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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About domain of $(-\Delta)^{\frac 12}$, do not follow a paper

I'm reading this paper on arxiv link. So far OK. Now this I don't understand. Take $s=\frac 12$. They say that by density the operator $(-\Delta)^s$ is defined on $\mathbb{H}^s(\Omega)$. ...
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21 views

Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
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20 views

The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
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43 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...
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27 views

How to derive this Sobolev-type inequality in $\mathbb R^3$?

Does anyone know a simple way to derive the following inequality for smooth, compactly supported functions in $\mathbb R^3$? $$ \max \,\{ |u(x)| \mid x \in \mathbb R^3 \} \;\leq\; K\|\, Du ...
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19 views

Multiplication by a Cutoff and Convergence in $H^s(\mathbb R^n)$

I'm trying to teach myself some things about Sobolev spaces out of McLean, Strongly Elliptic Systems and Boundary Integral Equations. Exercise 3.14 has me stumped for no reason: Let $K_j \subset ...
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12 views

Sobolev space trace theory on $M \times [0,T]$

Let $M$ be a compact Riemannian manifold without a boundary. I wonder how the trace map $T:H^1(M \times [0,T]) \to H^{\frac 12}(M \times \{0,T\})$ is exactly.. can I split it into two trace maps for ...
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28 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
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64 views

How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$

It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that $$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$ The ...
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42 views

$L^2$-Sobolev space

I am looking at the proof of the following lemma and I don't understand the conclusion. Lemma: For $k\geq 0$ integer, let $f=\sum_{n\in \mathbb{Z^d}}{c_n\chi_n}\in L^2(\mathbb{T}^d)$. Then $$f\in ...
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26 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
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45 views

Existence of weak solution to $-\Delta u =0$, $u|_{\Gamma_1} = g$ and $u_{\Gamma_2} = 0$?

Let $\Omega$ be a compact Riemannian manifold with $\partial\Omega = \Gamma_1 \cup \Gamma_2$ a union of disjoint sets. Let $g \in H^s(\Gamma_1)$ and consider $$-\Delta u = 0 \text{ on $\Omega$}$$ ...
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42 views

$L^{\infty}$ estimate on the boundary by means of Sobolev norms in the interior

I am studying this paper http://143.248.27.21/mathnet/paper_file/washington/gunther/cpde.pdf and I have a doubt I have not been able to solve since yesterday. In page 14, just before the last bunch of ...
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48 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
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32 views

Prove weak derivative commutes with difference quotient

Let $U$ be an open set in $\mathbb{R}^n$,$f:U\to \mathbb{R},f\in W^{1,p}(U)$. Let $\tau_{h,i}f(x)=\frac{f(x+he_i)-f(x)}{h},h>0$ Given any compact $V\subset U$, show there exists $h_0>0$ such ...
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35 views

Does Gagliardo-Nirenberg inequality in unbounded domain still hold?

Often we have the following Gagliardo-Nirenberg inequality: Let $1\leq p_1, p_2\leq \infty$, $0\leq r<l (r, l\in Z_+)$. Suppose that the number $$ \theta=\frac{n/p-n/p_1-r}{n/p_2-n/p_1-l} $$ ...
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40 views

Compute the value of the Sobolev norm in $H^{-1/2}$

I am working with finite elements using domain decomposition in 2D and one of the solutions I need to obtain is the co-normal derivative of the solution along a segment that is the intersection of two ...
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59 views

When is the convolution of a product the product of convolutions?

Although the convolution of the product is not the product of the convolution, i.e. $$fg*h\neq (f*h)(g*h).$$ I am wondering if this true (for a suitable class of functions) in the limit when one ...
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48 views

The weak convergence between $L^2$ and $H^1$

If the sequence $u_n\in H^1(R^n)$, $n\geq 3$ and $u_n$ weak converges to $u$ in $H^1(R^n)$, if we can get $u_n$ weak converges to $u$ in $L^2(R^n)$? Furthermore, if we can have $\nabla u_n$ weak ...
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21 views

Dual of $L^\infty(I,H^1(M))$

What is the dual of $L^\infty(I,H^1(M))$? Any references? Where $H^1(M)$ is Sobolev space, and $I$ is some interval in $\mathbb{R}$, and $M$ is a compact manifold, like the $n$-dimensional torus.
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43 views

sobolev spaces integral estimation

if I have a sequence $f_k\in W_{1,p}(\Omega)$ which converge weakly to some function $ f $ and I know that $\nabla f_k-\nabla f\to 0$ in $L_{p}^{loc}(\Omega)$ I try to estimate the integral ...
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33 views

Hardy inequality punctured space

given the minimization problem: $inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$ infimum taken on all smooth functions with compact support in the ...
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28 views

How to integrate functions in n dimensional space?

I have a question about an example of functions in Sobolev space. But I think you can give a hint without knowing the Sobolev space because I just want to know how to integrate a function with ...
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41 views

Doubt about convergence of a sequence in $H^1(\mathbb{R}^3)$

Let's consider a sequence $\{f_n\}_n$ of $C^\infty_0(\mathbb{R}^3)$ complex-valued functions and suppose thet $f_n\to f$ strongly in $H^1(\mathbb{R}^3)$. What can I say about the convergence of the ...
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23 views

Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?

Let $M$ be a compact Riemannian manifold that is closed. Define $$W(0,T, H^1, L^2) = \{ u \in L^2(0,T;H^1(M)) \mid u_t \in L^2(0,T;L^2(M))\}$$ $$W(0,T, H^1, H^{-1}) = \{ u \in L^2(0,T;H^1(M)) \mid u_t ...
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54 views

Poincare inequality on a domain for functions with mean value zero

Let $C=\Omega \times (0,\infty)$ for a bounded $C^1$ domain $\Omega$. Consider a function $u \in H^1(C)$. Write $u=u(x,y)$ where $x \in \Omega$ and $y \in (0,\infty).$ Is it true that if $u \in ...
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49 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that ...
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33 views

Show that elements $u \in W^{1,\infty}(U)$ have continuous representatives

Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose that the boundary $\partial U$ is of class $C^1$. Suppose that $u \in W^{1, \infty}(U)$. I wish to prove that there exists a ...
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43 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
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16 views

what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
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39 views

Boundedness of Helmholtz projection

Let's cosider the Helmoltz projection $P$ into solenoidal subspace defined as $$P=I-\nabla div\Delta^{-1}$$ What can I say about its boundedness as an operator in $H^s(\mathbb{R^3})$?
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23 views

Relationships among different definitions of Sobolev spaces

In Tsybakov's book(Page 51), Sobolev space (or Ellipsoid) for positive smoothness parameter $s$ is defined with sequential model, i.e. the series of the Fourier coefficients is finite. On the other ...
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25 views

Compactness of Pseudo-differential Operators on $H^s(\mathbb R^n)$?

The Sobolev space of order $s\in\mathbb R$ in $\mathbb R^n$, denoted by $H^s(\mathbb R^n)$, is defined as follows: $$H^{s}(\mathbb R^n):=\{u\in\mathscr{S}^{'}(\mathbb R^n): \exists f\in ...
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62 views

Proof of weak derivatives in Evans PDE?

In the textbook of Partial differential equation of Evans. Why from $\int_U(v-\overline v)\phi dx=0$ for all $\phi \in C_c^\infty (U)$, we can get $v-\overline v=0$ a.e.? How to prove it? ...
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27 views

Proof of existence of trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ not using the Fourier transform

I'm looking for a proof of the existence of the trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ which does not use the Fourier transform. In particular, I want to prove the ...
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46 views

Is trace operator is invertible?

This question come to me when I try to find the weak solution of following problem \begin{cases} -\Delta u =f&x\in\Omega \\\ u=g&x\in\partial\Omega \end{cases} where $\Omega$ is open bounded ...
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29 views

$W^{1,p}(\Omega)$ V.S. $W^{1,p}_0(\Omega)$

Generally we know that $W^{1,p}(\Omega)$ is bigger then $W^{1,p}_0(\Omega)$ for arbitrary $\Omega\subset \mathbb R^N$ and also we have $W_0^{1,p}(\mathbb R^N)=W^{1,p}(\mathbb R^N)$. Today I found on ...
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37 views

Gigliardo-Nirenberg-Sobolev inequality for functions in $W^{k,p}$, without zero trace.

The G-N-S inequality can be stated as follows: Let $U\subset\mathbb R^d$, open bounded, with $C^1$ boundary, then for any $w\in W^{k,p}_0(U)$, $p<d$ $$\|w\|_{L^{p^*}(U)}\le C(d)\|\nabla ...
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72 views

The best constant of Poincare inequality can be determined by eigenvalue of Laplace operator

Given $\Omega\subset \mathbb R^N$ be open bounded and smooth boundary. Then for $u\in H_0^1(\Omega)$ we have well-known Poincare inequality $$ \int_\Omega \lvert u\lvert^2dx\leq C\int_\Omega ...
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49 views

The $p$-Laplacian is strongly monotone

I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$ E[u]=\frac{1}{p}\int_\Omega \lvert\nabla ...
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47 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
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81 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
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32 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
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37 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
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41 views

Counterexample for Trace operator in BV space.

Suppose $\Omega\subset R^N$ is open bounded with Lipschitz boundary. Then for $u\in W^{1,p}(\Omega)$, there exists a linear operator $T$: $W^{1,p}(\Omega)\to L^{p}(\partial\Omega)$ such that the ...
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24 views

Auxiliary Backward problems

I am reading a paper that deals with a certain finite element method. We have the weighted Sobolev space $\mathcal{W}_{\mu}$ that consists of functions for which the norm $\Vert u \Vert_{\mu}^{2} ...
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19 views

differential quotient in Sobolev space by Fourier transform

I am studying Michiel's PDE book. Here I have a question about one theorem in chapter about Sobolev space. At the beginning of this chapter, he define $\tau_y(u)(x)=u(x+y)$ and by assuming $u\in ...
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29 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
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32 views

Evans PDE mappings into better spaces

Evans PDE chapter 5.9 theorem 4 (mappings into better space), Evans wrote in the proof: In addition, $\bar{u}^{'}\in L^2(0,T;L^2(V))$ , with the estimate: \begin{equation} ...
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23 views

Show that it is a element of $(H^1(\Omega))'$

Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$. I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le ...