For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

12
votes
1answer
343 views

$\tilde{u}=0,\ a.e.\ x\in\Gamma$?

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in ...
10
votes
1answer
467 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
5
votes
1answer
144 views

the basis for the Sobolev space $H^1_0([0,1],\mathbb{R})$

According to the Sturm-Liouville theorem, for any continuous function $p\in\mathcal{C}^0([0,1],\mathbb{R})$, there is a Hilbert basis (normlised) $(\psi_n)_{n\geq1}$ of $L^2([0,1],\mathbb{R})$ such ...
4
votes
1answer
76 views

First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
4
votes
1answer
50 views

Existence of variation

Let $I[w] =\int_U L(Dw,w,x) dx$. Let $1<q<\infty$, and there exist constants $\alpha>0$,$\beta\ge0$ such that $$L(p,z,x)\ge \alpha |p|^q - \beta$$ This implies that if $I[w]$ exists, $$I[w] ...
3
votes
1answer
98 views

Differences between $-\Delta: H_0^1(\Omega)\to H^{-1}(\Omega)$ and $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$

I'll try to explain what I want to know: Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. When we look to $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$, the meaning of $-\Delta$ is ...
3
votes
1answer
46 views

Minimality in the case of partial derivatives and Sobolev spaces?

I am trying to understand this question here that considers Sobolev spaces apparently and hence partial derivatives. What is the definition of minimality there? Is the minimality defined by ...
3
votes
1answer
50 views

Nonlinear parabolic PDEs, what methods/techniques for existence?

I am curious what kinds of techniques one uses to show existence of PDEs with nonlinearities. I am aware of: 1) Minimisation problems 2) Semigroup (both of which I'd like to avoid) For linear ...
3
votes
1answer
136 views

Time dependent or Bochner space references?

Does anyone have any recommendations where I can learn about time dependent or Bochner spaces? I mean spaces like $L^p(0,T; H^{-1}(\Omega))$. I think one needs some knowledge of distributions, so any ...
2
votes
1answer
35 views

What is the dual of $A\cap B$

I encountered with some elliptic problem which admits a variational formulation in terms of space $X$ and I need to understand its dual. Suppose that $2<p<\infty$, $\Omega\subset {\mathbb R}^d$ ...
2
votes
1answer
41 views

Discontinuous function in $W^{1, 1}(\mathbb{R}^{2})$

What's an example of a bounded function in $W^{1, 1}(\mathbb{R}^{2})$ which is discontinuous? Can this function be discontinuous on a set of positive measure?
0
votes
0answers
26 views

Dual of a Sobolev space (ex)

Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$. 1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$? 2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$? For the first ...
0
votes
0answers
26 views

injection in H^{-1}

let $\Omega$ an open on $\mathbb{R}^n$. if $f \in H^{-1}(\Omega)$ and $g \in L^1(\Omega)$. Who is the Sobolev space $V$ who can contains $f-g$ such as $V$ is injected on $H^{-1}(\Omega)$? Thanks for ...
0
votes
0answers
13 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
0
votes
0answers
28 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
0
votes
0answers
34 views

Convergence in $L^2$ of difference quotients to derivative of function in $H^1$

Is it true that if $u\in H^1({\mathbb R})$, then $(u(x+h)-u(x))/h$ converges to $u'(x)$ in $L^2({\mathbb R})$, as $h\to 0$? It's hard for me to get a handle on this, since $u'$ doesn't have to be ...
0
votes
0answers
8 views

How to check whether f belongs to H^(\beta+t)

In the Lemma 3 of the paper"Wendland H. Multi scale analysis in Sobolev spaces on bounded domains" How to check whether f belongs to H^(\beta+t)? Thank you!
0
votes
0answers
62 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
0
votes
0answers
15 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
0
votes
0answers
32 views

Trouble understanding proof of the Poincare Inequality

In the proof of the Poincare Inequality in this article on page 4 the domain of integration is divided into two parts, $v< 0$ and $v>0$. I think this has to do with the absolute norm appearing ...
0
votes
0answers
35 views

Boundary value problem for two functions

The question is: Let $\mathcal{H}=H_0^1(\Omega)\times H^1(\Omega)$ and consider the solution $(u,v)\in\mathcal{H}$ to the differential problem \begin{equation} -\Delta u=f+a(v-u)\quad\text{in }\Omega ...
0
votes
0answers
26 views

Why do the following regularisations of a function in Sobolev space exist?

Suppose $v_1, v_2$ satisfy $\mu \leq v_1(x,t), v_2(x,t) \leq M$ a.e. in $Q:=\Omega\times(0,T)$ and $$(v_1, \eta_1) \quad\text{and} \quad (v_2, \eta_2) \in L^2(0,T;H^1(\Omega)) \cap L^2(Q).$$ Define ...
0
votes
0answers
85 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
0
votes
0answers
26 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
0
votes
0answers
30 views

inequality for linear functions in sobolev space

Ist the following Statement true for $f$ and $g$ linear? $\vert fg \vert_{H^2} \leq C \Vert f \Vert_{H^1} \Vert g \Vert_{H^1}$, where $\vert \cdot \vert_{H^2}$ denotes the seminorm. My Idea: It is ...
0
votes
0answers
47 views

Sobolev spaces in polar coordinates

I need some properties about Sobolev spaces in polar coordinates. To be precise, let $U = \{(x,y)\in\mathbb R^2 : x^2 + y^2 < R\}$ be an open disc and let $H_0^1(U)$ be the usual Sobolev space ...
0
votes
0answers
22 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
0
votes
0answers
22 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
0
votes
0answers
32 views

Simple question about the Gagliardo Niremberg interpolation inequality

Consider the Gagliardo Niremberg interpolation inequality : (Gagliardo Niremberg interpolation inequality)Let $q,r$ be any numbers satisfying $1 \leq q, r \leq \infty $ and let $j,m$ be any ...
0
votes
0answers
25 views

Showing a subspace of a Hilbert space is also Hilbert (please check my proof)

Let $V \subset H \subset V^*$ be a Hilbert triple. Let $$W = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*)\}$$ and let $$W_T = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*) \text{ and } u(0)=u(T)\}.$$ ...
0
votes
0answers
38 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
0
votes
0answers
40 views

Laplace equation with Dirichlet homogeneous BC

I'm studying the Laplace equation with homogeneous Dirchlet boundary conditions that is, $$ -\triangle u + u = f \quad \text{in } \Omega \quad u=0 \quad \text{on } \Gamma $$ With $\Omega$ an open ...
0
votes
0answers
62 views

Estimates for linear finite element nodal basis functions

Let $\Omega\subset\mathbb R^2$ be a domain with $\operatorname{diam} \Omega=H$ and $\mathcal T^h$ be shape-regular triangulation of $\Omega$ with triangular $\mathcal P^1$-elements. (That means we are ...
0
votes
0answers
14 views

Bessel Potential spaces

Let $\Omega_1,\Omega_2 \subset \mathbb{R}$ be bounded. The mapping $F: \Omega_1 \rightarrow \Omega_2$ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||(DF(x))^{-1}||$ ...
0
votes
0answers
28 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
0
votes
0answers
21 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
0
votes
0answers
45 views

A Sobolev norm inequality

Assume $\mu\in H^{1}$ is compactly supported in $\Omega$ and $\Delta_{g}\mu\in H^{m}$ for some $m\ge 0$. Then if $K\subset \Omega$, there is a constant $C=C(K,m)$ such that $$|\mu|_{H^{m+2}(K)}\le ...
0
votes
0answers
29 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
0
votes
0answers
50 views

Chainrule: “piecewise smooth” and Sobolev functions

since we had a lengthy discussion on my last question (Generalized chainrule for Sobolev functions with a cut-off I didn't find an answer to it yet, I'm going to post a related more specific question ...
0
votes
0answers
34 views

estimation in elliptic forms

I obtained before the in equality $$\left\|u\right\|_{H^{1}}\left\|\phi\right\|_{L^{2}}\leq \left\|F\right\|_{L^{2}}\left\|\phi\right\|_{L^{2}} \\ \left\|u\right\|_{H^{1}}\leq ...
0
votes
0answers
42 views

Definition of global weak solution to PDE

What is the definition of a global weak solution to a parabolic PDE? Is it a solution $u \in L^2_{loc}(0,\infty;V)$ with $u' \in L^2_{loc}(0,T;V')$ or is it a solution $u \in L^2(0,\infty;V)$ with ...
0
votes
0answers
67 views

Sobolev maps between manifolds.

Let $M, N$ be smooth compact Riemannian manifolds. I have a reference that defines the $k$th Sobolev space of maps from $M$ to $N$, denoted $H^k(M, N)$, by saying that one only needs to check that ...
0
votes
0answers
53 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 ...
0
votes
0answers
66 views

In what Sobolev classes are the following functions

I need a little help. In what Sobolev classes are the following (give the answer for both $H^{s}$ and $H_{\mathrm{loc}}^{s}$) a. $\delta(x)$ b. $H(x)=\left\{\begin{matrix} 1, x\geq 0\\ 0, x<0 ...
0
votes
0answers
106 views

Variational problem - continuity exercise

Let $\Omega$ a bounded connected open regular set, and let $f \in L^2(\Omega)$. We consider the variational problem: find $u \in H^1(\Omega)$ such: $$\displaystyle\int_{\Omega} A \nabla u \cdot \nabla ...
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 ...
0
votes
0answers
96 views

Integrating $\ln(1+|\ln|x||)$ in $B_1(0)$

I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$ Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and ...
0
votes
0answers
79 views

Compact embedding theorem of $W^{k,p}(R^n)$?

Is there some kind of function space $X(R^n)$ which satisfies the compact embedding relation as follows: $W^{k,p}(R^n)\hookrightarrow\hookrightarrow X(R^n)$? Could I guess the indeterminate function ...
0
votes
0answers
56 views

Is this family of projections $\nu\mapsto P_\nu$ Lipschitz continuous?

For $\nu\in (\epsilon,1)$ with $0<\epsilon<1$, let $P_\nu:H_0^1(\Omega)\rightarrow H_0^1(\Omega)$ with $\Omega\subset \mathbb{R}^N$ bounded Lipschitz domain, be the projection operator onto the ...
-2
votes
0answers
20 views

Sobolev space: $\vert u\vert^{p-2}u $ belongs to $u\in W^{1,p'}(\Omega)$

Show that if $p\geq2$ and $u\in W^{1,p}(\Omega)$ then the function $\vert u\vert^{p-2}u $ belongs to $u\in W^{1,p'}(\Omega)$