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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6
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92 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
6
votes
0answers
98 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 ...
5
votes
0answers
43 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
5
votes
0answers
111 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
5
votes
0answers
91 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
5
votes
0answers
112 views

Is this function in the Sobolev space $H^{2,-s}(\mathbb{R}^3)$?

I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order ...
5
votes
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98 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
5
votes
0answers
92 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...
4
votes
0answers
79 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
4
votes
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106 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 ...
4
votes
0answers
79 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 $u \in C^\infty(\Omega)$ ...
4
votes
0answers
128 views

Show that the following $u\in L^{\infty}\cap H^1(B)$ is a weak solution to the given system.

Let $B=B_{\exp(-2)}\subset\mathbb{R}^2$. I would like to show that a weak solution to the following system (in $B$): \begin{align*} \triangle u_1&=-2|Du|^2(u_1+u_2)/(1+|u|^2)\\ \triangle ...
4
votes
0answers
72 views

Extension of Rellich's theorem: Embedding “sort of” compact in the limit case?

Let $\Omega$ be a domain with a nice boundary (i.e., smoothness of the boundary shall not be central to my question). Now it well-known that the embedding $\iota \colon W^{1,p}(\Omega) \to ...
4
votes
0answers
180 views

Existence of weak solution in Sobolev space $W^{1,p}$

Let $B\colon W^{1,p}\times W^{1,q}\rightarrow \mathbb{R}$ and $\frac{1}{p}+\frac{1}{q}=1$ , $p>2$ where $$B[u,v]=\int \left( Du \,Dv+ h u v \right)\, dx$$ where $h>0$. I am interested to prove ...
3
votes
0answers
32 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
3
votes
0answers
32 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
3
votes
0answers
138 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
3
votes
0answers
49 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
3
votes
0answers
44 views

Regularity of a Weak Solution

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
3
votes
0answers
46 views

Missing step in Evan's PDE?

In the attached proof, I am having trouble justifying the presence of "$x_n^{p-1}$" in eh line immediately below "Thus" and "Letting $m\to\infty$". I understand he uses Minkowski's inequality, and ...
3
votes
0answers
43 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
3
votes
0answers
116 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
3
votes
0answers
55 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
3
votes
0answers
57 views

Sobolev Spaces and Convergence

I have a question about one of my homework question. I have been struggling for a while and I really need some help. Assume $N>2$ and $u_k$ is a bounded sequence in $W^{1,2}(\mathbb{R}^N)$ ...
3
votes
0answers
71 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
3
votes
0answers
55 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
3
votes
0answers
78 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
3
votes
0answers
60 views

$W^2_p$ regularity of solutions of linear elasticity

I want to prove the following statement: Let $\Omega$ be bounded, polygonal, and convex, then for the solution of the linear elasticity (elliptic) equation with inhomogenous Dirichlet boundary ...
3
votes
0answers
97 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
3
votes
0answers
62 views

What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
3
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0answers
70 views

A “straightforward” inequality.

In section 5.5 page page 117 of Fanghua, Quing's book, in order to obtain $W^{2,p}$ estimates the idea is to show that for $p \in (1,\infty]$ the condition $\theta \in L^p(\Omega)$ implies $D^2u \in ...
3
votes
0answers
134 views

Checking initial condition of PDE is satisfied in Galerkin method

The following is an argument I read, it is part of a proof of checking that the solution given by the Galerkin method satisfies the initial conditions. From our PDE $$\langle u', v \rangle_{V',V} + ...
3
votes
0answers
45 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
3
votes
0answers
72 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
3
votes
0answers
192 views

Weak solution to PDE boundary value problems

Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset ...
3
votes
0answers
112 views

A finely open set, not open up to polar set?

Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, which is not open up to a polar set (i.e. zero capacity), i.e., there does ...
3
votes
0answers
110 views

invertible operator Sobolev space

Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
3
votes
0answers
165 views

Is the functional $F:H^{1}_{0}(\Omega) \longrightarrow \mathbb{R}$ given below weak lower semicontinuous?

Is the functional $F:H^{1}_{0}(\Omega) \longrightarrow \mathbb{R}$ given by \begin{equation} F(u) = \int_{\Omega} \langle (A_1(x)\chi_{\{u>0\}}+A_2(x)\chi_{\{u\le0\}}) \nabla u, \nabla u ...
3
votes
0answers
100 views

Truncation in singular integrals

After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ ...
3
votes
0answers
221 views

Why is this functional coercive?

Let $h\in L^2(0,1)$, such that $|\int h \mathrm{d}x|<1$. On the space $H^1(0,1)$, consider $$J(v)=\frac{1}{2}\int_0^1 v'^2\,\mathrm{d}x+\int_0^1\sqrt{1+v^2}\,\mathrm{d}x-\int_0^1hv \,\mathrm{d}x.$$ ...
2
votes
0answers
25 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
2
votes
0answers
30 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
2
votes
0answers
31 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
2
votes
0answers
53 views

Positive part of $y$ with $y\in L^2(0,T; H_0^1(\Omega))$ and $y'\in L^2(0,T; H^{-1}(\Omega))$

Let $\Omega \subset \mathbb R^n$ be a domain, sufficiently smooth. Let $T>0$. Define the space $W(0,T)$ by $$ W(0,T) = \{ y \in L^2(0,T; H^1_0(\Omega)): \ y'\in L^2(0,T;H^{-1}(\Omega)),\ $$ where ...
2
votes
0answers
25 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} ...
2
votes
0answers
59 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
2
votes
0answers
29 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
2
votes
0answers
29 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
2
votes
0answers
51 views

Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset ...
2
votes
0answers
49 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...