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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2
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97 views

Compactness of an embedding between weighted spaces

I read somewhere that if: $N\geq 2$ is an integer, $p\in ]1,N[$, $r>N/p$, $m\in L^r(0,a)$ (with $a>0$) and $m>0$ a.e. in $(0,a)$, then the weighted Sobolev space $W^{1,p^\prime} ...
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88 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
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299 views

Adjoint of multiplication operator on Sobolev space

This is sort of an idle question, and I'll admit I didn't think very hard about it. Let $H^1 = H^1(\mathbb{R}^n)$ be the Sobolev space with norm $||f||_{H^1}^2 = ||f||_{L^2(\mathbb{R}^n)}^2 + ...
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15 views

Can one explicitely construct a sequence of functions of compact support approximating $u\in W_{0}^{1,p}(\Omega)$?

We define $W^{1,p}_{0}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the $W^{1,p}$-norm (or equivalently as the closure of the $W^{1,p}$-functions with compact support). Given $u\in ...
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23 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
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26 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
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30 views

Minimizing sequences and topology (direct method)

To show the importance of the choice of the topology for the direct method we have been assigned the following exercise which I've not been able to solve due to my lack of understanding on how strong ...
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26 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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47 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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13 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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67 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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43 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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45 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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49 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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36 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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46 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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67 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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51 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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0answers
29 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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26 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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56 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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50 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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34 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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45 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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0answers
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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40 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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24 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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26 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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63 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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49 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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30 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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79 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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0answers
48 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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82 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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0answers
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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0answers
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 ...