# Questions tagged [elliptic-equations]

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

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### Polyharmonic system

In the problem, with $\Omega \subset \mathbb{R}^3$ a smooth bounded domain, and the given functions $h_1, h_2 \in C(\partial\Omega)$ \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \...
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### Existence and uniqueness of $-\Delta u+u^2=f$

My question is very similar to the one given here: Existence and uniqueness of $-\Delta u + u^3 =f$ If we consider the equation $-\Delta u+u^2=f$ in 3 dimensions with $f\in L^2(\Omega)$ is there ...
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### How can I find $P$ s.t. $2P=\infty$?

Let me consider the elliptic curve $C: y^2=x^3+2x$. Now I want to finde $P\in C(\Bbb{Q})$ s.t. $2P=\infty$. I thought about writing $P=\left(\frac{a}{b}, \frac{c}{d}\right)$, then I can use the ...
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### Standing wave operator?

Everyone knows the d'Alembert wave operator acting on a scalar function $\Psi(\boldsymbol{r},t)$ as: $$( \partial_t^2/c^2 - \Delta ) \Psi(\boldsymbol{r},t) = 0$$ The homogeneous solution in 3 ...
1 vote
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### Solution of PDE on Lipschitz domain which is in $\{ u \in H^1_0 : \Delta u \in L^2\}$ but not in $H^2$

I am looking for an explicit example that can be verified that demonstrates this: there is a Lipschitz domain $\Omega$ such that given $f \in L^2(\Omega)$, the PDE $$-\Delta u = f \text{ on \Omega}$$...
1 vote
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### Existence and uniqueness without ellipticity condition

Consider a function $a:[0,1]\to[0,\infty)$, e.g., $a(x)=x$. Are there existence theorems for equations of the form: \begin{align*} u-\big(a(x)u_x\big)_x=f(x) ? \end{align*} what are the boundary ...
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### Orlicz spaces, young functions and regularity theory

I'm looking some book from which I could learn more about Sobolev spaces and Orlicz spaces. I'm interested rather in regularity theory and week solutions: some ...
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### connectivity condition of domain, when studying eigenvalues of elliptic operators

The following is the Theorem 6.5.1 stated in Evans' PDE book, which is slightly modified so that the statement is self-explanatory. Let $U \subset \mathbb{R}^{n}$ be an open bounded and connected ...
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### Elliptic regularity in a weird domain

I am curious about the regularity of the solution in $H^1$ of the following PDE: In $\mathbb{R}^n$ let $B = B(0, 1)$, $\Sigma = \{x_n = 0\}$ and $T = B(0, 1/2) \cap \Sigma$. Consider \begin{align} \...
1 vote
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### 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? ...
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### Strong maximum principle for a local maximum

Suppose $L$ is a strictly elliptic operator in the non-divergence form with $c \equiv 0$, $u \in C^2(\Omega)\cap C(\Omega)$ and $Lu \le 0$ in $\Omega$. Prove that if $u$ attains a local maximum at an ...
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### Regularity of solutions to Poisson's equation on part of the boundary

Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set and suppose $\partial \Omega$ is smooth on a relatively open subset $\Gamma \subset \partial \Omega$. Consider a weak solution of the ...
1 vote
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### Numerical Inversion of Elliptic Operator

I am trying to solve the following elliptic differential equation. $$\frac{\partial^2\psi}{\partial R^2} + \frac{\partial^2\psi}{\partial Z^2} - \frac{1}{R}\frac{\partial\psi}{\partial R} = S(R,Z)$$ ...
1 vote
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### About $\epsilon$-Regularity theorem in Leon Simon's Theorems on Regularity and Singularity of Energy Minimizing Maps

I am reading Theorems on Regularity and Singularity of Energy Minimizing Maps by Leon Simon. The problem arises in Section 2.3, the proof of Shoen-Uhlenbeck theorem: Let $u:\Omega \to N$ is a ...