# Questions tagged [nonlinear-eigenvalue]

The nonlinear-eigenvalue tag has no usage guidance.

17
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### Weak minimum principle for the p-Laplacian

Consider the $p$-Laplacian with Dirichlet boundary conditions on an open domain of $\mathbb R^d$ (with smooth boundary, if necessary). A weak maximum principle would state that
$$
\Delta_p u\le \...

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### Is there an analogue of this linear-eigenvalue problem result to the nonlinear case?

Sorry, I don't know the name of the result or if there is one, but I know that for an operator $A$ with for which $(A - \lambda I)^{-1}$ is bounded and normal for $\lambda \notin \sigma(A)$, we have
$ ...

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### Spectral theorem for symmetric real tensors

Is there a definition of eigenvalues that allows to use a spectral theorem?
Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...

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### Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3?
Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...

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### Phase angles of a complex eigenvector

I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$:
$$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...

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152
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### Bifurcation points in parametric Hammerstein Integral equation

I am a theoretical physicist working on integrable systems, hence I preemptively ask forgiveness for the eventual lack of mathematical rigor.
My question concerns the properties of a particular ...

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### soliton solution of nonlinear convection-diffusion-reaction equation system?

I am interested in a flame propagation in 1D which is governed by a nonlinear convection-diffusion-reaction equation system:
$$
u_t+C(u)u_x+(D(u)\cdot u_x)_x + R(u) = 0,x\in (-\infty,+\infty) \\
u(-\...

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### The nonlinear operator defined as the commutator of a matrix and a nonlinear operator

In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...

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### Is this "stretched eigenvector" studied? (If so, what are its properties?)

An eigenvector is defined by
$$
\lambda \mathbf{v} = A\mathbf{v}.\tag{1}
$$
But suppose I change this to
$$
\lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2}
$$
for real $\alpha\ne 1$, where $\mathbf{v}^...

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### Polynomial Eigenvalue Problem with few non-zero coefficients

Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...

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### First eigenfunction of $p=3$-Laplacian of a square domain in $\Bbb R^2$ : reference for any work on this?

In the last few decades, lots of work on first eigenfunction of $p$-Laplace with Dirichlet and other boundary conditions. But I couldn't find much on periodic boundary conditions. I have computed the ...

2
votes

1
answer

278
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### Simultaneous triangularization of two diagonal matrices and a symmetric matrix

I have the following quadratic eigenvalue problem:
$\det(\lambda^2M + \lambda D + J)=0$
where, M and D are both $n \times n$ real diagonal matrix with positive diagonal entries; $J$ is
1) a ...

5
votes

1
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127
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### How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...

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2
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194
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### significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...

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### Non-linear state-space model system stability using Lyapunov?

I have a non-linear system modelled in state-space as follow:
$$
\mathbf {\dot x} = \mathbf A(x) \mathbf x
$$
I need to find out if this system is stable, so I was thinking in using the Lyapunov ...

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265
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### Distinct eigenvalues of the quadratic eigenvalue problem

Suppose we have a quadratic eigenvalue problem $(A_{0}+\lambda A_{1}+ \lambda^{2} A_{2})x=0$. I'd to know if there are conditions under which the problem is known to have a small number of distinct ...

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### Nonlinear eigenvalue problem - sorta

Suppose you have an equation of the form $Ax=f(x)$, where $A$ is a $n \times n$ matrix, $x$ is a vector of length $n$ and $f(\cdot)$ is some function. Is there a name for this sort of problem?