For question about weak derivatives, a notion which extends the classical notion of derivative and allows us to consider derivatives of distributions rather than functions.

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1answer
20 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
1
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1answer
36 views

A weak derivative of a function of two variable which depends only on one variable

Assume that $f: \mathbb R^2 \rightarrow \mathbb R$ is locally integrable and has a locally integrable weak patrial derivative $\partial_1 f.$ Let moreover $f$ depends only on the first variable: ...
3
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1answer
66 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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1answer
19 views

Find a weak derivative of this function

Let $B(0,1)$ be the open unit ball in Euclidean space $\mathbb R^2$ and $(a_n)_{n=1}^\infty$ be a dense subset of $B(0,1)$. I wish to show for fixed $s\in (0,1)$ the function $$ ...
0
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1answer
66 views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in ...
1
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1answer
25 views

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ make sense?

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ exist, i.e., is $u^+$ weakly differentiable in time? By $u^+$ I mean the positive ...
2
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1answer
41 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
1
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2answers
64 views

Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
0
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1answer
50 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
0
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1answer
24 views

Heaviside function has no weak derivative on $(-1, 1)$

I want to proof that the Heaviside function $$H(x) = \begin{cases} 1 & \text{if }x >0\\ 0 &\text{if }x\leq 0\end{cases}$$ has no weak derivative on $(-1,1)$. If I assume it has a weak ...
0
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1answer
31 views

If $u \in L^2(0,T;L^2)$ and $u' \in L^2(0,T;H^{-1})$, is $u \in C^0([0,T];V)$ for some space $V$?

If $u \in L^2(0,T;L^2)$ has weak derivative $u' \in L^2(0,T;H^{-1})$, is $u \in C^0([0,T];V)$ for some Banach space $V$? For what $V$ Lebesgue spaces does this hold? I cannot find any results.
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0answers
17 views

If $v_i \in L^2(0,T;H^1)$ with $v_i' \in L^2(0,T;H^{-1})$, what can we say about $w(t) = \int_s^t (v_1-v_2)(\tau)\;d\tau$?

If $v_i \in L^2(0,T;H^1)$ with $v_i' \in L^2(0,T;H^{-1})$, what can we say about $w(t) = \int_s^t (v_1-v_2)(\tau)\;d\tau$? Is $w$ also of the same regularity? Heuristically, I thought: let $V_i$ be ...
4
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1answer
51 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
0
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1answer
21 views

Does weakly differentiable and $L^{\infty}$ imply continuity

Suppose $\Omega \subset \mathbb{R}^d$ is open, connected and bounded. Is $$W^{1,1}(\Omega)\cap L^{\infty}(\Omega) \subset C(\bar{\Omega})?$$ Here $W^{1,1}$ denotes the space of all weakly ...
2
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3answers
85 views

Is a function in $L^2$ which second derivative is in $L^2$ in $H^2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary. Assume that $f\in L^2(\Omega)$ and $f^{\prime\prime}\in L^2(\Omega)$. Does one have $f\in H^2(\Omega)$? Useless comments: ...
3
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0answers
110 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
0
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1answer
29 views

Example of weak derivative on multivariable function

In order to explain about the concept of weak derivatives, I plan to give examples on them. I already manage one example for the single-variable case, but I think it would be better if I can provide ...
2
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1answer
52 views

Chain rule for weak derivatives of $f(u)$ where $f'$ is not bounded but $u$ is?

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$. Suppose $u$ has a weak derivative $u_x$. I want the chain rule $$\partial_x (f(u)) = f'(u)u_x$$ to hold. We know this holds if $f'$ is bounded. But I don't ...
3
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1answer
42 views

Finite Element Method Weak Formulation

I have a question about the weak formulation of a PDE in finite element analysis. Suppose we have the following two-dimensional PDE: $$ \Delta \cdot u(x,y) = q(x,y) $$ where $q$ is given, $u$ is ...
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0answers
15 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prevot/Rockner [PR07]: $ \int_0^T { \langle \Psi dW(t), \Phi(t)\rangle }$ A few useful ...
1
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1answer
56 views

Notion for weak derivatives of $L^p(0,T,X)$-functions

A definition in Evan's PDE-book from chapter 5.9.2 says (let $X$ be a Banach space): Let $u\in L^1(0,T,X)$. We say $v\in L^1(0,T,X)$ is the weak derivative of $u$ provided $$\int_0^T \phi'(t)u(t)dt = ...
2
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2answers
58 views

Density in $H^1_0$

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$. For $h \neq 0$ we define $D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$ such as $\forall \varphi \in ...
1
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1answer
79 views

Sobolev space-exercice [closed]

Let $\Omega = \mathbb{R}^2_+$. My question is: how we prove that if $v \in H^2(\Omega)$ such as $v(x,0)=0$, then $\dfrac{\partial v}{\partial x} \in H^1_0(\Omega)$ ?
3
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1answer
64 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
2
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0answers
24 views

How to interpret $\langle (f(u))', v \rangle = \langle f'(u)u', v \rangle$?

We know that if $u$ has a weak derivative $u'$, and if $f \in C^1$ with $f'$ bounded then $(f(u))'= f'(u)u'$. But how interpret the duality pairing $$\langle (f(u))', v \rangle = \langle f'(u)u', v ...
1
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0answers
19 views

Showing $\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$

Let $u \in L^2(0,T;H^1)$ have weak derivative $u' \in L^2(0,T;H^{-1}).$ Let $c$ be a constant. I want to show that $$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$ where $(f)^+ = ...
0
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0answers
17 views

PDE weak solution , how could i prove it?

let be $ n \ge 3 $ and U a bounded set including the point $0 $ then i must show that a) the vector $ u= \frac{x}{|x|} $ belongs to $ H^{1}(U, R^{n})$ and it is an harmonic map into the sphere $ ...
0
votes
1answer
100 views

Establishing the n-th order weak derivative.

Consider $$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$ where $AC$ stands for the space of absolutely continuous functions. Now, let $f,g\in{L_{loc}^{1}(a,b)}$ and ...
2
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1answer
90 views

Approximation of a function with a polynomial of degree n.

Let $f\in{L_{loc}^{1}}$ such that $\int_{a}^{b}{f(x)}{\phi}^{(n)}(x)dx=0$ for all $\phi\in{C_{0}^{\infty}}(a,b)$. Then how do we show there exists a polynomial $P(x)$ of degree less or equal to $n-1$ ...
2
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1answer
58 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
3
votes
1answer
101 views

Weak derivative: Showing a function is equal to zero a.e.

I am very beginner in the theory of weak derivative. I am trying to fix the following problem: Suppose that $f\in{L_{loc}^{1}}$ and $\int_{a}^{b}f(x)\phi({x})dx=0$ for all $\phi\in{C_{0}^{\infty}}$ ...
1
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0answers
60 views

some sobolev norm estimation

I would like to show this inequality. I need help to show this inequality Let $F(\Phi)=\left|\Phi\right|^{\alpha}\Phi$ with even integer $\alpha>0$. Let $k$ be a positive integer satisfying ...
0
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1answer
39 views

Does this function has weak derivatives?

Let $\Omega\subset\mathbb{R}^n$ be a bounded "smooth" domain and $\Omega_\delta=\{x\in \Omega:\ d(x,\partial\Omega)<\delta\}$. Let $U_\delta$ denote the part of the boundary of $\Omega_\delta$ ...
0
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1answer
48 views

Basic Query On Sobolev Space

Okay, I am very new to the premise of Sobolev Spaces and there is one exercise here that's really grinding my gears. The premise is this- to what Sobolev spaces for each real number $\alpha$ does ...
7
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1answer
247 views

Justification of formal derivative

there I'm having the following problem and so far I didn't find anything in the literature on this. $\Phi\in C^1(\overline{\Omega}\times[0,1])$ with $\Phi'(x,0)=\Phi'(x,1)=\Phi(x,0)=0$ for every ...
1
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2answers
79 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
3
votes
1answer
135 views

If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
1
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1answer
68 views

weak differentiability of log log function

I want to understand why the following function has a weak derivative in two or three dimensions: $w(x) = \ln |\ln|x|| , x \in B_{1/2}(0)$. Can I say that if I have a strong derivative (except for ...
2
votes
1answer
79 views

For which real values of $\alpha$ PDE $\Delta u(x,y)+2u(x,y)=x-\alpha$ has at least one weak solution?

Problem. Consider boundary value problem: \begin{cases} \Delta u(x,y)+2u(x,y)=x-\alpha, & \text{in $\Omega$,} \\ u(x,y)=0, & \text{on $\partial\Omega$,} \\ \end{cases} where $\alpha$ is ...
0
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1answer
49 views

Uniqueness of the solution of the continuity equation with discontinuous vector field

I'm trying to prove the uniqueness of the following continuity equation with discontinuous vector field: $ \begin{cases} m_t(t,x) + [(-\alpha x + \sigma u^{*}(x) + c) \ m(t,x)]_x = 0 \qquad (t,x) ...
1
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0answers
40 views

Estimate difference of solutions of an equation with a bilinear symmetric continuous form

My question refers to differential equations in Sobolev spaces. It is as follows: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. Let $a: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow ...
1
vote
1answer
77 views

First order weak derivatives of $f(x)=|x|^r$

Let $f(x)=|x|^r$ for a given real number $r$. Show that $f$ has first order weak derivatives on the unit ball $B_1(0)\subset \mathbb{R}^n$ provided that $r > 1-n$. Does anyone have an idea on how ...
1
vote
1answer
47 views

Strong differentiability in Sobolev spaces

My question is: how can one prove that if $\phi\in H^{n}(a,b)$ for all positive integer $n$, then $\phi\in C^{\infty}(a,b)$? $H^{n}(a,b)$ denotes de Sobolev space of the real interval $(a,b)$. For my ...
1
vote
1answer
69 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
vote
0answers
22 views

norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$ \|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ? $$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
3
votes
1answer
211 views

Weak Derivatives and Lp spaces

I just want to know how to prove two properties of weak derivatives and $L^{p}$ spaces if they are true, the first one just involves weak derivatives: If we have a locally summable function $u: U ...
1
vote
1answer
111 views

Finding the weak derivatives

Finding the weak derivatives: a/ $f(x)=\left|x_1 \right|$, for all $x=(x_1,\ldots, x_n)$; b/ $f(x)=\operatorname{sign} (x_1)$, where $\Omega=\{\left|x \right|<1\}$. ...
1
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0answers
53 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
2
votes
1answer
144 views

Has a probability density function a weak derivative

Assume I have a probability density function $\rho$ on $R$. (e.g $\rho \geq 0$ $\int \rho dx =1 $ $\rho \in L^1(R)$ ...). So $\rho$ is the density wrt the lebesgue measure. Now I try to understand if ...
0
votes
0answers
50 views

Chainrule: “piecewise smooth” and Sobolev functions

since we had a lengthy discussion on my last question (Generalized chainrule for Sobolev functions with a cut-off I didn't find an answer to it yet, I'm going to post a related more specific question ...