All Questions
Tagged with nonlinear-analysis functional-analysis
132 questions
0
votes
0
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68
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One-dimensional monotone operator is cyclically monotone
I try to understand the proof in the book of Brezis: "Opérateurs Maximaux Monotones" in which it is claimed that a multi-valued monotone operator $T:D(T)\subset\mathbb{R}\to\mathbb{R}$ is ...
- 1,483
0
votes
0
answers
29
views
Reference for the Sobolev $L^2$ trace lemma for an arbitrary line $C\subset \mathbb R^d$?
I am trying to find a reference for the following Lemma found in a paper:
Let $g\in H^s(\mathbb R^d)$ with $s>\frac{d-1}{2}$ and $C\subset\mathbb R^d$ be an arbitrary straight line. Then, there ...
- 329
0
votes
1
answer
86
views
Show that the $T$ is a compact operator
Let $\Omega\subset\mathbb R^2$ be bounded. Consider the partial differential equation $(P)$
\begin{align}
-\Delta u+u^5&=h \ \ \text{on }\Omega,
\newline u&-0\ \ \text{on }\partial\Omega.
\end{...
- 651
2
votes
0
answers
62
views
Holomorphic implicit function theorem
Is there a version of the implicit function theorem for holomorphic functions between complex Banach spaces? If yes, do you know any reference?
- 153
5
votes
0
answers
100
views
Characterize Bifurcation in Nonlinear ODE
Consider the mapping
$F: H^1_0(\mathbb R_+) \times \mathbb R \to H^1_0(\mathbb R_+)$
for an ODE with solutions $(u_E,E)$ that satisfy
$$ F(u_E,E) = 0.$$
Suppose there is a solution iff $E > E_0 >...
- 303
0
votes
0
answers
40
views
Exercise 7.6 of Robinson, Rodrigo, Sadowski: Smoothness of Navier-Stokes on Bounded Domains
My question is about Exercise 7.6 of the excellent book 'The Three-Dimensional Navier-Stokes Equations' by Robinson, Rodrigo and Sadowski. More generally, it is about higher regularity in space of ...
- 3
2
votes
1
answer
60
views
Higher order Frechet derivatives viewed as bilinear maps, on Taylors theorem
So I have been studying some introductory non-linear analysis. I am currently looking at higher order Frechet derivatives and I want to proof-check/ make sure I got something right. So given $X,Y$ ...
- 394
0
votes
0
answers
42
views
The definition of a continuous semigroup
Here is the definition of a continuous semigroup
Let $C$ be a subset of a Banach space $X$. A semigroup on $C$ is a group $\{S(t):t\geq 0\}$ of a self-maps defined on the subset $C$ which satisfies ...
- 3
1
vote
0
answers
248
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, I heard that $\operatorname{Id}-K$ is a proper map. Proper map ...
- 11
1
vote
1
answer
107
views
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 ...
- 51
0
votes
1
answer
59
views
Hausdorff separation for the definition of Mackey topology
I am reading the definition of the Mackey topology relative to a dual system $(X,Y)$ and the author (Edwards-Functional Analysis) imposes the condition that the dual system must be separated in $Y$. ...
- 1,483
4
votes
1
answer
847
views
(Frechet) Differentiability of Implicit function in Banach spaces
I'm looking at the classical implicit function theorem in Banach spaces. So $X,Y,Z$ are Banach spaces and $F: U_{x_0}\times V_{y_0} \to Z$ continuous and continuously differentiable with respect to y. ...
- 340
2
votes
0
answers
99
views
Extending Atiyah-Singer for selfadjoint Fredholm operators
In the paper "Index theory for skew-adjoint Fredholm operators" by Atiyah and Singer, the corollary at page 3 states that the space $\mathcal{F}_{s}(H)$ has two contractible components, ...
0
votes
1
answer
58
views
Let $a$ be continuous such that $|a(x)| \le C |x|^{p/q}$. Then $A:L^p(\Omega) \to L^q(\Omega), u \mapsto a \circ u$ is continuous
Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Let $p, q \in [1, \infty)$. Let $a:\mathbb R \to \mathbb R$ be continuous such that
$$
|a(x)| \le C |x|^{p/q} \quad \forall x \in \...
- 17.9k
1
vote
0
answers
162
views
Power Series in a Banach Space
I'm looking at differentiability, analyticity and power series functions between Banach spaces, I'm using Soo Bong Chae's book "Holomorphy and Calculus in Normed Spaces".
The book has the ...
- 2,867
1
vote
1
answer
76
views
If $T_n\to T$ uniformly and $x_n\to x$ weakly, then $T_nx_n\to Tx$ weakly
We say a sequence $(T_k)$ with $T_k\colon H\to H$ is uniformly convergent to $T\colon H\to H$ if $\|T_k-T\|\to 0$. I am interested in the following claim:
For any weakly convergent sequence $(x_k)$ ...
- 485
0
votes
0
answers
134
views
Motivation of the definition of compact operators for the non-linear case
We know that we can study a compact operator between normed spaces $T: E \to F$ in two ways:
When $T$ is linear, it is only natural to ask that $\overline{T(A)}$ be compact in $F$ for all $A$ bounded ...
- 110
5
votes
2
answers
339
views
Second Derivative Test in Banach Spaces
According to $[$Exercise $12.8$, $1]$ we have the following version of the Second Derivative Test:
Theorem. Let $E=(E,\|\cdot \|)$ be a Banach space, let $D$ be a subset of $E$ and suppose $f: D \...
- 1,707
1
vote
0
answers
67
views
In banach space every closed convex bounded set is a retract of a closed ball
In a Banach space (not necessarily finite-dimensional), we can find a closed ball with a sufficiently large radius that contains $A$ where $A$ is a closed, convex, and bounded set. My question is does ...
- 4,593
1
vote
1
answer
185
views
Understanding the proof that monotone operators are locally bounded
I am trying to understand the proof that monotone operators are locally bounded:
The question I have is why is the inequality highlighted yellow true? The problem I have is that the inequality of 5.2 ...
- 4,593
0
votes
0
answers
54
views
How to prove weak sequential continuity of a continuous, bounded nonlinear mapping?
Let $f \in H^1_0(\Omega)$ and $J$ a bounded, continuous and nonlinear mapping $J:H^1_0(\Omega) \to \mathbb{R}$ such that $|J(f)| \leq C\lVert f \rVert$ for some constant $C$ and $|J(f_n)-J(f)| \leq L \...
- 66
1
vote
0
answers
84
views
Lipschitz continuity and differentiability implies bounded partial derivatives in $\mathbb{R}^{n^{2}}$
Consider the nonlinear differential operator $a(D^{2}u)$, where x varies in a $C^{2}$-smooth and bounded domain $\Omega \subset \mathbb{R}^{n},u:\Omega \rightarrow \mathbb{R}$ and $D^{2}u$ stands for ...
1
vote
1
answer
89
views
Do we need continuously differentiable here to apply the inverse function theorem for mapping between Banach Spaces?
I feel like the problem here is stated incorrectly:
I think we need $F$ to not just be (Frechet) differentiable but also continuously differentiable, as in the map that maps each $x \in O \to F'(x)$ ...
- 4,593
0
votes
1
answer
108
views
Leray-Schauder Degree for Periodic Functions
Let $X$ be a Banach space, $Z \subset X$ a closed, linear subspace and
$$
S : U \cap Z \to X
$$
where $U \subseteq X$ is open.
Question: Can I define the Leray-Schauder degree of $S$? If I had $S : U \...
- 335
1
vote
0
answers
56
views
szuflas theorem for ordinary differential equations in Banachspaces
For my Bachelor thesis i would like to proof the following statment. Let $E$ be a Banach space and $Q\subset E$. $\alpha$ is the kuratowski measure of noncompactness.
Let $f:[0,T]\times Q\rightarrow E$...
- 55
0
votes
1
answer
177
views
Derivative of Riesz Transform
I am trying to find a bound for the $L^p$ norm, $1<p\leq \infty$ of the Riesz transform of $f^2$, where $f \in S$.
$$\mathcal{F}[{\mathcal{R}_x f}](\xi,\eta) = -i \frac{\xi}{|(\xi,\eta)|}\hat{f}(\...
- 1,682
4
votes
2
answers
323
views
Is a continuous function on compact convex set where the boundary is mapped to the set a self mapping?
Let $K ⊂ R^n$ be convex and compact with $0$ in the interior of $K$. Let $f ∈ C(K, R^n)$ with $f(∂K) ⊂ K$.
If this is the case, do we in fact have $f(K) \subset K$. It is probably not the case that ...
- 4,593
0
votes
1
answer
47
views
Showing that a maximum exist for a semi lower sequentially continuous mapping from Hilbert space to R
Let H be a Hilbert space over $R$ , $r > 0$ and $F ∈ C^1(H, R)$ such that:
1)−F is weakly sequentially lower semicontinuous
2) $DF(u) = 0$ implies $u = 0$ (this is the Frechet derivative)
3) $F(0) =...
- 201
1
vote
1
answer
59
views
Can $f_1(x, y)-f_2(u, v)$ be written as $g(x-u, y-v)$?
when I do some calculation on the basic theory of diffractive neural networks,the question behind blocks my way.It's a pure math that i want to know, $f_1 (x, y)$ and $f_2(u, v)$ are both nonlinear ...
- 23
2
votes
1
answer
77
views
Show u^2|u|^(p-3) is Holder continuous (derivative of power nonlinearity)
I am trying to show that the function $f:\mathbb{C}\rightarrow \mathbb{C}$ given by $f(u)=u^2|u|^{p-3}$ is (uniformly) Holder continuous for $p\in(1,2)$ (so $p-3\in(-2,-1)$, this ensures that $f$ is ...
- 73
1
vote
1
answer
58
views
Potential Extension of variable metric Quasi-Fejér monotone: interesting convergence analysis tool with iteration-dependent norms?
In [1], there are many theorems and propositions on (quasi) Fejér monotonicity.
Let us focus on finite-dimensional spaces, for instance Euclidean space.
Theorem 3.3 and Proposition 4.1-4.3 in [1] (...
- 2,743
1
vote
0
answers
37
views
Regularity theorem of minimizers
During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R ...
- 1,682
1
vote
1
answer
84
views
Linearization of The Ginzburg-Landau (GL) equation
I am studying the LG equation,
$$\partial_t u = (1 + i \alpha) \partial_x^2 u + u - (1+ i \beta)|u|^2 u, \quad \alpha, \beta, x \in \mathbb R, \text{ and }\,\, u(x,t) \in \mathbb C. \tag{1} $$
To find ...
- 1,682
1
vote
1
answer
87
views
Help proving that the derivative $Df(x)$ is continuous
In what follows, $X, Y$ are Banach and $U \subseteq X$ is open. Consider the map $f: U \to Y$. We say that $f$ is locally uniformly differentiable at $x \in U$ and $h \in X$ if given $\epsilon > 0$ ...
- 1,009
1
vote
0
answers
65
views
Find the solution that satisfies the conditions of the Schrodinger equation
The Schrodinger operator with delta potential $H$ is defined as follows:
$$\begin{cases}
D(H) & = \{ u \in H^1 (\mathbb{R}) \cap H^2 (\mathbb{R} \setminus \{0\}); u'(0 +) - u'(0 -) = 2 q u (0)\},\\...
- 1,682
1
vote
1
answer
263
views
Proof that, if the norm of any normed space is (Fréchet) differentiable, then the derivative is continuous.
I want to prove the following lemma.
Let $X$ be a normed space and its norm $\|.\|$ be Fréchet differentiable on $X \backslash \{0\}$. Then the derivative is continuous there.
I've got this from a ...
- 374
2
votes
1
answer
495
views
In which points is the supremum norm on $C[0,1]$ and $c_0$, respectively, Gâteaux/Fréchet differentiable?
$f : U \to Y$ where $U\subset X$ is open and $X, Y$ are normed spaces is called Gâteaux differetiable at $u\in U$ if there exists a bounded linear operator $T$ from $U$ to $Y$ such that for $h\to 0$ ...
- 374
2
votes
1
answer
181
views
Example for an operator that is strictly monotone but not maximally monotone (or the other way)
While the definition of strictly monotone = nowhere constant operators seems intuitive, I find it hard to picture in which way maximal monotone operators ($\forall (u,u') \in X \times X', \langle u'-v'...
- 21
3
votes
1
answer
534
views
Is the sine operator on $L^2[0,1]$ Fréchet differentiable or not and why?
This problem has given me some trouble. Let $F$ be the operator on $L^2[0,1]$ defined by $F(g)(t)=\sin g(t)$. I'm trying to determine whether or not $F$ is (Fréchet) differentiable in that space. I ...
- 374
0
votes
1
answer
69
views
Intuition behind regularization of non-coercive variational inequalities
I am looking into the theory of non-coercive variational inequalities and I came accross the following approximation techinique:
Fix an infinite dimensional Hilbert space $H$ and let $K \subset V$ be ...
- 2,566
0
votes
1
answer
53
views
Prove that the element $(0,1)$ has infinite many best approximations in the linear subspace $B=\{ (x,x)| x\in\mathbb{R} \}$
I have tried this problem as:
By using the definition
For a subspace $S$ of a normed linear space $X$, for all $x$ belongs to $X,g\in S$ is a best approximation of $x$ if $\|x-g\|=\inf\{\|x-g'\|:g'\...
2
votes
1
answer
205
views
On continuity of the Gateaux derivative of p-Laplacian operator
Let $\Omega\subset \mathbb{R}^n, N\geq3$, be an open set. For $p\in(1,+\infty)$, define a functional $J:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ by
$J(u)=\int_\Omega |\nabla u|^p\,dx.$
Then $J$ is ...
- 525
1
vote
0
answers
36
views
On a compact imbedding problem
Denote $$\mathcal{H}=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3} V(x)|u(x)|^2\,dx<\infty\},$$ where $V(x)\in L_{loc}^\infty (\mathbb{R}^3)$, $V(x)\geq0$ and $\lim_{|x|\rightarrow \infty} V(x)=\...
- 525
2
votes
0
answers
139
views
What is the so-called "bootstrap" argument in Mathematics and its application to nonlinear Schrodinger system.
We have the following nonlinear Schrodinger equations ($n\leq3$):
$$\begin{cases}
\Delta u_1 -u_1+\mu_1u_1^3+\beta u_1u_2^2=0\\
\Delta u_2 -\lambda u_2+\mu_2u_2^3+\beta u_1^2u_2=0\\
u_1,\,u_2\in H^1(\...
- 525
1
vote
0
answers
62
views
$I-f$ is a nonlinear homeomorphism on an infinite dimensional Banach space. $P$ is a linear projection. Is $I-Pf$ a homeomorphism?
$f:X\to X$ is a nonlinear operator where $X$ is an infinite dimensional Banach space, $I-f$ is a homeomorphism, and $P:X\to X$ is a linear projection. Additionally, we may assume that $I-Pf$ is ...
- 131
1
vote
1
answer
53
views
Weak continuity of diagonal embeddings of Hilbert spaces
Let $\mathcal{H}$ be an infinite-dimensional separable Hilbert space and let $I: \mathcal{H} \rightarrow \mathcal{H}\otimes \mathcal{H} $ be the (non-linear) map given by the diagonal embedding $h \...
- 590
4
votes
0
answers
127
views
Transforming a nonlinear homeomorphism by linear contraction on one term; still a homeomorphism?
I have a continuous nonlinear operator $f:X\rightarrow X$ with $X$ an infinite dimensional Banach space.
$\lambda I - f$ is a homeomorphism for all $|\lambda|\geq 1$. I also have a linear operator $B$...
- 131
0
votes
1
answer
32
views
Submultiplicativity of Lipschitz seminorms
I have a textbook (Nonlinear Spectral Theory, Appell et al 2004) that gives the following definitions and properties.
Let $X$ and $Y$ be Banach spaces, and $f:X\rightarrow Y$ a continuous (possibly ...
- 131
1
vote
0
answers
174
views
Problem books for nonlinear functional analysis?
Recently I have started reading variational methods application to nonlinear PDE and hamiltonian systems written by Michael Struwe. The book doesn't contains any problems so I want any supplementary ...
- 618
1
vote
1
answer
113
views
Lower hemicontinuity of intersection of relations
This question is about the proof of a Lemma 7.1 of the paper
E. Michael, Continuous selections I.,Annals of Mathematics
Second Series, Vol. 63, No. 2 (Mar., 1956), pp. 361-382 (22 pages)
Published By: ...
- 41.8k