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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Determine if a function belongs to the sobolev space $W^{1,p}(\mathbb{R})$ and not to $L^q(\mathbb{R})$

I don't understand the first conclusion of the user Tomas in the exercise Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$....
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Minimizing the functional $\int (|\nabla u|^2- u^{2}V)$ on the Sobolev space $H^1$

I have a question about a function defined on a Banach space. Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$ and $V:\Omega \to [0,\infty]$ a bounded function on $\Omega$. Let $H^{1}(\...
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26 views

Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ ...
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If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u $ in $W^{1,p}(\bar{\Omega})$

I have already proven that for every $u \in W^{1,p}(\Omega)$ with $1 \leq p < \infty$ and every open set of class $C^1$ there exists a sequence $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ such that ...
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33 views

Uniqueness of solution to Helmholtz-style equation

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and connected with $C^1$ boundary. Suppose $q\in L^\infty(\Omega)$ and that $q\geq C>0$ almost everywhere. Consider the following PDE problem: $$-...
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Is the charateristic function $\chi _{\Omega }$ in the Sobolev space $W^{1,2}_{0}(\Omega)$?

Given $\Omega$ is a bounded, $C^1$ domain in $\mathbb{R}^n$. $\chi _{\Omega }(x)$ is the characteristic function of $\Omega$. I have done the followings: We can get $\chi _{\Omega }(x) \in L^2(\...
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18 views

Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde Q(x):=...
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23 views

Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
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28 views

Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class $C^1$...
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28 views

Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
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Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in H^1(...
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Given $u \in W^{1,p}(\Omega)$ then $\overline{\alpha u} \in W^{1,p}(\Omega)$, with $\alpha \in C^1 \cap L^{\infty},\nabla \alpha \in L^{\infty}$

Given $u \in W^{1,p}(\Omega)$ and $\alpha$ a function which verifies $\alpha \in C^1(\Omega)$, $\alpha \in L^{\infty}(\mathbb{R}^n)$, $\nabla \alpha \in L^{\infty}(\mathbb{R}^n)^n $ y $supp(\alpha) \...
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Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad \text{...
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53 views

Product of two $W^{1,p}_0$ functions

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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24 views

Help showing compactness of the support of a function in the Sobolev Space $W^{1,p}$

In Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, in the proof of Theorem 8.12, it is needed to show that the support of a function is compact. The function ...
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27 views

Help showing $u \in W_0^{1,p}(I)$ if and only if $u=0$ on $\partial I$

I am reading the proof the following statement provided in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by haim Brezis: If $u \in W_0^{1,p}(I)$, then $u=0$ on $\partial ...
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15 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
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Are compactly supported functions in $W^{1,2}(\mathbb R^n)$ also in $W_0^{1,2}(\mathbb R^n)$? See for proof?

The question is stated clearly in the title. On the one hand, it seems obvious (and I give an argument below). On the other hand, after a quick search I haven't been able to find the statement ...
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72 views

show that statements are equivalent (Sobolev Spaces)

Given that $u \in L^p(R), 1<p<\infty$ show that the following statements are equivalent: a) $u\in W^{1,p}(R)$ b) $\exists c>0 $ such that for $\forall h \in R$ the following inequality ...
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29 views

Multi-index notation and differentation

For example, let $\Omega \subseteq \mathbb{R}^n$ open, and $C^\infty(\Omega):=\lbrace f: \Omega \longrightarrow \mathbb{C} : f$ $\mathrm{regular}\rbrace$. For $\alpha = (\alpha_1,...,\alpha_n) \in \...
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34 views

Understanding multiindex notation and the Sobolev Space $W^{1,p}$.

The notation comes from Evans Partial Differential Equations. From Appendix A, we are given information about multiindex notation. Assume $ u : U \rightarrow R$, $ x \in U$. (a) A vector of the ...
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82 views

Results about Hilbert-Sobolev space with homogeneous boundary condition.

I am currently reading works about numerical method for solving differential equations. The main setting of the work revolves on the space $H^m_0[0,1]$, which is defined by $$ H^m_0[0,1]:=\{f\in W^{2,...
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Sobolev space: Prove a function is in $W^{1,\infty}$

I am reading the book: Fully nonlinear elliptic equations of Caffarelli and Cabre. In page 8 (Prop 1.2) they prove that if function $u$ in a convex domain locally has at least one paraboloid touching ...
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29 views

The eigenvalue for mollified function

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $E:=\{u_k\}_{k=1}^\infty$ forms a basis for $L^2$ and we ...
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29 views

Operators of order $k$ between Sobolev spaces

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two ...
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Estimate for gradient

Notation: $B_{1}$ is the unit closed ball in $\mathbb{R}^{n}$ $<.>$ is the canonical inner product of $\mathbb{R}^{n}$ Let $u \in H^{1}(B_{1})$, $\xi \in C^{1}_{0}(B_{1})$. Set $v=(u-k)^{+}$ ...
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40 views

What is the difference between $H^1_{loc}$ and $H^1$?

I have started studying Sobolev spaces and I came across a space referred to as $H^1_{loc}$. I am not sure what the $loc$ subscript infers? What is it that makes this space different from $H^1$? Why ...
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33 views

References for Laplace Equation, weak solutions and eigenvalues via the variational method

Could you point me towards books and/or papers which treat the weak form of Laplace (Poisson) equation and determine the eigenvalues and eigenfunctions of the Laplacian using the variational method (...
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Prove generalisation of Sobolev embedding theorem using induction.

I am trying to prove the following; I am doing it by induction and the case $k=1$ is already done. So suppose the above is true for all integers less than or equal to k, first we want to show that ...
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32 views

Why are Sobolev spaces $W^{m,p}$ denoted with the letter “W”?

Why are the Sobolev spaces $W^{m,p}$ where $m\in\mathbb{Z}$ and $p\in\mathbb{Z}$ with $1\le p\le \infty$ denoted with the letter "$W$"? I know they are denoted with the letter $H$ when $p=2$ because, ...
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37 views

Elliptic regularity

Given the problem, $f \in L^2(\Omega)$, $a$ is continuous, coercive (elliptic) and bilinear, $L$ is continuous suppose this problem has a unique solution $$\begin{cases} u \in V, \\ a(u,v) = L(v), \...
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38 views

Relationship between norm of a function and its supremum

Let $u \in H^{1}(\Omega)$ , $0<\tau <1$ and $B_{\tau R}$ the open ball of radius $\tau R$ under what conditions (I) $||u||_{L^{p}(B_{\tau R})} = \sup\limits_{B_{R}} u$ and (II) $||u||_{L^{p}(...
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How do we compute the Green's function of the Laplacian?

Let $D:=(0,1)$ $U:=L^2(D)$ and $$\phi_n(x):=\sqrt 2\sin(n\pi x)\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ $H:=H^2(D)\cap H_0^1(D)$ and $$A:=-\frac{\partial^2}{\partial x^2}u\;\;\;\text{for }...
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Behaviour of functions in weighted sobolev spaces

If $f$ and $Df$ are in $L^2(\mathbb{R}, e^{u^2} dx)$, can we say $f(u)e^{\frac{u^2}{2}}$ is bounded. Here $Df$ distributional derivatie of $f$. That is, If $\int_{\mathbb{R}} \lvert f(u) \rvert^2 e^{...
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Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
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39 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u =...
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35 views

When is it true that the Sobolev trace of a positive a.e. function is positive a.e?

Let $u \in H^1(\Omega)$ on a bounded smooth domain $\Omega$. Is it true that if $u \geq 0$ a.e., then $Tu \geq 0$ a.e. on $\partial\Omega$ where $T$ is the trace? I don't think it is, since $u$ can ...
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Convergence of the series $\sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha}$

Suppose $S=\{y_k\}_{k=1}^{\infty}$ is a countable dense subset of the open unit ball $B(0,1)$ in $\mathbb{R}^n$. We write \begin{equation} u(x) = \sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha} \...
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Chain rule in $\mathbb{R}^d$, with $d\ge 2$.

Given $\Omega\subset\mathbb{R}^d$ be an open bounded set with Lipschitz boundary, let $v\in (H^1(\Omega))^d$, $\psi\in H^1(\Omega)$ $T_K(x):=B_K x+b_K$, where $B_K$ is a non-singular invertible matrix....
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weak formulation of $u''=\psi'(u)+f$ with $ u\in W^{1,2}_0((a,b))$.

Let $u\in W^{1,2}_0((a,b))$, $(a,b)=I$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. Consider $f\in L^2(I)$ and the differential equation $$u''=\psi'(u)+f.$$ I want to ...
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Equivalence of norms on $W^{1, p}(I)$

Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev ...
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Confusion about the definition of Sobolev spaces on manifolds

Let $(M,g)$ be a manifold with metric $g$ parametrized by the mapping $S$ and parametric domain $\Omega$. The sobolev space of order one with respect to the $L_2(M)$-norm $H^1_2(M)$ is defined as ...
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43 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
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52 views

Derivatives of mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
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35 views

Zero extension of a $W^{2,\infty}$ Sobolev function outside its domain

Let $O$ be a non empty open subset of a bounded open set $\Omega\subset \mathbb{R}^n$ and let $f\in L^2(\Omega)\cap W^{2,\infty}(O)$. Let $u: \Omega \to \mathbb{R}$ be a function such that the ...
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22 views

A limit about measure set

Assume $\Psi\in W^{1,\frac{3}{2}}_{loc}(\mathbb{R^2})$, satisfies: $$\lim\limits_{r\to 0}\frac{1}{r}\int_{B_r(x)}|\nabla\Psi(y)|^{3/2}dy=0$$ Then for each $(x_0,y_0)$, and for every $\epsilon>0$ ...
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23 views

Sum of normed function spaces equipped with two-parameter family of norms, Dual inequality

Let $(\cdot,\cdot)$ be the $L^2(\Omega)$ scalar product, and let $V=L^1(\Omega)$, $W=H^{-1}(\Omega)$ (the dual space of $H_0^1(\Omega)$). My question is if there exists a constant $C$, such that for ...
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35 views

Integrability, Sobolev space

Hello I have a question about Sobolev spaces. Let $n \in \mathbb{N}$, $\alpha >\frac{n}{2}$, $c :\mathbb{R}^{n} \to \mathbb{R}$ be a $\alpha$-integrable function. I want to show the following: \...
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22 views

Sobolev inequality on $\mathbb{R}^3$

I know that for $u \in H^q(\mathbb{R}^d)$ with $d>q$ we have for $p = \frac{qd}{d-q}$ that $\|u\|_p \le C \|u\|_{H^q}.$ Now, I have somewhere back in my mind that it is also in the unbounded case (...
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34 views

Weak differentiability and diffeomorphisms

Let $U,V\subset\mathbb{R}^n$ be open sets and assume the existence of a $\mathcal{C}^1$-diffeomorphism $\phi:U\rightarrow V$. Let $u\in W^{1,p}(U)$, $1\leq p\leq\infty$, and define $v=u\circ\phi^{-1}$....