Questions tagged [bochner-spaces]

For question involving Bochner space, which are generalization $\mathbb L^p$ spaces in the sense that the values of the functions are themselves in function spaces.

66 questions with no upvoted or accepted answers
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8
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159 views

Is there a notion of a continuous basis of a Banach space?

If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$. And a Schauder basis of $...
6
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214 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $\frac{\...
5
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0answers
47 views

How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space $(X_t)_{...
4
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219 views

Positive part of functions from Sobolev space involving time

Assume that $u \in W^{1,2}(0,T; W_0^{1,2}(\Omega))$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $T <+\infty$. Let $A \subset \Omega \times (0, T)$ be a Lipschitz domain such that $u(\...
4
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133 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in L^...
4
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339 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in $L^...
4
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0answers
157 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times \...
3
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0answers
48 views

Integral of a lsc function is lsc?

Let $f:\mathbb{R}^d\times \mathbb{R}^D\rightarrow \mathbb{R}$ be a lower semicontinuous function. Then is it true that $$ \begin{aligned} L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)&\...
3
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1answer
57 views

On DiPerna-Lions compactness' arguments.

In DiPerna Lions, Ordinary differential equations, transport theory and Sobolev spaces (1989), the authors used topological arguments that remains obscure to me. Page 515 the authors used an ...
3
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70 views

Post-Composition By Diffeomorphism And Integrability

Let $d,D$ be positive integers, $p \in [1,\infty)$, $U$ an (non-empty) open subset of $\mathbb{R}^D$, and suppose that $f$ is: In the Bochner-Lebesgue space $L^p_{\mu}(\mathcal{B}(\mathbb{R}^d);\...
3
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78 views

Sequential Banach-Alaoglu theorem for a Bochner space

I have a question concerning the sequential weak-$*$ compactness in a Bochner space. Let $H$ be some non-separable Hilbert space. Consider the Bochner space $L^{\infty}(0,T;H)$. It is known that this ...
3
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79 views

Can someone help me obtain these a-priori estimates?

I am trying to obtain some estimates on a PDE where the unknown is: $u(t) : \mathbb{R}^n \rightarrow \mathbb{R}^n, t \in [0,T]$. The variational formulation is given below, where the test function ...
3
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159 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{q_2}(I;L^{p_2}(\Omega))\hookrightarrow L^...
3
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160 views

Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

Let $\Omega$ be a bounded domain, and consider the equation $$u_t - \Delta u = f$$ $$u(0) = u_0 \in L^1(\Omega)$$ with Neumann BCs (or Dirichlet if convenient) where $f$ is smooth. Using energy/...
3
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115 views

Is there a proof that $L^p([0,1],X)$ has the Radon Nikodym Property that uses the characterization via Lipschitz continuous functions?

I'm looking for an alternative proof of the following theorem: Let $1<p<\infty$ then $L^p([0,1],X)$ has the Radon Nikodym property iff $X$ does. The theorem is due to Sundaresan and an ...
2
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1answer
40 views

Density of simple functions in Bochner space; approximation from below

Let $f \in L^2(0,T;V)$ where $V=H^1_0(\Omega)$ or $V=L^2(\Omega)$. Suppose that $|f| \leq M$ for some constant $M$ almost everywhere. Is it possible to find a sequence of simple functions $f_n(t) = ...
2
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68 views

Bochner integral of a function in $L^2$

This might be obvious, but I don't see why... Let $\lambda \in \mathbb{C}\backslash\mathbb{R}_+$ and $m(x) = \lambda/(\lambda - e^x)$. If $f \in L^2$, then $$ mf = \int^\infty_{-\infty} \frac{\...
2
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0answers
276 views

Product rule and integration by parts in Bochner Sobolev space $W^{1,1}(U)$

Assume $v(t)\in H_0^1(U)$ for all $0\leq t\leq T$. The weak derivative $v'$ of $v$ is defined by \begin{equation} \int_0^T \phi'(t)v(t)\,\mathrm{d}t=\int_0^T\phi(t)v'(t)\,\mathrm{d}t\quad\text{for all ...
2
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0answers
306 views

Dual of an operator

A linear operator $A$ is defined from $L^2(0,T)$ to $C([0,T);\mathbb{H})$ where $\mathbb{H}$ is a Hilbert space. At some point, dual of this operator is needed; a detour to calculate the dual, and ...
2
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0answers
114 views

$L^2$ space with values in vector bundle.

Let $E$ be a vector bundle over $S^2$ with inner product, $\pi$ its bundle projection and each fiber has dimension $m$. Let $\Omega \subset \mathbb{R^n}$. Consider space of all square integrable ...
2
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0answers
58 views

Identification of a weak limit using pointwise a.e. convergence

Let $X$ be a bounded domain. We have a sequence $b_n(t) \to b(t)$ pointwise a.e. (no dependence on space), and functions $f_n \to f$ in $L^2(0,T;L^2(X))$ such that $b_n\nabla f_n \rightharpoonup g$ ...
2
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0answers
46 views

Measurability of $u:[0,T] \to X$ where $u$ is in Bochner space

Let $f \in L^p(0,T;X)$ where $X$ is a separable Banach space. So $f$ is a Bochner function and hence Bochner measurable, meaning that there is a sequence of measurable countably-valued functions that ...
2
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0answers
107 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
2
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0answers
175 views

Want a dense subset of a Sobolev-Bochner space!!

Let $U \subset \mathbb{R}^n$ be a bounded domain. Let $$W=\{ u \in L^2(0,T;H^1(U))\cap L^\infty(0,T;L^\infty(U)) : u' \in L^2(0,T;H^{-1}(U))\}$$ and let $$D=\{u \in L^2(0,T;H^1(U)) \cap L^\infty(0,T;...
2
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0answers
83 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
2
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0answers
109 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
2
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0answers
45 views

$\langle f, u \rangle = 0$ for all $u \in C_c^\infty(0,T;H)$ implies $f=0$? Please check proof

Let $f \in L^2(0,T;H)$ where $H$ is a Hilbert space. Suppose that $$\langle f, u \rangle = 0\quad\text{for all $u \in C_c^\infty(0,T;H)$}$$ where the dual pairing is the one between $L^2(0,T;H)$ and ...
1
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0answers
28 views

Compactness in Bochner-spaces

Consider a sequence of functions $f_n: [0,T]\times [0,T] \times \Omega \rightarrow \mathbb{R}^3 $, $\Omega \subset \mathbb{R}^3$, such that \begin{align} t &\rightarrow f_n(t,s,x) \ \ \ \text{is ...
1
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0answers
46 views

Differentiability of inner product involving Bochner functions

Suppose we are given $f\in L^2(0, T; H^1)$ and that there exists $\xi\in L^2(0, T; H^1)$ such that, for any $g\in L^2(0,T; H^1)$ with $\dot{g}\in L^2(0, T; H^1)$, it holds that $$t\mapsto (\nabla f(t)...
1
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1answer
52 views

Extension of weak derivatives in Bochner spaces

I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans: He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In ...
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0answers
37 views

Question on Bochner spaces: Convergence of the weak time derivative

Let $U$ be a $C^2-$compact manifold and consider two non negative sequences $f_n,g_n$ such that $f_n \overset * \to f$ weakly in $L^{\infty}(0,T;\mathcal M_{*}(U))$ $g_n \overset * \to g$ ...
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0answers
23 views

evolution equation satisfied by projection

Consider the linear parabolic problem (as in Evans) $$\begin{cases} \partial_t u+Lu=f\ \: \mbox{in} \:\Omega \times (0,T]\\ u=0\ \: \mbox{on} \:\partial \Omega \times (0,T]\\ u(\cdot,0)=g \ \: \mbox{...
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0answers
107 views

Role of Pettis measurability theorem in the construction of the Bochner integral

I am reading a brief introduction to Bochner integral, and before introducing it, it is proved the Pettis measurability theorem, which says that a function $f:A \to B$ (with $A$ a measure space with $\...
1
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1answer
172 views

Absolutely continuous and differentiable almost everywhere

I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it: "A strongly absolutely continuous function which is differentiable almost everywhere is ...
1
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1answer
81 views

Boundedness of an operator on a Bochner spaces

Given two Banach spaces $(X,\Vert \cdot\Vert_X)$ and $(Y,\Vert \cdot\Vert_Y)$ such that $X\subset Y$ with continuous embedding (i.e. there exits a constant $c>0$ such that $\Vert x\Vert_Y \leq c\...
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0answers
43 views

Show that the integral of a two-parameter function is Hölder continuous

Let $(\Omega,\mathcal A,\mu)$ be a measure space $(M,d)$ be a metric space $\Lambda\subseteq M$ $E$ be a $\mathbb R$-Banach space $f:\Omega\times\Lambda\to E$ with $$f(\;\cdot\;,x)\in\mathcal L^1(\mu;...
1
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0answers
56 views

Density of $C_c^\infty((0,T)\times R^d)$ in $L^2(0,T;H^1)$

I have been dealing with the following problem I can't find an answer to: Let $H^1$ be the usual Sobolev space and $L^2(0,T;H^1)$ the Bochner space of square integrable functions $[0,T]\to H^1$. I ...
1
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0answers
53 views

If a process $Y=\{Y_{t}:t\in[0,T]\}$ is progressive, then $\{\int_{0}^{t}Y(s)\text{ d}s\}$ is *surely* continuous

I am reading a textbook and I am stuck proving a seemingly trivial point (but I believe I am not the only one that was pondering this question and it shows similarity to this question). Let me sketch ...
1
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0answers
113 views

Weak convergence everywhere implied by weak convergence in Sobolev-Bochner-Space

I want to understand an implication made by M. Sofonea and A. Matei in their book "Variational Inequalities with Applications". The sequence $\lbrace u_n \rbrace_{n\in \mathbb{N}}$ be bounded in $L^\...
1
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0answers
28 views

Bound of mollified in $H^{-2}$

Let $f\in L^2((0,T); H^2) $ with $ \partial_t f \in L^{2}((0,T);H^{-2}) $ and let $ \eta_{\varepsilon} $ a standard mollifier sequence in $ (t,x) $, then there exists a constant $ C $ independent of $ ...
1
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0answers
111 views

Conditions for weak differentiability of composition of $C^1$ real function with weakly time-differentiable $H^1$-valued function

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and $\mathbb{R} \ni T > 0$. I will abbreviate $X=H^1(\Omega)$ and write $X'$ for its topological dual. Given $$u\in L^2\left(0,T;X \right)$$ ...
1
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0answers
30 views

approximation of weakly differentriable bochner functions

Given a function $u\in L^2(0,T;H^1(\Omega))$ with $u_t\in L^2(0,T;(H^1(\Omega))^*)$. Can we approximate $u$ by functions $u^k$ with $$u^k=\sum\limits_{i=1}^{n(k)}c_i^k\phi_i^k,\text{ where } c_i^k\in ...
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0answers
101 views

Stochastic integral and weak integral

Can the stochastic (Skorokhod) integral be seen as a special case of the weak of Pettis integral with the Banach space which win integrate into chosen appropriately?
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0answers
196 views

Ricci curvature and Bochner identity

Let $M ^ n$ a no boundery compact Riemannian manifold and $f \in C^\inf (M)$ a non-constant solution of the equation $\Delta f + \lambda f = 0$ a)Show the Green identity (as a result of the ...
1
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0answers
126 views

Reference request: $L^\infty(0,T;L^\infty(\Omega))$ is compactly embedded in $L^2(0,T;L^2(\Omega))$

On a bounded domain $\Omega$, I am looking for a reference saying that $L^\infty(0,T;L^\infty(\Omega))$ is compactly embedded in $L^2(0,T;L^2(\Omega))$. I tried all the usual texts (Showalter, Evans, ...
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0answers
68 views

Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?

Let $M$ be a compact Riemannian manifold that is closed. Define $$W(0,T, H^1, L^2) = \{ u \in L^2(0,T;H^1(M)) \mid u_t \in L^2(0,T;L^2(M))\}$$ $$W(0,T, H^1, H^{-1}) = \{ u \in L^2(0,T;H^1(M)) \mid u_t ...
1
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0answers
22 views

Measurability of $t \mapsto \chi_{\{x \mid u(t)(x) \geq C\}}(x)$

Let $\Omega$ be a bounded domain and let $u = L^2(0,T;L^2(\Omega))$. Is the map $$t \mapsto \chi_{\{x \mid u(t)(x) \geq C\}}(x)$$ measurable for a constant $C$? Here $\chi$ is the indicator function. ...
1
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0answers
123 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that $$\int_0^...
1
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0answers
146 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists $...
1
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0answers
64 views

Property of intersections of Bochner spaces

My question: Assume I have a function $ u \in H^2(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega))$. Now I want to bound the gradient of $u$. Can I deduce that $u \in H^1(0,T;H^1(\Omega))$ and under which ...