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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1answer
44 views

if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then$ f\in H^k (\Omega')$ [closed]

prove that if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then $f\in H^k (\Omega')$
-1
votes
1answer
78 views

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$? Justify your answer! Thanks.
0
votes
1answer
60 views

Mean value theorem in sobolov space under integral

Sorry this seems like a basic question, but I'm having trouble figuring out the answer. Let g(x) be the step function over [-1,1] and f(x) a function with $f\in H^1[-1,1]$, that is it has a square ...
2
votes
0answers
56 views

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
1
vote
1answer
87 views

Sobolev spaces - embedding - exercise

I have to show that $H^{2}(\Omega) \subset \subset H^1(\Omega)$. I think the Arzela - Ascoli theorem can help.. .I dont know how to start this exercise . I am beginner in Sobolev spaces.. someone can ...
5
votes
1answer
207 views

Proving an alternative norm on Sobolev space is equivalent to usual norm

I have this exercice and my problel is only in item 4, and i will desespere. Let $f \in L^2(\mathbb{R}^n).$ 1- Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique ...
5
votes
2answers
146 views

Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?

If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
6
votes
1answer
201 views

Why are weak solutions to PDEs good enough?

Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be ...
1
vote
1answer
107 views

Use difference quotient with not uniform bound to appoximate weak derivative

Suppose U is an open set,not necessarily bounded or has Lipschitz boundary, $f\in L^p(U)$ ,define the difference as usual: $$D^h_i f=\frac{f(x+he_i)-f(x)}{h},\ \ \forall x\in U'\subset\subset U$$ ...
1
vote
0answers
27 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 ...
1
vote
1answer
122 views

How to prove $\int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v$ is an inner product in $H^1$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
2
votes
2answers
95 views

Sobolev space boundary value in PDE

I often read this: Let $\Omega$ be a open bounded set. There is a unique $u \in H^1(\Omega)$ such that $$-\Delta u = f \text{ on $\Omega$}$$ $$u|_{\partial \Omega} = g$$ But how can we write ...
3
votes
3answers
239 views

How to prove that $(u-v)^+\in W_0^{1,2}(\Omega)$, if $u\in W_0^{1,2}(\Omega)$, $v\geq 0$.

Let $\Omega$ denote a open subset of $\mathbb{R}^n$, and $W^{1,p}(\Omega)$ the Sobolev space of weakly differentiable functions $u\in L^p(\Omega)$ (that is, for which $D_iu$ exists and belongs to ...
1
vote
2answers
84 views

When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?

Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth. Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
1
vote
1answer
145 views

sobolev space-equivalence of scalar product

Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prove that there exist a constant $C \geq 0$ ...
0
votes
1answer
90 views

Fourier transform in $L^2$

I have a function $\phi\in L^2(\mathbb{R}^3)$ and I know that its Fourier transform satisfies the following equation: $$(p^2-A)\hat{\phi}(p)=Q\frac{A+\lambda}{p^2+\lambda}$$ where $Q$ is a constant, ...
2
votes
1answer
598 views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
0
votes
2answers
78 views

Laplacian inequality in Sobolev space

Is the following assertion true? For all $\alpha>0$ there exists some $\theta \in H^2(\Omega)\cap H_0^1(\Omega)$ such that $\|\frac{\Delta \theta}{\theta}\|_\infty \le \alpha.$ Thanks!
2
votes
2answers
213 views

Examples of truly abstract evolution PDEs?

Let $V \subset H \subset V^*$. Consider the parabolic PDE $$y' = A(t)y + f$$ which is found in many books. Usually under some assumptions on $A(t)$ and $f$, there is a solution $y \in L^2(0,T;V)$ with ...
1
vote
1answer
32 views

sobolev spaces on cartesian products

We have a canonical isomorphism $$ C^0(X,C^0(X,Y)) \simeq C^0(X \times Y, Z)$$ given by $f \mapsto \hat{f}$, where $\hat{f}(x,y) = (f(x))(y)$. Is there a similar statement for Sovolev space? For ...
0
votes
1answer
43 views

Elimination of a singularity

Let $z=a+ib$ with $b>0$; the function $$f(x)=\frac{e^{iz|x|}}{|x|}$$ is in $L^2(\mathbb{R}^3)$; in fact ...
1
vote
0answers
106 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. ...
0
votes
1answer
88 views

Sum of Banach spaces

Let $H^2(\mathbb{R}^3)$ the usual Sobolev space and consider the following set $$X=\bigg\{u\bigg|u=\phi+\frac{Q}{|x|},\phi\in H^2,\,\, Q\in\mathbb{C}\bigg\}$$ I observe that the decomposition is ...
0
votes
0answers
67 views

Supremum of norms of line integrals

I have the following problem: Let's say $\Omega\subset\mathbb{R}^2$ is a bounded open connected set with Lipschitz boundary and $f\colon\Omega\rightarrow\mathbb{R}$ is a function such that $f\in ...
3
votes
2answers
108 views

Boundedness of functions in $W_0^{1,p}(\Omega)$

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Suppose that $u\in W_0^{1,p}(\Omega)$ and $u$ is locally essentially bounded. Does this implies that $u$ is globally ...
2
votes
1answer
69 views

Laplacian inequality in $L^\infty$

Let $\Omega$ be a bounded domain of $R^n$ and let $y\in H^2(\Omega)\cap H_0^1(\Omega)$ such that the set $[x\in \Omega/ y(x)\ne 0]$ has non nul measure and $ \; \frac{\Delta y}{y} 1_{\{x\in \Omega/ ...
3
votes
0answers
108 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 ...
1
vote
1answer
35 views

Does this embedding holds?

Could you please tell me whether the following continuous embedding holds? $L^1(R^d)\hookrightarrow H^{-s}(R^d)$ $(s>\frac{d}{2})$
1
vote
1answer
52 views

How to show this equation holds for all $H^1_0$ by density? Don't understand the proof.

I read this: Suppose $a$ is a bounded (in $H^1_0$) coercive bilinear form and it holds that $$\langle u_t, w \rangle + a(u, w) = 0$$ for all $w \in E_M$, where $\cup_{M \in \mathbb{N}} E_M$ ...
3
votes
1answer
474 views

Trace inequality

Could you please give me a hint on how to prove the following inequality $$\|u\|_{L^2(\Gamma)}\le C\|u\|^{\frac12}_{L^2(\Omega)}\|u\|^{\frac12}_{H^1}, \quad \forall u\in H^1.$$,
1
vote
0answers
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 ...
1
vote
0answers
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 ...
2
votes
1answer
66 views

Inequality in Sobolev Space

Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that $ | \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| ...
2
votes
1answer
216 views

Estimating Poincare constant for unit interval

I want to show that the Poincare constant for the $W^{1,2}_0(0,1)$ is smaller than $1$. More specifically, I want to show that there is a constant $C<1$ such that for any $f\in C^\infty_c(0,1)$ ...
2
votes
1answer
68 views

Doubt about Sobolev space norm

I consider the space $H^2(\mathbb{R}^3)$. I have a function and I have to verify that it belongs to this space. In the text I'm reading the author verifies that the function and its Laplacian are in ...
3
votes
1answer
288 views

Is $H^1_0(\Omega)$ dense in $L^2(\Omega)$?

Is $H^1_0(\Omega)$ dense in $L^2(\Omega)$ for bounded domains? It is true for $H^1$ functions of course but what about this subset? Sorry for the elementary question but I never see this so I think ...
2
votes
1answer
45 views

Why the properties of $W^{m,p}(\Omega)$ cannot be extended to $W^{m,p}(\mathbb{R}^d)$

The text books on the topic of Sobolev Spaces and PDE etc., they treate the case $W^{m,p}(\Omega)$ with $\Omega\subset \mathbb{R}^d$ and $\Omega = \mathbb{R}^d \text{ or }\mathbb{R}^d_+$ separately. ...
0
votes
1answer
102 views

Does elliptic regularity result depend on boundary conditions?

Let $\Omega$ be a domain with boundary $\partial\Omega$. Suppose I am given a weak formulation: $$b(u,v) = (f,v) \quad\forall v \in H^1(\Omega)$$ Assuming $b$ is nice enough, does the elliptic ...
2
votes
1answer
290 views

Using Galerkin method for PDE with Neumann boundary condition?

I am wanting to show existence of solutions to $$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$ with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ...
2
votes
1answer
137 views

Is this set dense in $H^1(\Omega)?$

Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$ dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$ Here $f$ is some linear functional so that $V_1$ is also ...
1
vote
1answer
129 views

Poincare inequality on $H^1_0(M)$

Is it possible to deduce the Poincare inequality for functions in $H^1_0(M)$ from the Poincare inequality for functions in $H^1(M)$ with mean value 0? $M$ is a hypersurface with non-empty boundary.
1
vote
1answer
78 views

Is the gradient of a function in $H^2_0$ in $H^1_0$?

Suppose we have $f\in H^2_0(U)$, so $f$ is the limit of some sequence $(g_n)$ of smooth compactly supported functions on $U\in\mathbb{R}^n$ (assume bounded & smooth boundary) and $f$ is in the ...
4
votes
1answer
272 views

Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space

Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case? ...
2
votes
1answer
128 views

Proving that a weak solution of a BVP satisfies the boundary condition

I am given the smooth function $u$ which satisfies $\int_U (\nabla u \cdot \nabla v +uv)\,dx = \int_U fv\,dx$ for all functions $v$ in the Sobolev space $H^1(U)$, where $f\in ...
1
vote
0answers
64 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 ...
2
votes
1answer
44 views

Regularity and the Varitational Inequality

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
2
votes
1answer
120 views

Does $Du=0$ a.e. implie $u=c$ a.e.?

Let $W^{1,p}(U)$ be the Sobolev space. Suppose that $U$ is connected bounded domain in $\mathbb{R}^n$ and $u \in W^{1,p}(U)$ satisfies $Du=0$ a.e. in $U$. How can I prove that $u$ is constant a.e. in ...
2
votes
0answers
89 views

A question on weakly convergence and norm convergence.

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
2
votes
1answer
93 views

A question on a bounded sequence in $H^1(\mathbb{R}^n)$.

Let $r>0$, and $2 \le q \le 2^*$. Suppose that $\{u_k\}_k$ is a bounded sequence in $H^1(\mathbb{R}^n)$ and $\lim_{k\to \infty} \sup_{y \in \mathbb{R}^n} \int_{B_r(y)}|u_k|^q dx \rightarrow 0.$ ...
6
votes
2answers
230 views

Trace regularity result $\lVert n \times u\rVert_{H^{-1/2}}$

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...