Questions tagged [nonlinear-analysis]

Question 1

The Henon map $H_{a,b}$ is a smooth one-to-one mapping ${\mathbb R}^2 \ni (x,y) \rightarrow H_{a,b} (x,y):= (1 - a x^2 + y, b x) \in {\mathbb R}^2 $ of the plane to itself. This mapping exhibits a ...

Question 2

Consider $\Omega \subset \mathbb{S}^n$ a star shaped domain contained in an open hemisphere and with smooth boundary $\partial \Omega.$ Here star shaped means that there exists a point $p \in \Omega$ ...

Question 3

Let $X$ be a Hilbert space with an orthonormal basis $f_j \in \mathcal D(C)$ where $C:\mathcal D(C)\subset X \to X$ is a linear (unbounded) operator (possibly generating a $C_0$ semigroup; you may ...

Question 4

Let assume that $f: V \rightarrow U$ is a non-convex functional from function space $V$ to $U$. And let assume that linear operator $A \in \mathcal{L}(V;U)$ is the Fréchet derivative of $f$ on $v \in \...

Question 5

I have two systems of non-linear equations $F(\mathbf{x})=0$ and $G(\mathbf{x})=0$, where $\mathbf{x}=[a \quad b\quad c\quad d\quad e\quad f]'\in \mathbb{R}_+^6$. Both of them have a unique solution. ...

Question 6

Suppose we have 2 variables $x$ and $y$ and their relationship is: Suppose that we have another equation such as $y = dx/dt$ and we want to solve that nonlinear system of equations. The time ...

Question 7

I'm taking a course on Nonlinear Analysis, and one of the proposed exercises is to find the Fréchet derivative of the following operator: Let $\Omega$ be a bounded open set in $\mathbb{R}^N$, with ...

Question 8

Let $A=(a_{ik})\in\mathbb{R}^{N\times K}$. Consider the system of equations: $$x_{i}=\sum_{k=1}^{K}a_{ik}\tanh\left( \sum_{j}a_{jk}x_{j} \right)$$ for the vector $\mathbf{x}=(x_{i})\in\mathbb{R}^{N}$, ...

Question 9

After discussion about linearization of differential operators here, I found another problem. That is, the uniqueness of extension of nonlinear operator. Here, I consider especially nonlinear ...

Question 10

While reading Peter Topping's “Lectures on the Ricci Flow”, I came across the term “linearization” and was unsure of its precise meaning in this context. The book considers a nonlinear differential ...

Question 11

Consider the following system of two nonlinear equations: $h'_{x}( x) -C y -C_{1}=0$ $h'_{y}( y) -C x -C_{2}=0$ With $x,y\in[0,1]$ The $h$ functions both satisfy $h(0)=0$, $\lim _{t\rightarrow 1} h( t)...

Question 12

I was reading the proof of Lemma 8.30 of the book "Nonlinear Partial Differential Equations with Applications by Tomáš Roubíček" (https://link.springer.com/book/10.1007/978-3-0348-0513-1). ...

Question 13

Consider the following nonlinear PDE- $$-\Delta_{p} u = |u|^{q-2}u\,\,\, \text{in $\Omega$,}$$ where $\Delta_{p} u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $u\in W^{1,p}_{0}(\Omega)$, $\Omega \subset \...

Question 14

Let $\mathbb{C}^N$ be the set of all complex-valued vectors, $\mathbb{C}^{N \times N}$ the the set of all $N \times N$ complex-valued matrices, and, for all unit vectors $x \in \mathbb{C}^N$, let $$ \...

Question 15

Let $B$ be an orthonormal basis of $\mathbb{C}^N$, and, for all $\vec{v} \in \mathbb{C}^N$, let $U_B : \mathbb{C}^N \rightarrow \mathbb{C}^N$ be the (non-linear) operator $$ U_B\vec{v} = \sum_{\...

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Questions tagged [nonlinear-analysis]

Period-eight transition critical parameter for the Henon map.

Pohozaev non existence result in a starshaped domain of the sphere

Let X Hilbert, C linear, $F:X\to D(C)$ s.t. $CF\in C^0$, $P_n$ projection. When $CP_n F :X \to X$ is uniformly continuous on compact, UNIFORMLY in n?

Non-convex functional and it Fréchet derivative [closed]

Analytically compare the sum of unknowns between two systems of non-linear equations without solving it explicitely

Time constant for system described by a set of nonlinear equations

Well-definedness of an energy functional induced by a Nemytskii operator, under a growth condition on its partial derivative

Solutions of $x_{i}=\sum_{k=1}^{K}a_{ik}\tanh\left( \sum_{j}a_{jk}x_{j} \right)$

Uniqueness of extension of nonlinear operator to an appropriate Sobolev space

Exact definition of “Linearization” of nonlinear differential operators.

Proving Uniqueness of Nonlinear Equation Solution

Concerning a dense subset of Banach space valued test functions.

Relations among various weak solutions of $-\text{div}(|\nabla u|^{p-2}\nabla u) = |u|^{q-2}u$.

Proving that particular infimum exists for approximating a non-linear operator by linear operator

Approximating particular non-linear operator with linear operator

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