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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1answer
13 views

Show that $||u||_{L^{1}(I)} \le \frac{1}{k} ||f||_{L^{1}(I)}$

Suppose there exists a unique $u \in H_0^{1}(I)$ satisfying $$\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$$ Show that $$||u||_{L^{1}(I)} \le \frac{1}{k} ||f||_{L^{1}(I)}$$ In the ...
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1answer
15 views

prove that there exists a unique $u \in H_0^{1}(I)$ satisfying $\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$

Let $I=(0,1)$ and fix a constant $k \gt 0$. Given $f \in L^{1}(I)$ prove that there exists a unique $u \in H_0^{1}(I)$ satisfying $$\int_{I}u'v'+k\int_{I}uv=\int_{I}fv, \forall v \in H_0^1(I)$$ For ...
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0answers
9 views

Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
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2answers
22 views

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with ...
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0answers
17 views

Representation of the delta distribution as an element of the dual of $H^1$

I'm working with some Sobolev spaces and I just wanted to consider the elements of $H^{-1}$ as elements on $H^1$ (Riez Theorem). Since the delta function $\delta(f) = f(0)$ is an element of the dual ...
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0answers
16 views

Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
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1answer
24 views

On the trace theory and restrictions of Sobolev space functions

Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...
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1answer
14 views

Show that the following is true for a bounded sequence in $W^{1,p}(I)$

Let $I=(0,1)$. Assume that $u_n$ is a bounded sequence in $W^{1,p}(I)$ with $1 \lt p \le \infty$. Show that there exist a subsequence $(u_{n_k})$ and some $u$ in $W^{1,p}(I)$ such that ...
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0answers
7 views

Prove the Poincare's inequality on $B^{0}(0,1)$. [duplicate]

Fix $\alpha >0$. Let $U=B^{0}(0,1)$. Show that there exists a constant $C$, depending only on $n$ and $\alpha$ such that $\int_{U} u^{2}\mathrm{d}x\leq C\int_{U} |Du|^{2}\mathrm{d}x,$ provided ...
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0answers
18 views

Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class ...
4
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1answer
35 views

Prove that following are true for $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$

Fix a function $ \phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$ and set $u_n(x)=\phi(x+n)$. Let $1 \le p \le \infty$. Then Check that $u_n$ is bounded in $W^{1,p}(\mathbb{R})$ Prove that there ...
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0answers
13 views

Fractional Sobolev space on an interval

Consider a fractional Sobolev space $H_p^t(I)$ defined on an interval $I\subset \mathbb{R}$. When $I=\mathbb{R}$ the space can be defined via Fourier transform. Is it possible to do it when ...
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0answers
9 views

Sobolev spaces on non-compact manifolds — independence on charts

Are there some standard references where basic facts about fractional-order (or at least integer-order) Sobolev spaces on non-compact manifolds are treated? More precisely I would like to be able to ...
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0answers
22 views

Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
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0answers
20 views
2
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1answer
32 views

An example of open set $\Omega$ in $\mathbb{R^n}$ for which $C^{\infty}_c(\Omega)$ is dense in $W^{l,p}(\Omega)$

We know that the statement $C^{\infty}_c(\mathbb{R^n})$ is dense in $W^{l,p}(\mathbb{R^n})$ is always true for any $l\in \mathbb{N}$ and $p\geq \infty$, $p\neq \infty$. My professor told me that it ...
1
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1answer
19 views

Estimation of gradients in Poisson's equation

I am trying to show the following result. Let $D\subset\Bbb R^3$ be a bounded open set with smooth boundary. For any $f\in H^{-1}(D)$, let $\phi$ be the unique weak solution to the following ...
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0answers
18 views

Counterexample for the density of smooth functions in Sobolev spaces on a manifold

I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The ...
0
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1answer
15 views

Does uniformly boundness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this ...
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0answers
13 views

Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in ...
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0answers
21 views

How do we define $H^{-1}$? [duplicate]

In class we defined $H^{-n}$ on $\mathbb{R}^n$ via the Fourier transform of tempered distributions. But unfortuntely, on subset $\Omega \subset \mathbb{R}^n$ there are Schwartz functions. So let ...
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1answer
73 views

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
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0answers
15 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
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0answers
9 views

Question on weighted Sobolev spaces

Let us define a weighted Sobolev space $W^{k,p}_\delta(\Omega)$ as \begin{equation} W^{k,p}_\delta(\Omega) = \left \{ u \in L^p(\Omega): (1+r^2)^{\frac{1}{2}(-\delta-\frac{3}{p}+|\beta|)}D^{\beta}u ...
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2answers
11 views

$u_n \rightarrow u$ in $W^{1,2}$ implies $u_n \rightarrow u$ and $u'_n \rightarrow u$ in $L^2$

I report the following excerpt from a textbook: "By the usual density argument we can find for every $u \in X = \left\{ u \in W^{1,2}:u(-1)=u(1) \text{ and } \int_{-1}^1 u = 0 \right\}$ a sequence ...
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1answer
19 views

Hyperbolic energy estimate in Evans PDE book

Before Theorem 6 in Chapter 7.4 in Evans' PDE book Evans claims that there exists $\beta > 0$ such that $$ \beta\|u\|_{H^1(\Omega)}^2 \leq B[u,u]\,, \quad \forall u \in H_0^1(\Omega)\,. $$ From how ...
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0answers
21 views

Approximation, Truncation argument, Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
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1answer
45 views

Rellich's theorem fails at the end point.

If $\{f_k\}\subseteq H^s(\Bbb{R}^d)$ is a bounded sequence and support of each $f_k$ is contained in a common compact set, then there exists a subsequence that converges in $L^q$ for $\forall$ $q, ...
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0answers
16 views

Bounded sequence in Sobolev space has a convergent subsequence

Suppose $\{\phi_k\}$ are a bounded sequence of distributions on in $H^s(\Bbb{R^d})$, where $0<s<\frac{d}{2}$ and $H^s$ is the Sobolev space. Then the sequence has a subsequence that converges in ...
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0answers
11 views

Given $u \in W^{1,p}(\Omega)$ then $\overline{\alpha u} \in W^{1,p}(\Omega)$, with $\alpha \in C^1 \cap L^{\infty},\nabla \alpha \in L^{\infty}$

Given $u \in W^{1,p}(\Omega)$ and $\alpha$ a function which verifies $\alpha \in C^1(\Omega)$, $\alpha \in L^{\infty}(\mathbb{R}^n)$, $\nabla \alpha \in L^{\infty}(\mathbb{R}^n)^n $ y $supp(\alpha) ...
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2answers
22 views

Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$

Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this: ...
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1answer
36 views

About two subspaces of (1,2)-Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
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0answers
12 views

A approximation problem of bounded Sobolev function

Given a function $u\in L^\infty(\Omega)\cap H_0^1(\Omega)$, where $\Omega$ is a bounded domain in $R^n$, could we select a sequence $\{u_k\}_k\subset C_c^\infty(\Omega)$ such that $u_k\rightarrow u$ ...
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0answers
15 views

Regularity of $u$ and $v$ satisfying $u_t - \Delta u = \Delta v - v_t$

I have that there are two functions $u$ and $v$ in $H^1(0,T;L^2)\cap L^2(0,T;H^2)$ satisfying weakly the equation on a bounded domain $\Omega$ $$u_t - \Delta u = \Delta v - v_t$$$ given some ...
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0answers
20 views

Function Singularity in a Sobolev Space

For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$? UPD: actually I am ...
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1answer
16 views

Examples in $W^{1,p}(U)\setminus C(\overline{U})$ and $C(\overline{U})\setminus W^{1,p}(U)$

The following is the trace theorem in Partial Differential Equations by Evans: Let $U$ be a domain (open connected subset) of $\mathbb{R}^n$. Suppose $U$ is bounded and $\partial U$ is $C^1$. Then ...
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1answer
31 views

Intuition of weak solutions of elliptic equations in divergence form

Let $\Omega \subset \mathbb{R}^{n}$ a domain, and consider the following equation (1) $-D_{j}(a_{ij}D_{i}u) = 0$ (Einstein notation) The function $u \in H^{1}(\Omega)$ is a weak solution of (1) if ...
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0answers
14 views

Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
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1answer
22 views

Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f'' $ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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0answers
16 views

Prove, for an open $\Omega \subset \mathbb{R}^n$ with $x\in \Omega$, that $u\in W^{1,p}(\Omega-\{x\})\implies u\in W^{1,p}(\Omega).$

Let $n\geq 2$, $\Omega\subset \mathbb{R}^n$ open, $x\in \Omega$ and $p\geq1$ I want to prove the above implication. We just need to show that the weak derivative of $u$ on the punctured domain remains ...
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1answer
24 views

Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ ...
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0answers
24 views

$B=B(0,1)\subset\mathbb{R}^N$ and $\Omega=B\setminus\{0\}$. Does $H^1_0(\Omega)=H^1_0(B)$?

$B=B(0,1)\subset\mathbb{R}^N$ and $\Omega=B\setminus\{0\}$. (i) Assume $N=1$ and prove $H^1_0(\Omega)\neq H^1_0(B)$. (ii) Take $N\ge 2$. Does $H^1_0(\Omega)=H^1_0(B)$? I don't even know where to ...
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0answers
14 views

Prove that the following function is in $H^1(\Omega)$

Let $\Omega$ be such that $$ \overline{\bigcup_{k=1}^\infty \{b_k\}}^{|\cdot|} =\Omega:=\left\{(x_1,x_2)\in\mathbb R^2; \sqrt{x_1^2+x_2^2}<1/2\right\}, $$ where $|\cdot|$ denotes the Euclidean norm ...
2
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1answer
41 views

$|v|_{2,\Omega}=0$ implies $v=0$

I am stuck on this computation: let $\Omega$ be a domain in $\mathbb R^2$ and let $\Gamma_0$ be a relatively open proper subset of $\Gamma:=\partial\Omega$. Define $$ V=\{v \in H^2(\Omega); ...
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1answer
23 views

Neumann Laplacian heat kernel or semigroup representation

I have the equation $$u_t - \Delta u = f\text{ on $\Omega$}$$ $$\partial_\nu u = g\text{ on $\partial\Omega$}$$ $$u(0) = u_0$$ for $f \in L^2(0,T;H^1)$, $g \in L^2(0,T;H^1(\partial\Omega))$ and $u_0 ...
2
votes
1answer
30 views

Coercitivity of an elliptic operator with constant coefficients

We are given an elliptic operator $P=\sum_{|\alpha|\leq m}a_\alpha\partial^\alpha$ that is elliptic in $\Omega$. $a_\alpha$ are constants. I am supposed to show that $$\|u\|_s\leq ...
0
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1answer
31 views

Show that there is an operator on $H^{2}$ and it's compact.

Let $H^{2}=W_{0}^{2,2}(\Omega)$. Define $(u,v)=\int_{\Omega} (\triangle u\triangle v+2v\triangle u)\mathrm{d}S$ as an inner product on $H^{2}$. Define $a(u;v)=\int_{\Omega} (\nabla u\cdot \nabla ...
0
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1answer
30 views

Sobolev space, continuous function

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
1
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1answer
75 views

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$?

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$? Where $\mathcal{D}(\Omega)$ is the space of test functions with support compact and $\mathcal{D}'(\Omega)$ is the ...
0
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1answer
25 views

Sufficient conditions that guarantee the integrate by part formula?

Suppose, for simplicity, that $f,g$ are functions defined on $[0,1]$. Under suitable hypotheses on $f$ and $g$, the integrate by parts formula yields $$ \int_0^1 f'(t)g(t)dt = f(t)g(t)|_{t=0}^1 - ...