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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13
votes
1answer
359 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 ...
11
votes
1answer
515 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
157 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
107 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
52 views

Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
3
votes
1answer
123 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
56 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
152 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
51 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?
2
votes
1answer
67 views

Proving $u\mapsto |u|^2u$ is Lipschitz on bounded subsets of $H^2(\Omega)\cap H_0^1(\Omega).$

I'm reading a paper and am stumped verifying two details. Let $\Omega$ be a bounded region in $\mathbb{R}^2$ with smooth boundary. I'd like to show that the map $u\mapsto |u|^2u$ is a map from ...
2
votes
1answer
24 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
2
votes
1answer
30 views

Functions of Sobolev space with asymptotic decay

Define a subspace of the Sobolev space $H^1(\mathbb R^d)$ to be $$ X=\{u\in H^1(\mathbb R^d), |u(x)|=O(|x|^{1-d}), \text{ as } |x|\rightarrow +\infty\} $$ Is there a norm $\|\cdot\|_X$ such that $(X, ...
2
votes
1answer
78 views

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
2
votes
1answer
69 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
2
votes
1answer
99 views

An example of self-adjoint and positive operator

Let $\Omega\subset R^2 $ and let $H=H_0^1(\Omega)\cap H^2(\Omega)$ with inner product : $\langle u,v\rangle_H =\langle u,v\rangle_{L^2(\Omega)} + \langle\Delta u,\Delta v\rangle_{L^2(\Omega)}$. I am ...
2
votes
1answer
148 views

Dual space norms and equivalence

Suppose $S(r)$ is set parametrised by $r \in [0,T]$. Let $\phi_t^s : H^1(S(t)) \to H^1(S(s))$ is a linear homeomorphism. Suppose $\lVert \cdot \rVert_{H^1(S(t))}$ and $\lVert \phi_t^s(\cdot) ...
2
votes
1answer
181 views

About Sobolev Embedding Theorem

I want to know that how the statement below holds. The statement : There exists a constant $C = C(s)$ such that the continuous embedding of $W^{s,2}$ into the space of uniformly bounded, continuous ...
1
vote
1answer
33 views

Two theorems about approximation by smooth functions

Let $U$ be an open subset of $\mathbb{R}^{n}$. The following are two theorems taken from the chapter about Sobolev Spaces of the Evans' book. Theorem 1 Assume $u\in W^{k,p}(U)$ for some $1\le ...
1
vote
1answer
34 views

tempered distribution and sobolev spaces

The Schwartz space $\mathcal S(\mathbb R^d)$ is the set of all complex-valued function $f \in C^{\infty}(\mathbb R^d)$ such that $\sup_{x\in \mathbb R^d}|x^{\alpha}D^{\beta}f(x)|<\infty$ where ...
1
vote
1answer
29 views

poincare-sobolev inequality

How can we prove this inequality? For $q=\frac{np}{n-p}$ and $1\leq p<n$, there is a constant $c=c(n,p)$ such that if $u\in W^{1,p}(B_r)$, then ...
1
vote
1answer
44 views

Completeness of Sobolev space constructed from seminorm

Define $W^{p,r}(\mathbb{R}^d):=\{f\in L^p(\mathbb{R}^d) : D^\alpha f\in L^p(\mathbb{R}^d), \forall 0<|\alpha|\le r\}$ where $1\le p\le\infty$. Let the seminorm on $W^{p,r}(\mathbb{R}^d)$ be ...
1
vote
1answer
95 views

Hardy-Littlewood-Sobolev fractional integration inequality fails at endpoints

Here's a version of the theorem: If $1 < p, r < \infty$ and $ 0 < \alpha < n $ be such that $ \frac{1}{p} + \frac{ \alpha }{ n} = \frac{1}{r} + 1 $. Then for any $ f \in L^p ( \mathbb R ^n ...
1
vote
1answer
43 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 ...
1
vote
1answer
28 views

Extension of a function from the edge.

How can I extend the function $f\in W^{1-\frac{1}{p},p}(\mathbb R\times\lbrace0\rbrace)$ up to $g\in W^{1,p}(\mathbb R\times(0,\infty))$?
1
vote
1answer
68 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
0
votes
0answers
24 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
35 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
27 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
42 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
43 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
70 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
16 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
37 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
25 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
50 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
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
44 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
84 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
70 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
110 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
81 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 ...
-1
votes
0answers
5 views

Examples on Weak derivatives and Sobolev Spaces

I know that $u(x)=\log\log(1+|x|^{-1})$ is unbounded function.But how to show that $u$ $\epsilon$ ${W^{1,n}(B(0,1))}$ for $n\geq1.$ I know that the weak derivatives $D_{j}u$ for ...
-1
votes
0answers
26 views

Inequality in $H^2$

I have tried to prove this result, but it seems too hard. Need Help. Let $U\subseteq\mathbb{R}^n$ a bounded set with smooth boundary, and the differential operator: ...