For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

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Stream Function in Sobolev Space

Assume $\Omega$ is a multi-connected bounded domain in $\mathbb{R}^2$, and function $v\in W^{1,2}_{0}(\Omega)$ satisfies: $$\text{div} \; v=0$$ Then there exists a stream function $\psi\in ...
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17 views

Diagonalizing operators with the aid of wave packet transformation?

In a class on wave packet transformation, our teacher gave the definition of wave packet transformation $W^h\colon L^2(\mathbb R^n)\to L^2(\mathbb R^{2n})$ for $h>0$ as: $$W^hf(p,q)=\left(\frac ...
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24 views

Surface gradient definition

Let $\Omega$ be a bounded domain with $C^2$ connected boundary $\partial\Omega$. For a function $p\in H^1(\partial\Omega)$, we define the surface gradient $\nabla_{\partial\Omega}$ as $$ ...
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1answer
42 views

Use of regularity of the PDE solution to argue the smoothness of a function

I asked the question below before. $\Delta u$ is bounded. Can we say $u\in C^1$? I thought I understood the discussion using the PDE theory at the time but now I am lost. I am going to a similar but ...
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1answer
20 views

The upper bound of $L^2$ norm of the minimizer in an minimizing problem.

I am considering the following minimizing problem: $$ u_m:= \operatorname{argmin}_{u\in BV(\Omega)}\{ \frac{1}{2} \|u-u_0\|_{L^2}^2 + t |u|_{TV}\} $$ where $u_0\in BV(\Omega)\cap L^\infty(\Omega)$ and ...
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53 views

Is the image of gradient map from Sobolev space to Lebesgue space weakly closed?

Suppose f is a map defined between $W_0^{1,p}(\Omega)$ and $L^{p'}(\Omega)$ as follows - $u \mapsto |\nabla u|^{p-1}$. Is the range of this map weakly closed in $L^{p'}$?.
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14 views

The normalization of gradient in weak convergence.

Given $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary, $u_n\in BV(\Omega)$ is bounded in $BV$ norm and in addition we have $$0<\inf |u_n|_{TV}\leq \sup |u_n|_{TV}<+\infty$$ where ...
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1answer
19 views

The zero convergence of total variation

Let $\Omega\subset\mathbb R^2$ be open bounded, smooth boundary. Given a sequence $(u_\epsilon)\subset BV(\Omega)$ such that $$ \|u_\epsilon-u_0\|_{L^2}^2+\epsilon |u_\epsilon|_{TV(\Omega)}\to 0 $$ ...
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34 views

Coercive bilinear form from Poisson equation with Robin boundary conditions

I need to find if the bilinear form arising from the following problem is coercive or not : $$-v\Delta u = f \quad \text{in } \Omega$$ $$v\frac{\partial u}{\partial n} +hu = 0 \quad \text{on } ...
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17 views

Sum of normed function spaces equipped with two-parameter family of norms, Dual inequality

Let $(\cdot,\cdot)$ be the $L^2(\Omega)$ scalar product, and let $V=L^1(\Omega)$, $W=H^{-1}(\Omega)$ (the dual space of $H_0^1(\Omega)$). My question is if there exists a constant $C$, such that for ...
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1answer
33 views

Agmon's Inequality in higher dimensions

We have the Agmon inequality on $[0,a]$ $$\| u \|_{L^{\infty}} \leq \|u \|_{L^2}^{1/2} \|u_x\|_{L^2}^{1/2}$$ Is there a version in two dimensions, say on $[0,a]^2$? I know there is a multidimensional ...
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30 views

Weak chain rule the other way round.

If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function $\varphi \in ...
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24 views

Difference between $H^1[0,1]$ and $H_0^1[0,1]$

I want to show that $H^1[0,1]= \text{span}(e^{it}) + \text{span}(e^{-it})+H_0^1([0,1]).$ So $H^1$ is the space of square-integrable functions and square-integrable derivative on $[0,1]$ and ...
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1answer
91 views

Trying to understand a proposition before getting into the proof

Trying to understand For (b), I understand that there are two claims are proposed in respective the cases $[1,\infty]$ and $[1,\infty)$. i. For $p\in[1,\infty]$, one has $$\|\rho_k * ...
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28 views

Integrability, Sobolev space

Hello I have a question about Sobolev spaces. Let $n \in \mathbb{N}$, $\alpha >\frac{n}{2}$, $c :\mathbb{R}^{n} \to \mathbb{R}$ be a $\alpha$-integrable function. I want to show the following: ...
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1answer
23 views

Embedding of Sobolev space W_2^1[a,b] in C[a,b]

Let's define $$||f||_{1,2}=[\int_a^b(f(x)^2+f'(x)^2)dx]^{\frac{1}{2}}$$ and the Sobolew space $W_2^1[a,b]$ to be the completion of $C^1[a,b]$ with respect to $||f||_{1,2}$ norm. How can we show that: ...
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1answer
30 views

About weak convergence in Sobolev spaces

Let $\Omega$ a bounded domain in $R^n$ with smooth boundary. I am reading a paper, and I have the following situation: Consider $\varphi \in W^{1,p}(\Omega)$ and $v_j$ a sequence in ...
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1answer
55 views

Sobolev embedding theorem, inequalities

I have a question about an inequality. Let $n \in \mathbb{N}, \alpha >n$ and $b:\mathbb{R}^{n} \to \mathbb{R}$ be a $\alpha$-integrable function. (i.e. $b \in L^{\alpha}(\mathbb{R}^{n})$). I want ...
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1answer
34 views

Replacing $C_c^\infty$ by $H_0^1$ in the definition of weak subsolution.

Consider the elliptic operator $$Lu=-\sum_{i,j} (a^{ij}u_{x_i})_{x_j},$$ where $a^{ij}\in L^\infty(\Omega)$ and $a^{ij}(x)\xi_i\xi_j\geq\lambda |\xi|^2$ for all $\xi\in\mathbb{R}^n$, $x\in\Omega$. ...
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1answer
19 views

Best possible constant for Poincare inequality.

Consider Poincare inequality in the form of: $|u|_{0,\Omega} \leq C|u|_{1,\Omega} \text{ for every } u\in H^1_0(\Omega),$ where $\Omega \subset \mathbb{R}^n$ is a bounded open set. Find the best ...
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1answer
26 views

Sobolev embedding

In an exercise I am asked to prove the following statement: The embedding $$T:W^{1,1}(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n), u\mapsto u$$ is continuous. Using the Gagliardo-Nierenberg inequality ...
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21 views

Sobolev inequality on $\mathbb{R}^3$

I know that for $u \in H^q(\mathbb{R}^d)$ with $d>q$ we have for $p = \frac{qd}{d-q}$ that $\|u\|_p \le C \|u\|_{H^q}.$ Now, I have somewhere back in my mind that it is also in the unbounded case ...
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Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
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1answer
24 views

composition of Lebesgue integrable function with smooth function

If I have a function $f\in L^{1}_{loc}(\mathbb{R})$ and I compose it with $g:\mathbb{R^2}\to\mathbb{R}$ which is smooth. Then is it always true that $f\circ g\in L^{1}_{loc}(\mathbb{R^2})$? thank ...
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1answer
42 views

Lower semicontinuous representative of positive Sobolev function?

For a function u in the Sobolev space $W_0^{1,p} (\mathcal O )$, ($p \in [ 1, n ]$), having $u > 0$ inside $\mathcal O$, where $\mathcal O$ is an open bounded connected set in $\mathbb R^n$, can ...
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11 views

Estimate a integral over a surface S with an integral over a volume V containing S in H^1

Given a function $v \in H^1(\Omega)$, a volume $ \Omega_e \subset \Omega $ and a surface $ S \subset \Omega $ contained in $\Omega_e$. Namely, $S \cap \Omega_e = S$. The statement $$ \| v ...
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1answer
36 views

Holder inequality in proving sobolev interpolation inequality

I am reading the book "Elliptic Partial Differential Equations of Second Order" by D. Gilbarg and N.S. Trudinger. In theorem 7.27, it stated that after we obtained $$|u'(x)| \leq \int_{a}^{b}|u''| + ...
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1answer
25 views

Counter example: Sobolev Spaces

I need to show that the embedding of $H^1(\mathbb{R}^N)$ in $L^2(\mathbb{R}^N)$ is not compact. I need to find a sequence $(u_n)\subset H^1(\mathbb{R}^N)$ bounded such that there is no subsequence ...
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2answers
36 views

Finding a bound for the first derivative of a C^2 function within the p-norm

Let $u\in C^2([0,1])$ and $\epsilon >0$ arbitrary. Show that there exists an absolute constant C>0 such that $$ \int_0^1 |u'|^p dx \leq \epsilon^p \int_0^1 |u''|^p dx + \frac{C}{\epsilon^p} ...
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1answer
32 views

Convergence of vector-valued truncation in $H_0^1(\Omega)^m$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. For $m \in \mathbb{N}$ and $M > 0$ we denote by $T_M$ the truncation of vectors in $\mathbb{R}^m$ to length $M$, i.e., $$T_M(x) = ...
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1answer
68 views

Hermite Functions and Multiplication Operators: Why is the “obvious” equation correct?

Question: Let $T$ be the unbounded operator on $L^2(\mathbb{R})$ defined by $$Tf(x) = \frac{1}{\sqrt{2}} \left( x f(x) - f^{\prime}(x) \right), $$ with domain $\mathfrak{D}(T)$ restricted to the set ...
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10 views

Proof: $\left\|A\vec{u}\right\|_{0,K}\leq C\|A\|_{0,K}\left\|\vec{u}\right\|_{\infty,K}$

Let $u$ and $v$ be continuous functions defined over a compact $K$. We define: $\left\|\begin{pmatrix}u\\ v\end{pmatrix}\right\|_{0,K}^2:=\|u\|_{0,K}^2+\|v\|_{0,K}^2=\int_K u^2\,dx+\int_K v^2\,dx$. ...
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0answers
18 views

The slicing argument for jump set

Given $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $\nu\in \mathbb S^{N-1}$ be a fixed direction. We define $$ \pi_\nu = \{x\in\mathbb R^N:<x,\nu>=0\},\,\Omega_x = ...
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1answer
22 views

Show that the functional $F(u)=\int ((u_x^2-1)^2+u_y^4)$ has zero infimum on a Sobolev space

Let $\Omega=(-1,1)\times (-1,1)$. We consider the functionnal $$F(u)=\int_{\Omega}\left[\left(\left(\frac{\partial u}{\partial x}\right)^2-1\right)^2+\left(\frac{\partial u}{\partial ...
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0answers
34 views

Weak differentiability and diffeomorphisms

Let $U,V\subset\mathbb{R}^n$ be open sets and assume the existence of a $\mathcal{C}^1$-diffeomorphism $\phi:U\rightarrow V$. Let $u\in W^{1,p}(U)$, $1\leq p\leq\infty$, and define ...
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1answer
85 views

Some intuition on a specific problem on Sobolev's embedding theorem with its relation to Fourier transform of restricted functions

I have recently encountered this problem in my studies of Sobolev spaces and generalized functions (distributions), on which I can say I might have some intuition but cannot stumble across a final ...
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1answer
25 views

If $\int_\Omega |\nabla (u-M)^+|^2 = 0$ where $u \in H^1(\Omega)$, and $u|_{\partial\Omega} \leq M$, does it follow that $u \leq M$ a.e?

Let $\Omega$ be a bounded domain. If $\int_\Omega |\nabla (u-M)^+|^2 = 0$ where $u \in H^1(\Omega)$, and $u|_{\partial\Omega} \leq M$, does it follow that $u \leq M$ a.e? I do not think this is ...
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37 views

Dirichlet eigenfunction cannot be extended to a continuous function on the closure

I need to show that there exist a bounded domain $ \Omega \subset \mathbb{R}^2 $, and a Dirichlet eigenfunction $u$ on $ \Omega$ such that u cannot be extended to a continuous function on $ ...
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1answer
62 views

A question involving sharpening the bound on Sobolev type inequality with Sobolev spaces in terms of distributions of Schwartz functions

I have met this problem recently in my real analysis class involving sharpening the bound on a Sobolev type inequality, from Folland's real analysis, but first I should mention the notations used ...
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1answer
30 views

Helmholtz Decomposition in Sobolev Space

Let $\Omega$ is a $C^2$-smooth bounded domain in $R^2$ (not assume to be simple connected) then $f\in W^{1,2}(\Omega) $ can be represented as: $$f=\nabla^{\perp}b+\nabla\varphi$$ With $b,\varphi\in ...
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1answer
42 views

BMO space and log-lipschitz regularity

Let $\Omega$ an open bounded domain in $R^n$ with smooth boundary and consider a function $u \in W^{1,p}(\Omega) (2<p < +\infty)$ . Suppose that for every ball $B \subset \subset \Omega$ exists ...
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1answer
118 views

Extension theorem Sobolev spaces

Basically, I have the same question as in Extension of $W^{1,\infty}(\Omega)$: Given a bounded, open set $\Omega\subset\mathbb{R}^n$ with $\mathcal{C}^1$-boundary and another open bounded set ...
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1answer
67 views

Composite of a $\mathcal{C}^1$ real function and a $W^{1,p}$ function

The problem: Assume $F:\mathbb{R} \to \mathbb{R}$ is $\mathcal{C}^1$, with $F'$ is bounded. $U$ is bounded and $u \in W^{1,p}(U)$, for $1 < p < \infty$. Show $v := F(u)$ is still in ...
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19 views

Boundedness in $W^{-1, \infty}_{loc}$

For the solution of vanishing viscosity equation: $u^{\epsilon}_{t} + f(u^{\epsilon})_{x} = \epsilon u^{\epsilon}_{xx}$ in $\mathbb R$ X $R^{+}$ with bounded measurable initial data: ...
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1answer
29 views

A Sobolev function

I want to prove that the function $$u(x)=\sum_{k=1}^\infty 2^{-k}|x-r_k|^{-\alpha}$$ is in the Sobolev space $W^{1,p}(B_1(0))$ if $0<\alpha<\frac{n-p}{p}$ where the sequence $ (r_k)$ is dense in ...
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1answer
107 views

Smoothness of function 1D sobolev space

I came across the following statement in my reading and I need some help to understand it If $u \in W^{2,2}(a,b)$, then $u\in C^1([a,b])$ What I know is that if $u\in W^{2,2}(a,b)$, then the ...
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1answer
22 views

Orthogonality in $H^1(T)$ with the inner product $(\cdot,\cdot)_{L^2(T)}$

Take the Sobolev space $$H^1(T)=\left\{f \in L^2(T) ~|~ f' \in L^2(T)\right\}$$ where $T$ is the 1-torus (that is a circle) and $f'$ the weak derivative. Take then a function $v\in H^1(T)$ and ...
2
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1answer
35 views

IHow to apply the density of $C_0^1(\Omega)$ in $W_0^{1,1}(\Omega)$ to derive the identity in the following?

I'm reading the book by Gilbarg and Trudinger. In Lemma 7.14, the authors proved that if $u \in C_0^1(\Omega)$, then $$u(x)=\frac{1}{nw_n}\int_{\Omega}\frac{(x_i-y_i)D_iu(y)}{|x-y|^n}dy.$$ Then ...
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1answer
61 views

keeping a.e. convergence when composing a nice sequence with ln.

I want to prove a theorem which I urgently need, but i am kind of stuck at a certain point. My Problem is to prove the following: EDIT I found a proof, but I'm not sure if there are any mistakes in ...
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0answers
21 views

Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...