Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.

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What is the minimum about the function $u$ we must have to use Green's formula?

In Evans' book Partial Differential Equations, appendix C.2 Theorem 3, Green's formulas are established: Let $u, v \in C^{2}( \overline{U})$. Then $$ \int_U Du \cdot Dv dx = - \int_U u \Delta v dx + \...
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Exercise 8.15 Brezis - Interpolation inequality

I have a problem with this exercise (see the text in the following link). Interpolation like inequality ,Question from Brezis' book exercise 8.15 The link practically solves it. Only one last step ...
Seurat's user avatar
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Maximum principle to get $C^0$ estimates in terms of $L^1$ norms

I'm reading this paper https://arxiv.org/abs/1401.7366 and trying to prove Corollary 4.6. The result essentially says the following. Let $B_r \subseteq \mathbb R^4$ be the ball of radius $r$ with the ...
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Is the weak topology in separable Sobolev spaces induced by a metric?

I know that the spaces $W^{k,p}(\Omega)$ are separable and reflexive ($\Omega \subset \mathbb{R}^n$ open and bounded) for $p \in (1, \infty)$. I also learned that in a separable Banach space $X$ there ...
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Integration by Parts for not so regular Sobolev functions

I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$ and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta ...
micha's user avatar
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$v(x) := \sum_{i\in\Bbb N}\frac1{2^i}u(x − x_i) \in W^{1,p}(B(0, 1))$

We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then: (i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ ...
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L^p bounds for function on double annulus

The problem is Consider the open disc $B_r=B(0,r)\subset \mathbb{R}^2$, $\phi=\max\{|x|-1,0\}$ is the distance from $B_1$. For $u\in C^1(\overline{B_3-B_1})$. Show that $\exists$ $c$ such that (i)$1\...
vegetabledoge's user avatar
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Best constant for Sobolev inequality from Sobolev space to Hölder space

Let $u(x) \in W^{1,p}(\mathbb{R}^n)$ such that the Sobolev embedding from $W^{1,p}(U)$ to $C^{0,\alpha}(U)$ holds for some $n$, $p$, $\alpha$ and bounded domain $U$. We have the Sobolev inequality $$ \...
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Concentration-compactness Lemma [closed]

Let $N\geq 3$ y $2^{*} := 2N/(N - 2)$. The space $\mathcal{D}^{1,2}(\mathbb{R}^N) := \left\lbrace u \in L^{2^{*}} (\mathbb{R}^N) : \nabla u\in L^2(\mathbb{R}^N)\right\rbrace $, with the inner product $...
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Book recommendation: Sobolev spaces.

I have already read the standard part in Evan's PDE text book. However, in his book, the theorems like Sobolev embeddings only involve bounded domain, and it talks just a little about fractional ...
Xinlin Wu's user avatar
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For $f \in H^s$, then $\exists g \in C_c$ such that $f=g$ a.e.

Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous ...
Mr. Proof's user avatar
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An integral inequality involving exponentials and $H^1(\mathbb{R}^2)$ [closed]

Let $\beta, \alpha>1$, and $u \in H^1(\mathbb{R}^2$). I'm trying to show that $$\int_{\mathbb{R}^2}\left(e^{\beta u^2(x)} -1\right)^{2\alpha}dx \leq \int_{\mathbb{R}^2}(e^{2\alpha \beta u^2(x)}-1)...
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Comparing $(-\Delta)^{1/2}$ and $\nabla$ in $L^p$ space for $p \in (1,\infty)$

Let $\mathbb{T}^n := \bigl(\mathbb{R}/\mathbb{Z} \bigr)^n$ be $n$-dimensional torus. Then, I wonder if $(-\Delta)^{1/2}$ and $\nabla$ are "equivalent" in $L^p(\mathbb{T}^n)$ for each $p \in (...
Keith's user avatar
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Extension of Sobolev functions on non-connected set [closed]

Let $D = D_1 \cup D_2 \cup...\cup D_m$ be a subset of $\mathbb{R}^d$, where each $D_j$ is a (connected and) bounded domain with Lipschitz boundary, and $\min_{i,j} dist(D_i,D_j) \geq c > 0$ for ...
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ould you help me to understand and complete the proof of this theorem of P.L. Lions about the existence of a minimizer?

Could you help me to justify better and in detail the proof of the theorem? This is the theorem 1.40 of the book Minimax Theorems by M. Willem Let $N\geq 3$ y $2^{*} := 2N/(N - 2)$. The space $\...
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A Basis of $L^2$ in $H_0^s$ orthogonal in both spaces.

If $V$ and $U$ are infinit dimensional seperable Hilbert spaces and $V$ is compactly and densely embedded in $U$, then one can show, and it is known, that there exists positive self adjoint Operator $...
Furkan's user avatar
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Is it possible to exploit higher integrability in the trace theorem?

Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain and $n>1$ and let $T_p\colon W^{1,p}(\Omega)\to L^p(\partial\Omega)$ denote the trace theorem for $p\in[1,\infty]$. Suppose, we are given $...
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Evans PDE Ch5 Problem 3 - for what p is the function in W^1,p (U)

I was recently attempting a problem from Evans' PDE text and I solved it, but I'm not sure my answer is correct since it looks a bit fishy. I was hoping someone could verify it and correct me if I'm ...
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If $v$ is a bounded linear functional on $C^{\infty}_0$, then $v \in H^1$.

In a proof I am reading, we are given a function $v \in L^2(\omega)$, and we must show that $v \in H^1(\omega).$ They argue that "since $|\int_{\omega} v \partial_i \psi dx |\leq C||\psi||_{L^2(\...
ali's user avatar
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Boundedness condition in Theorem 5.3.2 of Evans

Thanks! I was going through this proof, and it mostly made sense to me. However, I don't understand where in the proof, boundedness of the open set $U$ is required. The construction of a smooth ...
Geometer's user avatar
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Functional continuity with respect to strong $W^{1,p}$-convergence

Suppose $p\in [1,\infty), f\in C([a,b],\Bbb R^m,\Bbb R^m)$ such that $f$ satisfies the following growth bound: There exist $C_1,C_2 <\infty$ such that $|f(x,y,z)|\leq C_1|z|^p+C_2$ for all $x,y$. ...
Mathemann's user avatar
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$W^{1,p}(\Omega)$ estimates of solutions of elliptic equation

Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$: $$ -\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega $$ with Neumann boundary conditions in $\...
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Is the function $ f $ here Lipschitz?

Assume that $ f:\mathbb{R}^n\to\mathbb{R} $ is a continuous function and $ E\subset\mathbb{R}^n $ is a closed set with zero Lebesgue measure, i.e. $ m(E)=0 $. Suppose that $ f\in C^1(\mathbb{R}^n\...
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Density of compactly supported functions in a Sobolev space

Denote by $W^{k,p}(\mathbb{R})$ the Sobolev space of functions in $L^p(\mathbb{R})$ whose all weak derivatives up to order $k$ belong to $L^p(\mathbb{R})$ with the norm $$ \| f \|_{W^{k,p}(\mathbb{R})}...
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A problem of continuity at an endpoint

Let $\alpha>0$ and $1\leq p \leq \infty$. We define a weighted Sobolev space $$X^{\alpha,p}(0,1):= \{u\in W_{loc}^{1,p}(0,1):u\in L^p(0,1), x^\alpha u'\in L^p(0,1)\},$$ where the notation $u\in W_{...
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Definition of weak directional derivative

I have recently come across the notion of weak directional derivatives in the context of Sobolev functions. Let $u\in W^{1,p}(\Omega)$ denote a Sobolev function for arbitrary exponent $p\geq 1$ and a ...
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Gaussian Sobolev spaces $W_0^{1,2}(\mathbb{R},e^{-x^2}dx)$

I define $\gamma(x)=e^{-x^2}$. I am interested in the $W_0^{1,2}(\mathbb{R},\gamma)$ which I define as the completion of compactly supported functions, for the $\|.\|_ {W_{0,\gamma}^{1,2}}$ defined by ...
BlueCharlie's user avatar
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Validity of pointwise estimate involving spatial derivative of Sobolev function

We consider a Sobolev function $u\in W^{1,p}(\Omega)$ for some $p>1$ and a bounded domain $\Omega\in\mathbb{R}^n$. Assuming we have shown the inequality $$|\partial_s u(x)|=|\nabla u(x) \cdot s|\...
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compact embeddings in manifolds

Do we have compact Sobolev embeddings in complete noncompact Riemannian manifolds (under certain assumptions)? In Hebey's book we see that for $n>kp$ the embedding $W^{k,p}(M)\hookrightarrow L^q(M)...
am_11235...'s user avatar
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Does the product rule always hold in one dimensions? [duplicate]

Let $f \in W^{1,p}\bigl([0,1], \mathbb{R} \bigr)$ with $p \in (1,\infty)$. Then, we can assume that $f$ is absolutely continuous with the classical derivative a.e. equal to the weak derivative, which ...
Keith's user avatar
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Existence of a (weak) antiderivative for any given $f \in L^1(\mathbb{R}^n, \mathbb{R}^n)$

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be an integrable function. Then, I wonder if there exists any integrable $F : \mathbb{R}^n \to \mathbb{R}$ such that \begin{equation} \nabla F = f \end{equation}...
Keith's user avatar
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$\Delta (u^m)=\mathrm{div}(m u^{m-1} \mathrm{grad}(u))$?

How can one show that for a suitable $u$ the weak laplacian $\Delta (u^m)$ equals $\mathrm{div}(m u^{m-1} \mathrm{grad}(u))$ both in the weak sense ($m>1$) and what conditions need one to impose on ...
Furkan's user avatar
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Confusion about definition of Fractional Sobolev space II

I'm studying fractional Sobolev spaces and, similarly to a previous question of mine, I have some troubles to understand some definitions. Consider the Bessel potential spaces, defined as $$ H^{s,p}\...
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Notation for a subspace of $H^1(\mathbb{R})$

In a work of mine, I need the following subspace of $H^1(\mathbb{R})$: $$ \{u \in H^1(\mathbb{R}):u(0)=0\}. $$ I first used the notation $H_0^1(\mathbb{R})$ for it but a referee did not like that ...
Gateau au fromage's user avatar
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Showing this inequality in a Sobolev space

This is Problem 9.24 from Teschl's "Partial Differential Equations". Show that for $f \in H^{1}_0((a, b))$ we have $$\lVert f \rVert^{2}_{\infty} ≤ 2\lVert f \rVert_2 \lVert f'\rVert_2.$$ ...
MathGeek's user avatar
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Confusion about Fourier transform and fractional Sobolev space definition

Consider the fractional Sobolev spaces defined as $$ W^{s,p}\left(\mathbb{R}\right):=\left\{ u\in L^{p}\left(\mathbb{R}\right):\int_{\mathbb{R}}\left(1+\left|\xi\right|^{sp}\right)\left|\widehat{u}\...
User's user avatar
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Does restricting variables of a Sobolev function produces a Sobolev function?

To simplify I will ask in $\mathbb{R}^2$, but I think it will work in any dimension, if I have an open and bounded domain $\Omega$ and a function $u \in W^{1,p}(\Omega)$, is it true that for almost ...
Raul Fernandes Horta's user avatar
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Brezis Proposition 9.3 alternative proof

Let $u \in L^p(\Omega)$ with $1<p \leq \infty$. Proposition 9.3 states two characterization of a function being in $W^{1,p}(\Omega)$. One is that, if $f\in W^{1,p}(\Omega)$, then there exists $C>...
Crash Bandicoot's user avatar
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Equivalent norm on sobolev spaces

One way to define the sobolev-space with dominating mixed smoothness is $$H^s_{\text{mix}}(\mathbb{T}^d) :=\left\{f \in L^2(\mathbb{T}^d)\ \middle|\ \|f\|_{H^s_{\text{mix}}}^2 :=\sum_{m\in\mathbb{N}...
David's user avatar
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Showing that Sobolev Space $H^m$ is in $L^\infty$

I'm very new to Fourier analysis/Sobolev spaces and am stuck on this exercise. I found proofs of more general embedding theorems for Sobolev spaces and some similar questions on here, but they are too ...
singleton-set's user avatar
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Analyzing the Laplace Operator in $W_0^{2,2}(\Omega)$: Integral Identities, Inequalities, and Weak Solution

So I have these questions: Let $\Omega \subset \mathbb{R}^d$ be open and bounded. Consider the Laplace-Operator given by $$ \Delta u=\sum_{j=1}^d \partial_j^2 u \quad \text { for all } u \in W^{2,2}(\...
CanDoMajoringMath's user avatar
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Problems with the proof of theorem 2 in section 5.3 in LC Evans PDE book (second edition)

Note: In the following, $U\subseteq \mathbb R^n$ is an open, simply connected set. The notation $A\subset\subset B$ means $A$ is "compactly contained" in $B$, that is $A\subseteq \bar A\...
K.defaoite's user avatar
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For $g \in H^1(\mathbb{R}^n)$, is it always true that $\lVert \Delta g \rVert_{H^{-1}} = \lVert \nabla g \rVert_{L^2}$?

Let $g \in H^1(\mathbb{R}^n)$, we may define $\Delta g$ as an element of $H^{-1}$ by \begin{equation} \langle \Delta g, f \rangle_{H^{-1} \times H^1}:= -\langle \nabla g, \nabla f \rangle_{L^2} \end{...
Keith's user avatar
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Question on the definition of fractional sobolev norm

The Sobolev spaces with fractional order $H^s(\Omega)$ ($s>0,n\in\mathbb{N}, \Omega\subseteq \mathbb{R}^n$ open) in Hitchhikers guide is defined wrt. the norm $$\|u\|^2_{s}:= \|u\|_{H^m(\Omega)}^2+\...
Furkan's user avatar
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Second mixed weak derivatives of function not absolutely continuous

Let $u,v:[0,1]\to\mathbb{R}$ functions not absolutely continuous in $[0,1]$ but integrable in $(0,1)$. Let $w:(0,1)\times(0,1)\to\mathbb{R}$ such that $$ (x,y)\mapsto u(x)+v(y) $$ then $w$ is locally ...
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Brezis' exercise 8.32.2: given $f \in H$ and $\varepsilon>0$, there exists a unique $u \in V$ satisfying a condition

Let $I$ be the open interval $(0, 1)$. I am trying to solve a problem in Brezis' Functional Analysis Exercise 8.31 Let $V=\{v \in H^1 (I) : v(1)=0\}$. Let $H$ be the vector space of real-valued ...
Akira's user avatar
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Brezis' exercise 8.31.2: if $\int_I f=0$ then $\|u\|_{L^2(I)} \leq \frac{1}{(1+\pi^2)} \|f\|_{L^2(I)}$

Let $I$ be the open interval $(0, 1)$. I am trying to solve a problem in Brezis' Functional Analysis Exercise 8.31 Consider the Sturm-Liouville operator $A u=-u^{\prime \prime}+u$ on $I$ with Neumann ...
Akira's user avatar
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A basic question about $H^1(\Omega)$ Sobolev space

How can I prove that the function: $$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
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Interchanging limits and integrals for Cauchy sequences in $L^p$

NOTE: $U\subseteq \mathbb R^n$ is an open, simply connected set. I am reading L.C Evans's PDE book. I am on page 263. In order to show that Sobolev spaces are a kind of Banach space, we need to show ...
K.defaoite's user avatar
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Pointwise convergence up to a subsequence and Sobolev spaces

Let $u_m\to u$ in $W^{k,p}(\Omega)$ for some domain $\Omega\subset\mathbb{R}^n$. I feel like, up to a subsequence, i can say that $D^\alpha u_m\to D^\alpha u$ a.e. for every $|\alpha|\leq k$. The ...
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