solving evolution equations.

in elleptic PDE's i used Min-Max methods and critical points researche to find my weak equations and i learnd about their techenics however, i didn't find up to this time references speaking of this ...
43 views

• 189
31 views

### On the existence of global classical non-zero solutions of a linear elliptic equation

Does the equation $$-\Delta u +u=0$$ have any non-zero classical, i.e., $C^2$, solutions on $\mathbb{R}^d$? How about if $\mathbb{R}^d$ is replaced with half-space? How about solutions of polynomial ...
• 245
23 views

### Angle of line at known point on known ellipse

I have an oval/ellipse, $x^2/a^2 + y^2/b^2 = 1$. The ellipse is always centered at $(0,0)$. I have a point on that ellipse, $(x, y)$. I'm looking for the angle of the tangent at that point relative to ...
• 133
1 vote
42 views

109 views

### temperature distribution using Fourier/Laplace

Consider room that occupies quarter space That is formed by two walls. One wall is fully insulated and has a constant temperature $0$. Another wall has a window of length L that with one edge at the ...
51 views

### What are the equations of rotated and shifted ellipse, parabola and hyperbola in the general conic sections form?

How will look the general conic sections equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ in case of rotated and shifted from coordinates origin ellipse, parabola and hyperbola? I need a formulas for ...
• 189
1 vote
145 views

### How to approach this semilinear system of PDEs?

Cross-posted to MathOverflow (link) My question concerns a system of $n$ $n$-dimensional PDEs. \begin{equation*} -\frac{1}{2}\sum_{jk}a_{jk}x_{j}x_{k}u^{i}_{x_{j}x_{k}} +\sum_{jk} u^{j}\pi_{jk}u^...
• 714
1 vote
81 views

• 123
51 views

### Solving Laplace's Equation on Complex Geometries Using Conformal Maps

I am interested in solving Laplace's equation $$\Delta u = 0.$$ with Dirichlet boundary conditions but on complex geometries. To my understanding, the standard approach is to (if possible) map your ...
• 1,366
1 vote
59 views

### Linearisation of a Quasilinear Elliptic PDE

I have noticed that the word 'linearisation' can have different meanings in different places in the literature. For example, if one has a second-order quasilinear elliptic PDE, what would be the ...
• 1,076
1 vote
Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then $$u(x) = \int_\Omega \log |x-y| f(y)\ dy$$ ...