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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66 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 ...
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41 views

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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27 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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16 views

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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25 views

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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31 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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33 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) ...
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24 views

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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20 views

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 ...
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33 views

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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32 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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34 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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19 views

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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28 views

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 ...
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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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28 views

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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31 views

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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38 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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41 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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29 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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20 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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29 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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21 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 ...
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38 views

Question about the interpret of Picone inequality for non-regular functions.

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
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31 views

Find the infimum value of a functional.

Consider the functional $$F(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ \ u\in W^{1,p}(0,1)$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of $$\inf\{F(u): u\in ...
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22 views

Understanding the embedding of $W^{\infty, 2}$

I am trying to understand Sobolev's embedding theorem, more precisely to understand when a Sobolev generalized function of infinite order is smooth of some order. Consider the following statement ...
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29 views

Bessel potential action on a product of function

Let's consider the Bessel potential $$J^s:=(I-\Delta)^{\frac{s}{2}}$$ Does it exist some kind of Leibniz rule for $$J^s(fg)$$?
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28 views

Can one explicitely construct a sequence of functions of compact support approximating $u\in W_{0}^{1,p}(\Omega)$?

We define $W^{1,p}_{0}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the $W^{1,p}$-norm (or equivalently as the closure of the $W^{1,p}$-functions with compact support). Given $u\in ...
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58 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
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44 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
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37 views

Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
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The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
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70 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...
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28 views

How to derive this Sobolev-type inequality in $\mathbb R^3$?

Does anyone know a simple way to derive the following inequality for smooth, compactly supported functions in $\mathbb R^3$? $$ \max \,\{ |u(x)| \mid x \in \mathbb R^3 \} \;\leq\; K\|\, Du ...
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24 views

Multiplication by a Cutoff and Convergence in $H^s(\mathbb R^n)$

I'm trying to teach myself some things about Sobolev spaces out of McLean, Strongly Elliptic Systems and Boundary Integral Equations. Exercise 3.14 has me stumped for no reason: Let $K_j \subset ...
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21 views

Sobolev space trace theory on $M \times [0,T]$

Let $M$ be a compact Riemannian manifold without a boundary. I wonder how the trace map $T:H^1(M \times [0,T]) \to H^{\frac 12}(M \times \{0,T\})$ is exactly.. can I split it into two trace maps for ...
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32 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
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89 views

How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$

It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that $$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$ The ...
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46 views

$L^2$-Sobolev space

I am looking at the proof of the following lemma and I don't understand the conclusion. Lemma: For $k\geq 0$ integer, let $f=\sum_{n\in \mathbb{Z^d}}{c_n\chi_n}\in L^2(\mathbb{T}^d)$. Then $$f\in ...
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26 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
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54 views

Existence of weak solution to $-\Delta u =0$, $u|_{\Gamma_1} = g$ and $u_{\Gamma_2} = 0$?

Let $\Omega$ be a compact Riemannian manifold with $\partial\Omega = \Gamma_1 \cup \Gamma_2$ a union of disjoint sets. Let $g \in H^s(\Gamma_1)$ and consider $$-\Delta u = 0 \text{ on $\Omega$}$$ ...
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69 views

$L^{\infty}$ estimate on the boundary by means of Sobolev norms in the interior

I am studying this paper http://143.248.27.21/mathnet/paper_file/washington/gunther/cpde.pdf and I have a doubt I have not been able to solve since yesterday. In page 14, just before the last bunch of ...
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66 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
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35 views

Prove weak derivative commutes with difference quotient

Let $U$ be an open set in $\mathbb{R}^n$,$f:U\to \mathbb{R},f\in W^{1,p}(U)$. Let $\tau_{h,i}f(x)=\frac{f(x+he_i)-f(x)}{h},h>0$ Given any compact $V\subset U$, show there exists $h_0>0$ such ...