Questions tagged [sobolev-spaces]

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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Gauss - Green theorem for Sobolev $H^1$ space

I know the Gauss-Green theorem: Let $U \subset \mathbb{R}^n$ be an open, bounded set with $∂U$ being $C^1$. Suppose $u ∈ C^1(\bar U)$, then $$∫_U u_{x_i} dx = \int_{∂U} u \nu^i dS,$$ where $\nu=(\...
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32 views

Compactness of Sobolev Space

Let $\Omega \subset \mathbb{R}$ be a bounded domain and $2<p<\infty$. Assume $u \in C^{2,1}(\Omega \times (0,\infty))\cap C^{1}((0,\infty);L^{2}(\Omega))\cap C([0,\infty);H_{0}^{1}(\Omega))$ ...
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Extension of weak derivatives in Bochner spaces

I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans: He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In ...
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29 views

Limit cases in sobolev embedding

I am trying to find coumterexamples for the critic cases in sobolev embedding theorem for all $\mathbb{R}^N$. For instance, a function $u\in W^{1,p}(\mathbb{R}^N)$ s.t. $u\notin L^q(\mathbb{R}^N)$ for ...
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Existence of convergent sequence implies convergence of function as $t\to\infty$

Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in C([0,\infty);H_{0}^{1}(\Omega))$. Assume there exists a sequence $\{t_{n}\}_{n\in\mathbb{N}}$ such that $t_{n}\to\infty$ and $||u(t_{n}...
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29 views

Isometric embedding of $L^2$ onto $H^{-1}$

Let $X$ be a Banach space. Many sources in the literature identify $L^2(X)$ with $H^{-1}(X)$ through the identification $$ \varphi: L^2(X) \to H^{-1}(X); \quad \quad \varphi(u)(v) := (u,v)_{L^2}, \...
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40 views

Intuition Sobolev spaces and smoothing splines

With inputs $X_1, \dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $\hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of ...
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79 views

Confusions in Evans book regarding weak derivatives in Banach spaces

I am studying PDE using Evans' book and I have two main confusions (probably stupid questions to experts) regarding weak derivatives in Banach spaces. First confusion: $\def\u{\mathbf u}$ $\def\v{\...
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$ W_{0}^{2}(\Omega)=\{ f\in W_{0}^{1}(\Omega):\Delta f\in L^{2}(\Omega)\}? $

Let $\Omega\subset\mathbb{R}^{n}$ be an open bounded domain. Let $W^{2}\left(\Omega\right)$ be the usual Sobolev space $$ W^{2}\left(\Omega\right)=\left\{ f\in L^{2}\left(\Omega\right):f,\partial_{i}...
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Existance of $\phi \in L^2$ such as $L(\vec v)=\int_{\Omega}\phi \;\text{div} \vec v \;dx$

[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere] Let $\Omega \in \mathbb{R}^N$ an open bounded connected set such as $\partial \Omega$ is $\mathcal{C}^1$. And ...
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37 views

what is $W^{-1,2}$?

I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet. Let $\Omega$ be a simply connected domain in $\mathbb R^n$...
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42 views

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$). So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous ...
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Different topologies in Sobolev space $W^{1,p}$

In paper [1], L.Ambrosio talks about the space $W^{1,p}(\Omega),\ 1\leq p<+\infty$ endowed with four different topologies: The strong topology, denoted by $W^{1,p}(\Omega)$. The weak topology, ...
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30 views

Fundamental theorem of calculus for function composition of Lipschitz functions

Let $\Omega$ be a bounded open subset with $C^2$ boundary, $f:\mathbb{R}^d\to \mathbb{R}$ be a Lipschitz function, and $u,v:\Omega\to \mathbb{R}^d$ be measurable functions such that $u_i,v_i\in L^2(\...
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38 views

Divergence operator: is it a linear continuous map on Sobolev Spaces?

Let $\Omega $ be a bounded open set in $\mathbb{R^n}$ Let $H_{}^{1}(\Omega)=\{v\in L^2(\Omega) ; \frac{\partial v}{\partial x_i}\in L^2(\Omega)\}$ and $H_{0}^{1}(\Omega)=\{v\in H_{}^{1}(\Omega) ; v=...
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27 views

Weak convergence and compacity

Please I dont understand this. I have: $ \parallel \nabla m_n \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C$ $ \parallel \frac{\partial m_n}{\partial z} \parallel_{L^{\infty}(\mathbb{R}^+,...
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29 views

Weak Convergence from Strong Convergence

Let $\Omega \subset \mathbb{R}$ be a bounded domain, $v \in H_{0}^{1}(\Omega)$ such that $||u_{n}-v||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$ for a bounded sequence $\{u_{n}\}_{n\in\mathbb{N}} \...
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Confirmation of convergence in subdomain

Let $\{u_{n}\}_{n\in\mathbb{N}} \subset L^{p}(\Omega)$ be a sequence of function such that $u_{n} \to u$ in $L^{p}(\Omega)$ for $\Omega\subset \mathbb{R}^{N}$ bounded domain. I want to show that for ...
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48 views

Computing Euler Lagrange Equation for a Certain Functional

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where ...
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27 views

Mean Value Property for Linear Partial Differential Equation

Given any linear second order partial differential equation. I would like to know the steps to follow in order to obtain the mean value property of the equation. For example, I was studying a book by ...
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46 views

$H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$?

In a paper I see that the authors used $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary. I think that this imbedding holds ...
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26 views

An inequality in Sobolev spaces

Suppose that $ f\in C^{\left[\frac{n}{2}\right]+1+s} $ and $ v(x,t)\in H^{\left[\frac{n}{2}\right]+1+s} $ differentiable. Then $ f(v(x,t)) \in H^{\left[\frac{n}{2}\right]+1+s} $, is differentiable and ...
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Why is $|\log(\|x\|)|^k$ in $H^1(\Omega)$ when $0<k<1/2$ and $\Omega$ is the $2D$ unit ball?

Let $\Omega= B_1(0)$ the unit ball in $\mathbb{R^2}$. How to prove that for $0<k<1/2$ the function $v(x)= |\log(\|x\|)|^k$ is in $$H^1(\Omega)=\left\{ f\in L^2(\Omega) : \frac{\partial f}{\...
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29 views

Density of a subspace

Let $\Omega\subset\mathbb{R}^d$ be open, bounded and simply connected with smooth boundary $\partial\Omega$. Define $\mathcal{H}:=\{u\in H^1(\Omega)~|~\Delta u\in L^2(\Omega)\}$ with norm $$\|u\|_\...
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Convergence in $W^{1,\infty}$ implying convergence in $W^{-1,\infty}$.

I'm looking for references or proofs of the following facts: Weak star convergence in $L^\infty$ implies strong convergence in $W^{-1,\infty}_{\text{loc}}$, and Strong convergence in $L^\infty$ + ...
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Sobolev spaces embedding on $ R^n$ [closed]

It's true this Sobolev spaces embedding? if $ k < s $ then $ H^{s}(R^n)\hookrightarrow H^{k}(R^n) $ ? Thank you.
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Norm estimate of $(-\lambda A+ I)^{-1}$ for strictly elliptic operator

Let $\Omega$ be a smooth domain in $\mathbb{R}^n$, and $A$ be a strictly elliptic operator $$ Au=\partial_i(a^{ij}(x)\partial_j)u, $$ where $a^{ij}$ are bounded functions satisfying $$ a^{ij}(x)\...
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Gagliardo–Nirenberg–Sobolev inequality for weighted Sobolev space with exponential weights

Consider the weighted $L^p_\omega(\mathbb{R}^d)$ space on $\mathbb{R}^d$ be the set of Lebesgue measurable functions such that $$\|f\|_{L^p_\omega}=\int_{\mathbb{R}^d}|f|^p\omega_\mu(x)\,dx< \...
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1answer
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Morrey's Inequality in 1D

Morrey's Inequality in 1D for $p=2$: There exists a constant $C$ such that $||u||_{C^{0,1/2}(\mathbb{R})} \leqslant C ||u||_{H^{1}(\mathbb{R})}$ for all $u \in C^{1}(\mathbb{R})$. Of course, for ...
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Monotonicty of the trace on Sobolev spaces.

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded domain with smooth boundary and suppose $u\in W_0^{1,p}(\Omega)$. If $v\in W^{1,p}(\Omega)$ satisfies $\lvert v \rvert \leq \lvert u \rvert$ in all of $...
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38 views

Decay properties for the Sobolev spaces

Suppose we have a $f$ belonging to the Sobolev space $W^{1,1}(0,\infty)$, i.e., on $\mathbb R^{+}$. Then why does $f(x)$ decay to zero as $x$ goes to infinity?
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Proving a Certain Functional is Coercive

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where ...
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21 views

Integration in first Sobolev space

Let be $H^1(R)$ the first Sobolev space: $$H^1(R)=\{f\in L^2(R)\mid \quad f'\in L^2(R) \quad \}$$ where the derivative is intended in distributional sense. I have to show that the following equations ...
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Formula for the Sobolev norm for functions defined on the boundary of a domain

I know that given a function $u\in H^k(\Omega)$, we can easily compute its Sobolev norm via a formula involving the $L^2(\Omega)$ norms of its derivatives up to order $k$. I was wondering if there ...
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What means norm $\|u'\|$ and $\|u\|$ in Sobolev space $W^{1,2}(I)$ and why is functional continuous?

We have functional $F: W^{1,2}(I),I\langle 0,1\rangle \rightarrow \mathbb{R}$, I proved that functional is linear, I also counted that is bounded and I get this state: $$|F(u)|\leq {C_1} \|u\|+{C_2}\|...
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How to define a norm of the Cartesian product of two Sobolev spaces when solving PDE

Let $ \Omega \subset \mathbb{R}^3 $ be a convex polyhedron with connected boundary $ \partial \Omega $. $ \Omega $ can be divided into finite subdomains. For simplicity, it is assumed that $ \Omega $ ...
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25 views

On the existence of a function satisfying a certain inequality

Let $D$ be an open disk in $\mathbb{R}^n$ ($n\ge 1$). I am wondering if there is a nonnegative function $f:D\rightarrow \mathbb{R}$ with $$f\in H^1(D)\cap L^\infty(D)$$ satisfying $$|\nabla f(x)|\le ...
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Understanding Robin Laplacian definition through Friedrichs extension on compact manifolds

Let's consider $\Omega$ to be either a compact manifold with boundary (as good as needed) or a bounded domain in $\mathbb{R}^n$). Also $H^k(\Omega)$ is the Sobolev Hilbert space of order $k$. There ...
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34 views

Is it true that if $f\in C^1_c(\mathbb{R}^n)$ then $f\in W^{1,p}(\mathbb{R}^n)$, $1\leq p <\infty$?

Is it true that if $f\in C^1_c(\mathbb{R}^n)$ then $f\in W^{1,p}(\mathbb{R}^n)$, $1\leq p <\infty$? Well, I know that $W^{1,p}(\mathbb{R}^n)=\overline{C^\infty(\mathbb{R}^n)}^{{\lVert\cdot\rVert}_{...
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Question on parabolic smoothing of nonhomogeneous heat equation

Suppose $\Omega $ is a bounded domain in $\mathbb R^n$ and let $u$ be the weak solution of the initial value problem for the nonhomogeneous heat equation: $\begin{cases} \partial_t u-\Delta u=f,\;\...
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36 views

Laplacian Operator + Closed graph theorem

I know that the Laplacian operator defined as $$\Delta:(L^2(\Omega),\|\cdot\|_{L^2(\Omega)}) \to (L^2(\Omega),\|\cdot\|_{L^2(\Omega)})$$ is unbounded. But under other settings like $$\Delta:(H^2(\...
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Poisson problem on unbounded domain

I am looking at the weak Poisson problem. For $f \in L^{2}(\Omega)$ find $u \in H^{1}_{0}(\Omega)$ so that for all $v \in H^{1}_{0}(\Omega)$: $$\int_{\Omega}\nabla{u}\nabla{v} = \int_{\Omega}fv$$ In ...
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A question about the relation between weak and strong derivatives in Sobolev spaces

I am attaching photos from the chapter "Sobolev Spaces". What the author does is that first he approximates a $W^{1,1}$ function by smooth functions that converge in the $\|\|_{W^{1,1}}$ norm, and ...
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1answer
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Why $H^1_0(U) \subset L^2(U) \subset H^{-1}(U)$?

I'm reading by myself the PDE's book by Lawrence Evans and he denoted the dual of $H^1_0(U)$, which is the closure of $\mathcal{C}^{\infty}_c(U)$ in $W^{1,2}(U)$ ($H^1_0(U) = W^{1,2}_0(U)$ ), by $H^{-...
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1answer
30 views

Product of $W^{1,p}(\mathbb{R}^n)$ Functions

Let $u,v\in W^{1,p}(\mathbb{R}^n)$, and $p > n$. Show that $uv \in W^{1,p}(\mathbb{R}^n)$, and moreover that $uv$ satisfies the inequality: $$ \|uv\|_{W^{1,p}} \leq C\|u\|_{W^{1,p}}\|v\|_{W^{1,p}} $...
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A continuity result for functions on a Sobolev space

Let $$W^{1,p}_T = \{u \in W^{1,p}([0,T];\mathbb{R}^N) \mid u(0) = u(T)\},$$ where $W^{1,p}([0,T],\mathbb{R}^n)$ is the usual Sobolev space of functions from $[0,T]$ to $\mathbb{R}^N$. Let $$F:[0,T] \...
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84 views

global approximation by smooth functions theorem

how the yellow shades comes and what is the use of this theorem and how can we say that $supp(U^i) \subseteq supp(W_i)$
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25 views

$f\in C_c^{\infty}(U) \implies D^{\alpha}f\in C_c^{\infty}(U)$, $U$ is open in $\mathbb{R}^n$ and $\alpha$ is multi-index

If $f\in C_c^{\infty}(U)$ then $D^{\alpha}f\in C_c^{\infty}(U)$ here $U$ is open in $\mathbb{R}^n$ and $\alpha$ is multi-index what I know is since $f\in C_c^{\infty}(U)$ then $f$ is infinitely ...
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1answer
185 views

How to show a tempered distribution is a Schwarz function?

Given a tempered distribution $T$, in order to show that it is a Schwartz function, does it suffice to prove for any $f$ Schwartz, $T(f) = \int g f$ for some $g$ Schwartz? Now if $T$ is a tempered ...
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1answer
21 views

Sobolev Chain rule in domains with infinite measure

In chapter 4 of "Measure Theory and Fine Properties of Functions" by Evans and Gariepy, part (ii) of theorem 4.4 states: If $f \in W^{1,p}(U)$ and $F \in C^{1}(\mathbb{R})$, $F' \in L^{\infty}(\...