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
75 views

$C^1$ domains vs. Lipschitz domains in PDEs, do we need transformation of coordinates?

I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within ...
0
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1answer
26 views

Show the example belong to the Bessel potentials space (fractional order sobolev space), where $p=2$

If $\delta>-\frac12$, show that $(1-x^2)_+^\delta\in W^{s,2}$, where $s\in (0,\delta+\frac12)$. Thanks in advanced.
0
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1answer
64 views

Sobolev trace operator bounded from below??

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary $\partial\Omega.$ Is the trace operator $$T:H^1(\Omega) \to H^{\frac 1 2}(\partial\Omega)$$ bounded from below: $$|Tu|_{H^{\frac 1 2}} ...
0
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0answers
38 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
6
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0answers
96 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
2
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1answer
31 views

If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $|f(x,t)| \leq C$ for all $x$ and $t$?

Here $\Gamma$ is a bounded closed $C^k$ hypersurface. If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $f$ uniformly bounded on $[0,T]\times \Gamma$? Or even does it hold that $|f(t)| \leq C_t$ for ...
2
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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
37 views

Question about a Sobolev norm

The Sobolev seminorm on $H^s$ is $$|f| = \int_{\Omega}\int_{\Omega} \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2s-1}}dxdy$$ My question is, is this integral a double integral or a iterated integral? i.e. can I ...
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1answer
52 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
2
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0answers
39 views

Properties of the first eigenvalue of the $p$-Laplace operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $p\in (1,\infty)$, $p\neq 2$. Consider the usual Sobolev space $W_0^{1,p}(\Omega)$ and its dual $W^{-1,p'}(\Omega)$, where $1/p+1/p'=1$. Define ...
1
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1answer
77 views

property of local Sobolev space

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if ...
1
vote
1answer
67 views

Sobolev spaces on boundaries

Is the following a good definition for a Sobolev space on a boundary: Can anyone show me another source where such a space is defined? In the definition, $v \in W^{s,p}(\partial\Omega)$ if $v \circ ...
2
votes
1answer
84 views

Why is partition of unity required in definition of Sobolev space on manfolds?

Why do we need to use $\phi_i u$ in the expression for the norm? Why not just $u$? The range of integration is over $R(x_i)$ anyway, so I don't understand why it is necessary. If you check Kendall, ...
1
vote
1answer
20 views

Descomposition on temporal sobolev space

Let $\Omega$ an open subset of $\mathbb{R}^2$ with Lipschitz boundary. Can I descompose in a unique way any $u\in L^2(0,T;L^2(\Omega))$ such that for all $t\in [0,T]$, $u(t)=u_1(t)+u_2(t)$ with ...
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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 ...
3
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1answer
92 views

Using Nemytskii Theorem for Sobolev Spaces

The Nemytskii mappings in Lebesgue spaces theorem is as follows: If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory ...
0
votes
1answer
42 views

limit of function in Sobolev space

Let function $f(x)$ and $f(x)g(x)$ belong to $\mathcal{W}^{s+1}$ with $s\ge 1$, where $\mathcal{W}^{s+1}$ be the Sobolev space of regularity $s+1$ in $\mathbb{L}^2$-norm. We also have $g\in ...
2
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2answers
164 views

The normal derivative in PDE problems (how is the weak form defined?)

For smooth functions, we know that $$-\int_{\Omega}\Delta v w = \int_\Omega \nabla v \nabla w - \int_{\partial \Omega}\nabla v \cdot \nu w.$$ Thus we can define a generalised normal derivative ...
2
votes
2answers
96 views

The Poisson problem $-\Delta u =0$ with $u=g$ on the boundary where $g \in H^{\frac{1}{2}}$

Consider $$ \begin{align} -\Delta u =0 & \text{on $\Omega$} \\ u = g & \text{on $\partial\Omega$} \end{align} $$ where $g \in H^{\frac{1}{2}}(\partial\Omega)$. It seems there exists a ...
0
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2answers
49 views

Some doubts on the Trace Theorem

The Trace Theorem in Evan's Book (1st edition) says that, Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exsits a bounded linear operator $T$, $$T:W^{1,p}(U)\rightarrow L^p(U)$$ such ...
0
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1answer
48 views

A problem about Sobolev Spaces

I meet a problem in homework: Assume $\Omega\subset\mathbb{R}^n$ is bounded and $\partial\Omega$ is $C^1$. Does $H_0^2(\Omega)$ equal to $H_0^1(\Omega)\cap H^2(\Omega)$? Obviously, ...
1
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0answers
62 views

Product rule for Banach space-valued differentiable functions?

Let $\Omega \subset \mathbb{R}^n$ be a bounded open set and let $f(\cdot,\cdot)$ and $g(\cdot,\cdot)$ be functions from $[0,T]\times \Omega$ into $\mathbb{R}$. Suppose that $f \in ...
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0answers
44 views

The space $C^1([0,T]\times \Omega)$ for $\Omega$ open and bounded

Let $\Omega$ be open and bounded. Is there anything nice I can say about the space $C^1([0,T]\times \Omega)$ and its inclusion in some Bochner like spaces? If $f \in C^1([0,T]\times \Omega)$ then ...
3
votes
1answer
101 views

Integration by parts with few regularity

I'm having problems proving an integration by parts formula presented in the work of Alt and DiBenedetto on porous media flow (Remark 3.4.2). Essentially, the problem is the following. Let $s\in ...
4
votes
0answers
105 views

If $u$ has a weak derivative and $f$ is $C^1$ does $fu$ have a weak derivative (fractional Sobolev space and weak time derivatives)

Let $\Omega$ be an open bounded set. Let $s \in (0,1)$ and $H^s(\Omega) := W^{s,2}(\Omega).$ Let $f \in C^1([0,T]\times \Omega)$ and $u \in L^2(0,T;H^s(\Omega))$ with weak derivative $u' \in ...
0
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1answer
37 views

Showing $u \in H^s$ and $\varphi \in C^1$ implies $u\varphi \in H^s$ (product rule)

Let $\Omega$ be bounded and open set in $\mathbb{R}^n$. As a start, I pose this question: For $u \in H^s(\Omega)=W^{s,2}(\Omega)$, define the Holder seminorm type quantity $$F(u) = ...
1
vote
1answer
36 views

Please explain this notation of mapping into a set and product space (related to Sobolev spaces)

So does this mean that I can say that, for example, $\gamma \frac{\partial u}{\partial \nu}$ has a unique continuous extension as an operator from $W^s_p(\Omega)$ onto $W^{s-1-{\frac 1 ...
2
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1answer
86 views

A question about a Sobolev space trace inequality (don't understand why it is true)

Let $\Omega$ be an open set with boundary $\partial\Omega$. Let $u \in H^1(\Omega)$. There exists a $\lambda \in \mathbb{R}$ such that $$\int_\Omega |\nabla u |^2 + ...
2
votes
2answers
211 views

Is fractional Sobolev space $H^s$ Hilbert?

For $s \in (0,\infty)$ a fractional number, define $H^s(\Omega) = W^{s,2}(\Omega)$ on good domain $\Omega$. Every textbook doesn't say that $H^s$ is Hilbert. Is it? I have only seen this fact when ...
0
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0answers
40 views

Laplace equation with Dirichlet homogeneous BC

I'm studying the Laplace equation with homogeneous Dirchlet boundary conditions that is, $$ -\triangle u + u = f \quad \text{in } \Omega \quad u=0 \quad \text{on } \Gamma $$ With $\Omega$ an open ...
0
votes
1answer
44 views

Show that $L$ it is a continuous operator.

Let $L(.)$ a linear operator in $W$ so that $$L(\phi)=\int_0^T<f,\phi>_{H^{-1}, \ H_0^1}dt+(u_0,\phi(0))_{L^2}.$$ Then $L(.)$ it is continuous in $W$. Where $f \in L^2(0,T; H^{-1}(\Omega))$ ...
1
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2answers
70 views

Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$

I have a problem: For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$. ...
1
vote
1answer
75 views

Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. Denote $L_{2,\ 0}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}$, with norm in $L_2(\Omega)$. We consider: ...
3
votes
1answer
119 views

Show that a Bilinear form is Coercive

I'm reading through Brezis' book on functional analysis, Sobolev spaces and PDE, and I'm having trouble showing that the Bilinear form: $a(u,v) = \int_{0}^{2}u'v'dx+\left(\int_{0}^{1}u dx\right) ...
1
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1answer
58 views

Show that $b(\ ., \ .)$ it is not coercive.

Let $b(\ .,\ .)$ a bilinear operator so that $$b(u, \phi)= \int_0^T((u(t), \ \phi(t)))dt- \int_0^T(u(t), \ \phi'(t))dt,$$ where $((\ ., \ .))$ it is the inner product of $H_0^1(\Omega)$ (Sobolev ...
1
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1answer
107 views

Gagliardo Nirenberg Sobolev inequality for n >= 2

I have a quick question regarding the Gagliardo-Nirenberg-Sobolev inequality. It states that: Assume $1 \leq p < n$. There exists a constant $C$, depending only on $p$ and $n$, such that ...
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0answers
63 views

Estimates for linear finite element nodal basis functions

Let $\Omega\subset\mathbb R^2$ be a domain with $\operatorname{diam} \Omega=H$ and $\mathcal T^h$ be shape-regular triangulation of $\Omega$ with triangular $\mathcal P^1$-elements. (That means we are ...
0
votes
1answer
57 views

weak solution for a simple boundary problem

Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = 0 \}$ and $\Omega $ a bounded convex domain (not necessarily smooth) such that $\partial \Omega \supset K$. ...
0
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1answer
61 views

Using Cauchy Schwarz for functions in Sobolev Space

Hi I just want to confirm something simple and check that the following is allowed: The Cauchy-Schwarz inequality states if $A = ((a_{ij}))$ is a symmetric, non-negative $n \times n$ matrix then ...
5
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1answer
213 views

Folland PDE chapter 6.C problem 1

Problem 6.C.1: Suppose $0 \neq \phi \in C^\infty_c(\mathbb{R}^n)$ and $\{ a_j \}$ is a sequence in $\mathbb{R}^n$ with $|a_j| \to \infty$, and let $\phi_j(x) = \phi(x - a_j)$. Show that $\{\phi_j\}$ ...
0
votes
0answers
14 views

Bessel Potential spaces

Let $\Omega_1,\Omega_2 \subset \mathbb{R}$ be bounded. The mapping $F: \Omega_1 \rightarrow \Omega_2$ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||(DF(x))^{-1}||$ ...
4
votes
1answer
157 views

Weak solution $u(x,t)$ of heat equation converges as $t \in \infty$

Where can I find a proof that the weak solution $u \in L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$ of the heat equation $$u_t -\Delta u = f$$ converges as $t \to \infty$ to the solution of the elliptic PDE ...
2
votes
1answer
228 views

The proof of Morrey's Inequality in Evans Book

The proof of Morrey's Inequality in page 266 of Evans's PDE book really puzzles me a lot. I cannot get the general idea of the proof. I know a simple proof just in the case of $n=1$: For any ...
1
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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 ...
4
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0answers
73 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
5
votes
1answer
103 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
1
vote
1answer
47 views

Some chain rule in W1p

Question: I need to prove the following chain rule: Let $F:\mathbb{R}\rightarrow\mathbb{R}$, $F\in C^1$ with $F'$ bounded. Let $U$ bounded and $u\in W^{1,p}(U)$ with $1\leq p\leq\infty$. Show that ...
2
votes
1answer
142 views

Friedrichs's inequality?

Friedrichs's second inequality is stated as follows(see www.win.tue.nl/~drenth/Phd/friedrichs.ps): For all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n}\cdot\mathbf{u} = 0$ or ...
1
vote
2answers
116 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
0
votes
1answer
53 views

About convergence in Sobolev space

I have $u_{n}$ sequence of $H^{1}_{0}(\Omega)$ where $\Omega$ is open bounded and connected domain in $\mathbb{R}^{n}$ with $n>1$. $u_{n}\rightarrow u$ in $H^{1}_{0}(\Omega)$ norm. Let ...