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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39 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha ...
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29 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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32 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 ...
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39 views

A bound on $\nabla u$ in $L^\infty(0,T;L^2)$; how to make argument rigorous?

Suppose $u \in L^2(0,T;L^2)$, $u_t \in L^2(0,T;H^{-1})$ and $f \in L^\infty((0,T)\times\Omega)$. I have the weak form $$\langle u_t, \varphi \rangle_{H^{-1}, H^1} + \int_\Omega\nabla u \nabla \varphi ...
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27 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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50 views

eignvalues of Laplacian operator and distributions

Let $\Omega$ be open and bounded in $\mathbb{R}^n$ and $I$ an interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigenvalues ...
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60 views

Functions of fractional-order Sobolev spaces

In fracture mechanics one might end up dealing with functions such as $$w(r,\theta) = \sqrt{r} \sin \frac \theta 2,$$ which is defined, for example, on a cracked unit circle, $\Omega = ...
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57 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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43 views

Variable density in the equation of motion

At a fixed point in time, consider the equation of motion $$ \nabla \cdot \boldsymbol \sigma(u) + \boldsymbol f = \rho \ddot{\boldsymbol u} \quad \text{in $\Omega \subset \mathbb R^d$} $$ for a ...
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49 views

Extending by zero a Sobolev function

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some ...
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20 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)^+ = ...
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49 views

implications of convergence in sobolev spaces

If we are given that $O \subset \Omega$ is open and bounded and $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. We have a sequence $\{u_{m}\}$ satisfying $$ u_{m} ...
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61 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 ...
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29 views

Compactness in Sobolev spaces

I am looking for characterizations of compactness in the Sobolev space $H^{-1}$. In particular, I am looking for a characterization involving the Fourier transform. Can anyone suggest some results ...
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20 views

Characteristic Method for linear equation

I'm looking here for a reference to solve the following equation with the method of characteristic $$ \partial_t U + A\partial_x U = f$$ $$U(t=0,x)=U_0(x)$$ $$B(t,x=0)U = 0$$ where $(t,x)\in ...
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62 views

Versions of Trace Theorems

I have a quick question about the Trace Theorem. I have been using Evans book of Partial Differential Equations to study Sobolev Spaces. The Trace Theorem is given as "If $U$ is bounded and $\partial ...
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36 views

Aproximating a function on SO(3)

By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from ...
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41 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 ...
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65 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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47 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 ...
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36 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
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37 views

An weak type sobolev multiplication theorem

Suppose that $f_i\rightharpoonup f$ in $W^{1,p}$ and $g_i\rightharpoonup g$ in $W^{1,q}$ with $q=\frac{np}{n-p}$, what the best $r$, for which we have $f_ig_i\rightharpoonup fg$ in $L^r$ (maybe ...
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23 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 ...
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20 views

Weak convergence version of sobolev multiplication theorem

Suppose that $1/n-1/p+1/q=0$, and $f_i$ weakly convergent to $f$ in $W^{1,p}$, $g_i$ weakly convergent to $g$ in $W^{1,q}$, can we conclude that $f_i*g_i$ weakly convergent to $f*g$ in $W^{1,p}$?
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33 views

Trigonometric functions are dense in Sobolev Spaces

So I am trying to prove that the following functions $\{f_n=\frac{1}{\sqrt{2 \pi}}e^{inx}\}_{n=-\infty}^{n=\infty}$ are dense in the space $H^{n}[0,2\pi]$. For the proof it would be safe to assume ...
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72 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where ...
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23 views

A projection error estimation in Sobolev space of Fractional order (1D)

Let $\Pi_0$ be the $L_2$ projector that maps $u \in L_2(I)$ to a constant function, where $I$ is taken as the interval $(0,1)$. I would like to find an error estimation for $u \in H^{1/2}(I)$, such ...
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55 views

Continuous and dense embeddings and the density of sets in Hilbert space.

Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose ...
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119 views

If a sequence of smooth functions converge in the Sobolev norm, what can one say about the limiting function?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of $k$ times weakly differentiable functions on $\mathbb{R}^n$, having finite Sobolev $p$ norm. Suppose $\{f_i\}$ is a Cauchy sequence in ...
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28 views

Does $f(u)\in H^1$ imply that also the “inner average” $f(u_h)\in H^1$?

To add more details to the question: Let $u\in L^\infty(\Omega,[0,1])$ (bounded domain, space dimension $n$ larger than one), $f\in C^1([0,1])$ increasing but $f'(0)=f'(1)=0$ and such that $f(u)\in ...
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128 views

A contradiction to $C_c^\infty \subset H^1$ not dense?

Now we know that $C_c^\infty(\Omega) \subset H^1(\Omega)$ is NOT dense. We also know that (eg. from Lions' and Magenes) that if $V \subset H \subset V'$ is Hilbert triple, $$\mathcal{D}(0,T;V) ...
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36 views

$H^1$ estimate by $L^2$-norm

Let $(\tau_h)$ be a shape regular triangulation. Prove that there exists a constant $c>0$ such that $$\|v\|_{H^1(\Omega)}\leq \frac{c}{\min_{T\in\tau_h}} \|v\|_{L^2(\Omega)}$$ for all $v\in V^h$ ...
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49 views

Multipliers in trace spaces

I need a reference for the following fact. Let $\Omega \subset \mathbb R^n$ be an open domain with $C^{1,1}$ boundary (maybe, less regularity is needed). Let $H^{1/2}(\Gamma)$ be the trace space of ...
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54 views

Prove operator $T$ is onto

Consider the Hilbert spaces $X := H^{1}(\Omega)\times H^{1}(\Omega)$ and $Y:=L^2(\Omega)\times L^2(\Omega)$, where $\Omega =\ ]{-}\pi, \pi[$, and \begin{eqnarray*} \langle(u,v), (z,w)\rangle_X & = ...
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80 views

adjoint operator in Sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with inner product : $<u,v>_H ...
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28 views

“Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
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109 views

Sobolev trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
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35 views

about Sobolev imbedding theorem and dual sapce question

Let $D$ be an open and bounded subset of domain $\Omega$, let $f$ be a distribution on $\Omega$. Show that there is an integer $k$ such that the restriction of $f$ to $D$ is in $H^{-k}(D)$. The hint ...
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51 views

How to eliminate the regularizing function item?

Let $\mathbf{f} \in W^{1,q}_{loc}(\mathbb{R}^n)$, $J \in L^p (\mathbb{R}^n)$, with $1 \leq p,q \leq \infty$, and $\frac{1}{p}+\frac{1}{q} = 1$, $\rho_{\epsilon}$ be a regularizing kernel for ...
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67 views

Sobolev Spaces: The difference between $W^{k,p}$ and $W^{k,p}_0$

Let $U$ be an open set in $R^d$. I am confused about the differences between $$W^{k,p}(U):=\{u\in L^p(U): D^{\alpha}u\in L^p(U) \text{ for all } |\alpha|\le k\}$$ and ...
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38 views

Finding an optimal $p$ such that $u \in L^p$

We have an $L^2$ function $u$ defined on $\mathbb{R^2}$ with compact support such that $u \in H^{2/3}$ (H stands for Sobolev spaces, as always), $\partial_y u \in L^2$, and $(x\partial_y - ...
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85 views

Solving a Sturm-Liouville differential equation variationally

This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows: Let $q ...
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103 views

Hölder norm estimates

How do you prove the following estimate for composition of functions: If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that $$ \|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq ...
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61 views

How can I integrate this?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where ...
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43 views

How can I prove $\mathcal S$ is dense in $W^{s,2}$?

Let $\mathcal S (\Bbb R^n)$ be the Schwartz class and $W^{s,2}(\Bbb R^n)$ be the Sobolev space($s=0,1,\cdots$). In fact I know that $C_c^\infty(\Bbb R^n)$ is dense in $W^{s,2} (\Bbb R^n)$ and ...
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44 views

How to show that $u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n}))$?

If $u(t,x)\in L^{2}([0,T],H^{2}(\mathbb{R}^{n}))$, $\partial_{t}u \in L^{2}([0,T],L^{2}(\mathbb{R}^{n}))$, prove that $$ u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n})) $$
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30 views

Density of finite element functions in $W^{1,p}(\Omega)$

I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...
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33 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
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193 views

A continuous embedding.

If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
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42 views

One of conjectures of De Giorgi

conjecture: If $\exp(tw), \exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight. $w$ is regular if weighted Sobolev space $W^l_p(\Omega,w)$ is equal to the ...