A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

learn more… | top users | synonyms

0
votes
0answers
134 views

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak formulation $$a(u,v)+\int\limits_{\Omega}{f(u)vdx}=0,\,\forall v\in H_0^1(\Omega)\cap ...
0
votes
0answers
45 views

Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that ...
3
votes
2answers
210 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
1
vote
0answers
79 views
+100

Regularity of a Dirichlet form

I have a question about Dirichlet form. Let $\Omega$ be an Euclidean domain of $\mathbb{R}^{N}$ and $X=\bar{\Omega}$. The measure $m$ on the Borel $\sigma$ algebra $\mathcal{B}(X)$ is given by ...
2
votes
1answer
87 views

Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? [on hold]

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
1
vote
2answers
44 views

List of tensor product spaces with uniform crossnorms

Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products $H:=\bigotimes_{j=1}^n H^{(j)}$ and ...
1
vote
0answers
34 views

Trace of $u$ on bottom edge of a square if $u_x=0$ inside the square

I want to show that: Let $\Omega =(0,1)\times (0,1)$. For $u \in H^1(\Omega)$, if $u_x=0$ a.e. in $\Omega$, then the trace of $u$ on bottom edge $y=0$, i.e., $u\left|_{y=0}\right.$, is a constant. ...
6
votes
1answer
108 views

Best constant for a trace inequality

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla ...
1
vote
1answer
49 views

The dependence of constant in a trace theorem on the diameter of domain

The trace theorem says for a nice domain, say bounded Lipchitz domain, $\Omega$, we have $$\left\Vert Tu \right\Vert_{H^{1/2}(\partial \Omega)} \leq C \left\Vert u \right\Vert_{H^1(\Omega)},$$ where ...
4
votes
1answer
260 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
2
votes
1answer
42 views

What is the function space $H^1_{m, \sigma}$?

I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order". Theorem 3.1 is the first time in their paper that the function space ...
2
votes
1answer
90 views

condition for $f \in H^{1/2} \cap C^0$

I need to show that for $f \in H^{1/2}(S^1) \cap C^0(S^1)$ we have : $$ \iint\limits_{S^1 \times S^1}{ \frac{|f(x)-f(y)|^2}{\sin^2(\pi(x-y))}dx dy}< + \infty $$ and that, conversely, if $f$ is a ...
2
votes
0answers
65 views

Getting an a priori energy estimate from PDE weak formulation

On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying $$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int ...
1
vote
0answers
63 views

How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?

Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$. Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in ...
1
vote
0answers
64 views

An H2 estimate for Helmholtz equation

How to show the following statement? Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation, $$ -\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u ...
1
vote
0answers
15 views

Approximation error estimate

I would like to find a good reference for the following or a similar, probably well-known, approximation error result: Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in ...
8
votes
2answers
591 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
0
votes
1answer
337 views

Condition to obtain a not compact embedding

I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...
3
votes
1answer
89 views

$L^p$ norm of boundary values of holomorphic function

I am looking for an estimate of the following form: Suppose that $D\subset \mathbb{C}$ is a simply connected domain. Suppose that $F$ is holomorphic and bounded on $D$ and can be holomorphically ...
0
votes
0answers
57 views

Sobolev function and capacity

It seems that there is a theorem which claims that If $u\in L_1^2(B_1\setminus T)$ and $T\subset B_1$ is of capacity zero (in some sense, since the author says $(1,2)$-capacity, of which I don't ...
1
vote
1answer
99 views

Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step ...
3
votes
1answer
227 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
1
vote
0answers
60 views

Properties of a Sobolev bound

I am interested in computing $$ A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2} $$ where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound. ...
3
votes
1answer
79 views

Sobolev inequality involving summing from $j = 0$ to $m - 2$, exists constant

Let $I = (0, 1)$ and $1 \le q < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|D^{(m - 1)}u\|_{L^q(I)} + \sum_{j = 0}^{m - 2} \|D^ju\|_{L^\infty(I)} \le ...
0
votes
0answers
40 views

trace sobolecv inequality for $q=2(n-1)/(n-2)-\varepsilon$ in half space

Can I do the following inequality, for $ u\in D^{1,2}(R^n_+)$, we have $(\int _{R^{n-1}} |u|^{2(n-1)/(n-2)-\varepsilon}dx')^{\frac{ 1}{2(n-1)/(n-2)-\varepsilon } }\leq C ( \int _{ R^n_+}| \nabla ...
2
votes
0answers
49 views

Constant in a trace Sobolev theorem for concave domains

I wonder is the following inequality is true/known: Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then $$ \int_{\partial\Omega} |u|^2 ds ...
3
votes
0answers
57 views

Modify the jump set of $BV$ function

Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
6
votes
1answer
188 views

Simplicity of eigenvalues

Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
6
votes
1answer
250 views

Chain rule for weakly differentiable functions

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...
5
votes
1answer
89 views

If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result

Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have ...
5
votes
1answer
193 views

Are Sobolev spaces on non-compact manifolds separable?

If $M$ is a Riemannian manifold that is not compact, is it true that the Sobolev spaces on $M$, $W^{k,p}(M)$, still be separable (for $p < \infty$)?
4
votes
0answers
50 views

Sobolev spaces defined on non-compact Lie groups

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...
7
votes
0answers
46 views

Continuous inclusions Sobolev theorem, question [closed]

How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...
1
vote
1answer
71 views

$L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X)$ for $X \subset Y$ dense?

is the Inclusion stated in the title true? In my case the spaces (essentially) are $X = H^1(\Omega)$ and $Y = L^2(\Omega)$ for $\Omega \subset \mathbb{R}$ bounded. My first try was to show $\lim_{t_1 ...
1
vote
3answers
140 views

Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]

Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in ...
0
votes
0answers
64 views

Boundedness of a function that satisfies a PDE-type inequality

Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$. Suppose that $$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + ...
2
votes
0answers
150 views

$L^\infty$ bound on solutions of linear parabolic equations

We work on a closed Riemannian manifold $M$. Let $u$ and $v$ be the non-negative weak solutions of $$au_t - 2d\,\Delta au = cv - f$$ $$bv_t - d\,\Delta bv = f$$ $$u(0)=u_0, \quad v(0)=v_0$$ where $f$ ...
2
votes
0answers
74 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
1
vote
0answers
49 views

Trace theorem for magnetic Sobolev spaces

I am considering a smooth vector field $A \colon \mathbb{R}^N \to \mathbb{R}^N$ and I define the magnetic Sobolev space $H_A^1$ as the closure of $C_0^\infty(\mathbb{R}^N)$ with respect to the norm $$ ...
1
vote
0answers
56 views

Wavelet characterization of Sobolev spaces

We know that there exist wavelets generating orthonormal bases in Sobolev spaces $W^{p,s}(\mathbb R^n)$, where $p$ is the index of integral and $s$ is the index of smoothness. Consider the orthonormal ...
0
votes
0answers
48 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
7
votes
1answer
496 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here: Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
5
votes
1answer
79 views

Trace spaces on convex polyhedra: compatibility conditions at edges

Let $\Omega$ be a convex polyhedron in $\mathbf{R}^3$ with boundary $\partial\Omega$ consisting of $N$ polygons $\{\Gamma_j\}_{j=1}^N$. It's well known that in general $$ H^s(\partial \Omega) \neq ...
3
votes
0answers
97 views

Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
2
votes
0answers
120 views

$L^\infty-L^1$ smoothing effect for the heat equation

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ where $u_0$ is bounded initial data. Here ...
2
votes
0answers
50 views

Doubts regarding pre-compactness of bounded sequence of measure valued functions

Given: $\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ . ...
7
votes
1answer
853 views

Isoperimetry and Poincaré Inequality

What are the known relations between isoperimetric and Poincaré inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
2
votes
1answer
69 views

Convergence of energy of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by \begin{equation} f_r(x):= \begin{cases} ...
1
vote
2answers
129 views

Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$. Question 1: Can we ...