Questions tagged [elliptic-equations]
For questions about elliptic partial differential equations. If your question is specific to the Laplace equation, see (harmonic-functions).
659
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Doubts in lemma 9.12 of Gilbarg Trudinger
Lemma $9.2$: Let $u\in W_0^{1,1}(\Omega^+),f\in L^p(\Omega^+),1<p<\infty$ satisfy $\Delta u=f$ weakly in $\Omega^+$ with $u=0$ near $(\partial \Omega)^+$. then $u\in W^{2,p}(\Omega^+)\cap W_0^{1,...
- 907
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0
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29
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Finding slope of a given point on the elliptic curve given the points x and y coordinates [closed]
How to find the slope of a given point on the elliptic curve provided i have it's x and y coordinates without knowing how the x and y coordinate were formed.
The equation for the curve am looking for ...
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0
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32
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An isoperimetric-type inequality
I am reading some notes on de Giorgi's methods in the regularity of elliptic equations, and have come across a step which I can't make sense of. The claim is as follows (see Lemma 10 in the linked ...
- 61
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$-\Delta u =f $ for $f\in L^p(\Omega)$ implies $\nabla u \in L^p(\Omega)$
I wonder if there is a quick&dirty proof of the following statement:
Assume $\Omega \subset \mathbb{R}^n$ open and bounded with $C^1$-boundary and $p\geq 2$. The unique solution $u\in H^1_0(\Omega)...
- 31
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0
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13
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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{$P_{\varepsilon}$}
\...
- 493
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0
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43
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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 ...
- 31
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1
answer
81
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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 ...
- 2,669
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23
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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 ...
- 149
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0
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26
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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
$$-\Delta u = f \text{ on $\Omega$}$$...
- 1,384
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0
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22
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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 ...
-2
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27
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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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18
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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 ...
- 648
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40
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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}
\...
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1
answer
54
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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?
...
- 211
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0
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30
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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 ...
- 57
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1
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36
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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
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1
answer
42
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Elliptic regularity: Does the source have to be $L^{2}$
Let $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$ be an elliptic differential operator (formally self-adjoint for simpliciy), say the Laplacian for example. Is the following true:
Let $...
- 2,506
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Basic question about the Sobolev space $H^1_0(\Omega)$
Let $\Omega\subset\mathbb{R}^N$ be a bounded and connected Lipschitz domain, and $u\in H^1(\Omega)$ be any function. If I denote for each real $k\in\mathbb{R}$
$$\Omega_k=\{x\in\Omega\ |\ u(x)>k\}$$...
- 1,857
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0
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25
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What would be the eta quotient for the Weierstrass equation $x^2 = y^2$?
I am trying to find an as simple as possible example of a Weierstrass equation where the eta quotient exists and is not completely trivial.
What would be the eta quotient for the Weierstrass equation ...
- 7,326
3
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1
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Questions from Evans PDE problem 6.11
The original problem states that: Let $U\subset\mathbb{R}^n$ be a bounded open set. Assume $u\in H^1(U)$ is a bounded weak solution to the uniformly elliptic equation $$-\sum_{i,j=1}^n(a_{ij}u_{x_i})_{...
- 129
3
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1
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68
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Showing uniqueness of solution to a non-linear Poisson problem
I'm trying to prove that a non-linear Poisson problem has a unique solution. The context is the following:
Let $\Omega \subset \mathbb{R}^n$ be a bounded open subset of class $C^2$. Consider the ...
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3
votes
1
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70
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Dense subspace of radial functions of $H^1(\mathbb{R}^N)$
By definition the space of radial functions in $H^1(\mathbb{R}^N)$ is
$$
H^1_{rad}(\mathbb{R}^N) = \{u \in H^1(\mathbb{R}^N) : u = u \circ R, \forall R \in O(N)\}.
$$
I'm trying to find a dense ...
- 891
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20
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Papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional.
Can you recommend me some papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional.
I glance over some methods including mountain pass theorem and ...
- 463
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35
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Does a gauge-invariant Caccioppoli inequality hold?
(I suspect that this question has an elementary resolution. But perhaps it would be more appropriate on MathOverflow, and if so I would not be opposed to migrating it there.)
Let $V \Subset U$ be ...
- 719
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35
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Regularity of functions approximated by polynomials
I want to ask how to prove the following result:
Suppose $u\in C(\overline{B_{1}})$, if there exists a universal constant M>0, such that for any $x_{0}\in \overline{B_{1}} $, we have a quadratic ...
- 1
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1
answer
30
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Transformation of parametric oblique ellipse equation its normal form
I have an equation describing an oblique ellipse in parametric form
$$
\left(\begin{matrix} x\\ y \end{matrix}\right) = \left(\begin{matrix}
\cos(\alpha) & -\sin(\alpha);\\
\sin(\alpha) & \...
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answers
45
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Schauder estimates on boundary
I recently studied the Schauder estimates with the boundary and checked the wiki page. The following is the link (the boundary estimate is on the bottom part):
https://en.wikipedia.org/wiki/...
- 43
1
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1
answer
75
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Uniqueness of radially symmetric eigenfunctions
Given a smooth radially symmetric potential $V = V(|x|)$ in the unit ball $B_1\subset\mathbb{R}^n$, consider the eigenvalue problem
\begin{equation}
\begin{cases}
\Delta \varphi + V \varphi = \lambda \...
- 546
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vote
2
answers
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Euler Lagrange equation of a functional on the space of traceless symmetric matrices
Let $\mathcal{S}_0:\{Q\in \mathbb{R}^{3\times 3}:\text{tr}Q=0 \hspace{5pt}\text{and}\hspace{5pt} Q_{ij}=Q_{ji} \hspace{5pt} \text{for any $i,j=1,2,3$}\}$ and
$$ \widetilde{ \mathcal{E}}(Q)
=\int_{\...
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0
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Aircraft Wing Analysis - Elliptical Pressure Distribution
I'm having great difficulty with generating the shear and bending moment diagrams for a wingspan. I am using the elliptical pressure distribution equation; however, this problem doesn't consider a ...
- 11
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1
answer
71
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Quite general second order PDE
I know too little about solutions for PDEs in general, so I would be grateful if anyone has any idea if there is a solution.
I am trying to find a function $h$ of $n$ coordinates defined on a cube of ...
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0
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Question about the convergence of eigenvalue $\lambda_R$ and the eigenfunction $\phi_R$ of a schrodinger operator in $B_R$
I want to ask about a question about the convergence of eigenvalue $\lambda_R$ and the eigenfunction $\phi_R$ of a schrodinger operator in $B_R$.
Given an eigenvalue equation (2nd order elliptic PDE) ...
- 463
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Question in proof of Weyl's lemma on harmonic equations
I'm reading Jost's Partial Differential Equation Chapter 2.
To point out where I'm stuck, here is the excerpt of the textbook dealing with Weyl's lemma.
(The mollifier is defined as $\rho (t) = C exp(\...
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What is the finite Morse index solution?
I'm dealing with an elliptic PDE, it depends on the dimension of the domain, for some situations, I highly doubt that there is no $C^2$ solution, so I searched for some papers related to this equation ...
- 463
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0
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49
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Found the parameter of the elliptic integral of the first kind
Suppose I have:
$$\frac{p}{[K(p)-K(\frac{-\pi}{2},p)]^2} = x$$
$K(p)$ being the complete elliptic function of the first kind and $K(\theta,p)$ the incomplete elliptic function of the first kind.
How ...
2
votes
1
answer
76
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Is this a coincidece for harmonic equations.
Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $. Consider the functional problem
$$
\min_{v\in H_0^1(\Omega)}\int_{\Omega}|\nabla v|^2dx.
$$
It is easy to calculate the E-L equation of ...
2
votes
1
answer
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A problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $R^2$ and $\int_{R^2} \exp u(x) d x< +\infty$
I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $R^2$ and $\int_{R^2} \exp u(x) d x< +\infty$.
LEMMA 1.1 (Ding). If $u$ is a ...
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A problem about regularity and mean value property in the Merle and Brezis work on $-\Delta u = V(x) \exp u$ in $\mathbb{R}^2$ plane.
I'm reading the Theorem2 in UNIFORM ESTIMATES AND BLOW-UP BEHAVIOR FOR SOLUTIONS OF $-\Delta u=V(x) e^u$ IN TWO DIMENSIONS
They prove that for the solution of
$$
-\Delta u= V(x)\exp u \text { in } \...
- 463
1
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1
answer
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A detail in Evans's prove of THEOREM 3 in section 6.5 (about principal eigenvalue)
The following picture shows the statement of the theorem and the first part of Evans' proof.
In his prove, Evans defines an operater A, which I know is properly defined (because of the maximum ...
- 96
1
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1
answer
68
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existence and regularity of weak solution to elliptic type equation
I'm considering a elliptic type equation which is
$$-\Delta u+\int_{\Omega}u dx=f\mbox{ and }\partial_{n}u|_{\partial\Omega}=0$$
where $f$ is a given $L^2(\Omega)$ function and $\Omega$ is a open ...
- 389
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0
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34
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Gilbarg Trudinger Problem $13.1$
The authors asked us to use the interpolation inequality and improve the interior Holder estimate. Here $d=\text{dist}(U, \partial \Omega)$.
$$[Du]_{\alpha; U} \leq C d^{-\alpha}$$
to the following ...
- 878
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1
answer
63
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Green function for the upper-half space $\mathbb{R}^{n+1}_+$
Given $\mathbb{R}^{n+1}_+=\{X=(x,t): x \in \mathbb{R}^n , t>0 \}$ as domain, what is the explicit formula for the Green function $G(X,Y)$ for the Laplacian on $\mathbb{R}^{n+1}_+$? I know that ...
- 393
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0
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Proving that a $W^{1,2}$ type space is Hilbert
Split the $N-$dimensional Euclidian space as $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. A vector in $\mathbb{R}^N$ will be denoted by $z = (x,y)$. Let $\alpha > 0$ and consider the ...
- 891
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Weak maximum principle for $W^{1,p}$ weak solution to elliptic equations
Let $\Omega\subset \mathbb R^n$ be a bounded open subset and $a^{ij}\in L^\infty(\Omega)$ satisfy $a^{ij}\xi_i\xi_j\ge \lambda |\xi|^2$ and $\|a^{ij}\|_{L^\infty}\le \Lambda$. Let $1<p<2$ and $u\...
- 744
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1
answer
47
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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
0
answers
27
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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 ...
- 33
2
votes
1
answer
69
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References for regularity of solution to fourth order elliptic/parabolic PDEs
After searching, all I could find was regularity of solutions to second-order PDEs like in Partial Differential Equations by Evans and Functional Analysis, Sobolev Spaces and Partial Differential ...
- 75
2
votes
0
answers
61
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Poincaré inequality in Elliptic PDE
I'm having a problem figuring out the answer to this question.
Consider the problem
$$
\begin{cases}
-\Delta u = f & \mbox{in } \Omega = (0,1)^2, \newline
\nabla u \cdot \mathbf{n} = g_N & \...
- 21
4
votes
1
answer
71
views
Soft enforcement of boundary constraints in linear PDE
Suppose I have a region $\Omega\subset\mathbb R^2$ with smooth boundary curve $\partial\Omega$. Consider the PDE
\begin{cases}
\Delta u(x) = 0\ \forall x\in\mathrm{int}\ \Omega \\
au(x)+b\frac{\...
- 880
0
votes
1
answer
33
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Lemma 1.34 in Colding-Minicozzi
Recall the stability operator $L=\Delta_\Sigma+Ric(N,N)+|A|^2$ for a minimal hypersurface $\Sigma^n\subset M^{n+1}$. Stability of $\Sigma$ is equivalent to non-negativity of the Rayleigh quotient $$\...
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