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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0
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1answer
54 views

What is the Frechet derivative of $(u^+)^q$?

I know that if we define $E[u]=\int_\Omega u^+dx$, where $\Omega$ is compact in $R^n$ and $u\in H_0^1(\Omega)$, $u^+:=\max\{u,0\}$, then $E[u]$ is not Frechet differentiable. However, if now I define ...
2
votes
0answers
17 views

$T: H^{-\infty}(R^n) \to H^\infty(R^n)$ continuous iff $T: H^{-r}(R^n) \to H^s(R^n)$ bounded for all $r,s>0$?

Denote by $H^s(\mathbb{R}^n)$ the Sobolev space on $\mathbb{R}^n$ of order $s \in \mathbb{R}$ and recall that we have $H^s(\mathbb{R}^n)^\ast \cong H^{-s}(\mathbb{R}^n)$ for the dual space of ...
1
vote
0answers
17 views

Characteristic Method for linear equation

I'm looking here for a reference to solve the following equation with the method of characteristic $$ \partial_t U + A\partial_x U = f$$ $$U(t=0,x)=U_0(x)$$ $$B(t,x=0)U = 0$$ where $(t,x)\in ...
0
votes
1answer
53 views

A version of Rellich-Kondrachov's theorem

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$ , $k$ be positive integer, and $p \in [1,\infty)$ such that $kp < n$. Let $q\in[1,\dfrac{np}{n-kp}) $ and put $T(u) = u$ ...
0
votes
1answer
31 views

Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

Let $D$ be an open subset of $\mathbb{R}^n$ , $p$ and $q$ be in $(1,\infty)$ such that $p^ {-1} +q^ {-1} = 1$. Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$ ...
3
votes
1answer
80 views

Fractional Sobolev embedding into $L^\infty$

Are there any $t\in(0,1)$, $p\in[1,\infty)$ such that $W^{t,p}(\mathbb{R})$ is continuously embedded into $L^\infty(\mathbb{R})$? I have been looking several literatures, but I have not yet found ...
1
vote
1answer
52 views

Is $\nabla u \in L^{\infty}$ if $u$ is bounded $C^{0}$?

I would like to prove something of the form $|A_{1}(u)| \leq c \lVert u \rVert_{L^{\infty}}$ and $|A_{2}(u)| \leq c \lVert \nabla u \rVert_{L^{\infty}}$ for some operators $A_{1},\ A_{2}$ and ...
0
votes
2answers
41 views

There are $u$ in $W^{1,p}(D)$ and a subsequence $\left\{ u_{m_{k}}\right\} $ such that $\left\{ u_{m_{k}}\right\} $ weakly converges to $u$.

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $p \in (1,\infty)$, and {$u_m$} be a bounded sequence in $W^{1,p}(D)$. Then there are $u$ in $W^{1,p}(D)$ and a subsequence ...
1
vote
1answer
37 views

If $f\in L^2$, has the equation $u_{xx}=f$ an unique solution $u\in H^2\cap H_0^1$?

Let $-\infty<a<b<+\infty$ and $f\in L^2(a,b)$. Is it possible to prove that the equation $u_{xx}=f$ has an unique solution $u\in H^2(a,b)\cap H_0^1(a,b)$? If so, how can we prove it? ...
2
votes
2answers
68 views

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...
1
vote
1answer
36 views

If $(f_n)$ is Cauchy in the $L^2$-norm, then is $(f'_n)$ Cauchy in the $L^2$-norm?

Let $(f_n)$ be a sequence in $H^1(a,b)=\{f\in L^2(a,b);\;f'\in L^2(a,b)\}$, where $-\infty<a<b<+\infty$. If $(f_n)$ is a Cauchy sequence in the norm $\|\cdot\|_{L^2}$, is it possible to ...
0
votes
0answers
25 views

Showing a subspace of a Hilbert space is also Hilbert (please check my proof)

Let $V \subset H \subset V^*$ be a Hilbert triple. Let $$W = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*)\}$$ and let $$W_T = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*) \text{ and } u(0)=u(T)\}.$$ ...
7
votes
2answers
137 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
3
votes
1answer
53 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
1
vote
1answer
35 views

Is $L^\infty(0,T;V)$ a reflexive space? Question about weak convergence

Let $V$ an Hilbert space and $T>0$. Is $L^\infty(0,T;V):=\{v:[0,T]\to V: \text{ess}\,\text{sup}_{t\in [0,T]}||u(t)||<\infty$ a reflexive space? I think that since the $L^\infty$ isn't ...
1
vote
2answers
52 views

Compactness of the solution operator

Let $\Omega$ be a smooth open bounded subset of $\mathbb{R}^n$. The bilinear form $$a(u,v)=\int_{\Omega}\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}dx$$ is elliptic on ...
5
votes
2answers
97 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
4
votes
3answers
107 views

Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $\Delta u+\cos u=0$.

Let $D$ be the open bounded smooth subset in $\mathbb{R}^{n}$. Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $$\Delta u+\cos u=0.$$ Help me some hints to start. ...
4
votes
2answers
90 views

Prove that $\int_{D}\nabla u\cdot\nabla vdx=\int_{D}uv\,dx=0$

Let $D$ be the open bounded subset in $\mathbb{R}^{n}$ with smooth boundary, $\alpha$ and $\beta$ be different non-null real numbers, and $u$ and $v$ be in $W_0^{1,2}(D)\setminus\left\{ 0\right\} $ ...
4
votes
2answers
74 views

Let $\Omega$ be a bounded open subset of $\mathbb{R^3}$, and $f$ be in $L^2(\Omega)$ Does there exist a weak solution in $W^{1,2}_0(\Omega)$

Let $\Omega$ be a bounded open subset of $\mathbb{R^3}$, and $f$ be in $L^2(\Omega)$. Does there exist a weak solution in $W^{1,2}_0(\Omega)$ to the following equation: \begin{cases} \Delta ...
0
votes
1answer
66 views

Bounded subsequence in Sobolev Space

The following is an exercise. Let $I=(0,1)$ and let $(u_n)$ be a bounded sequence in Sobolev space $W^{1,p}$, First question: does "bounded" here means that (for a suitable $M$) $$ \| u_n \|_p ...
1
vote
1answer
40 views

Does there exist a weak solution in $W^{1,2}_0(\Omega)$ to the following equation: $-\Delta u+bu=f$?

Let $\Omega$ be a smooth bounded open subset in $\mathbb{R^3}$, $f$ and $b$ be in $L^2(\Omega)$ such that $b$ is non-negative on $\Omega$. Does there exist a weak solution in $W^{1,2}_0(\Omega)$ to ...
1
vote
1answer
74 views

Inequality for function in certain Sobolev space

I have to prove the following inequality for a function $u$ in $H^1(\mathbb{R}^3)$: $$\int_{B_r}\vert u\vert^q\leq C\bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^a\bigg(\int_{B_r}\vert ...
3
votes
1answer
27 views

A basic question about $W^{1,1}(\Omega)$

Let $\Omega$ be an open interval $(-1,1)$. Does there exist $u$ in $L^1(\Omega)$ such that $u$ is not in $W^{1,1}(\Omega)$? Let $I=(a,b)$ ba an open interval, possibly unbounded. ...
0
votes
1answer
76 views

Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
3
votes
1answer
132 views

If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
2
votes
1answer
37 views

Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
2
votes
2answers
105 views

A question on gradients and PDE

Let $\Omega$ be a smooth bounded domain in Euclidean space. Let $u \in C_c^1(\Omega)$. The subscript indicates compact support. Let $1 < p < \infty$. Can a $v \in C_c^1(\Omega)$ (or in its ...
1
vote
1answer
36 views

Solution regularity of the heat equation after $t>\varepsilon$

Consider the heat equation \begin{align} \partial_t u - \Delta u &= 0 && \mbox{ in }Q=\Omega\times[0,T] \\ u &= 0 && \mbox{ on }\partial \Omega \times [0,T] \\ u(\cdot,0) ...
0
votes
2answers
60 views

The space $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$

For a open Lipschitz domain $\Omega$, consider the space $$A =\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}.$$ Now I heard somewhere that all the second derivatives of a function $u$ are ...
0
votes
1answer
29 views

Canonical Separation of variables

Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.
2
votes
1answer
59 views

Convergence in Sobolev Spaces

Consider the bounded mapping $A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*}$ where $A$ is defined as: $\langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ...
1
vote
1answer
32 views

Equality of two traces

Here we suppose that $\Omega$ is a bounded subspace of $\mathbb{R}^n$ with $C^1$ boundary, denoted by $\partial \Omega$. If $u\in H^1(\Omega)$ (with values in $\mathbb{R}^n$) then $(u,u)$ (scalar ...
0
votes
2answers
69 views

How to prove $H^1(M) \subset H^s(M)$ is a continuous embedding for manifold $M$?

Let $M$ be a $C^k$ manifold for some integer $k$. How does one show that $$H^1(M) \subset H^s(M)$$ is continuous, where $s \in (0,1)$? I was planning to pull back the norms onto a subset $D_i$ of ...
2
votes
1answer
48 views

Is $f \in W^{1,1}[a,b]$ equivalent to $f$ absolutely continuous on $[a,b]$?

$f$ is a function defined on $[a,b]$. Then $f \in W^{1,1}$ is equivalent to $f$ is absolutely continuous?
3
votes
1answer
95 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...
1
vote
0answers
45 views

Versions of Trace Theorems

I have a quick question about the Trace Theorem. I have been using Evans book of Partial Differential Equations to study Sobolev Spaces. The Trace Theorem is given as "If $U$ is bounded and $\partial ...
2
votes
1answer
36 views

Sobolev spaces and integrability of Fourier transforms

I have a Lemma from a text that states that if $g\in W^{1,2}(\mathbb{R})$ ($W^{k,p}$ a Sobolev space) and the weak derivative $Dg\in L^2(\mathbb{R})$ then the Fourier transform $\mathcal{F}g\in ...
1
vote
0answers
32 views

Aproximating a function on SO(3)

By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from ...
1
vote
1answer
54 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
1
vote
1answer
58 views

weak differentiability of log log function

I want to understand why the following function has a weak derivative in two or three dimensions: $w(x) = \ln |\ln|x|| , x \in B_{1/2}(0)$. Can I say that if I have a strong derivative (except for ...
2
votes
1answer
60 views

Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain. It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite ...
2
votes
1answer
59 views

How to prove that the mapping $f\longmapsto f'$ from $H^1(\mathbb{R})$ to $L^2(\mathbb{R})$ is closed?

Let $D:H^1(\mathbb{R})\to L^2(\mathbb{R})$ be the operator given by $D(f)=Df$ where $Df\in L^2(\mathbb{R})$ is the weak derivative of $f$, that is, the function $Df$ satisfies ...
0
votes
1answer
37 views

Don't understand this proof of nonhomogenous Poisson problem (Sobolev space trace and inequality)

See this proof of existence of solution to nonhomogenous Dirichlet problem (from http://www.ann.jussieu.fr/~frey/cours/UdC/ma691/ma691_ch4.pdf page 7): How is the inequality marked with the red ...
0
votes
1answer
55 views

$C^1$ domains vs. Lipschitz domains in PDEs, do we need transformation of coordinates?

I am getting a bit confused. In the definition of $C^1$ manifold in Renardy and Rogers, they say that $\partial\Omega$ is of class $C^1$ if every point on $\partial\Omega$ has a neighbourhood within ...
0
votes
1answer
22 views

Show the example belong to the Bessel potentials space (fractional order sobolev space), where $p=2$

If $\delta>-\frac12$, show that $(1-x^2)_+^\delta\in W^{s,2}$, where $s\in (0,\delta+\frac12)$. Thanks in advanced.
0
votes
1answer
49 views

Sobolev trace operator bounded from below??

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary $\partial\Omega.$ Is the trace operator $$T:H^1(\Omega) \to H^{\frac 1 2}(\partial\Omega)$$ bounded from below: $$|Tu|_{H^{\frac 1 2}} ...
0
votes
0answers
36 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
6
votes
0answers
85 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
2
votes
1answer
30 views

If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $|f(x,t)| \leq C$ for all $x$ and $t$?

Here $\Gamma$ is a bounded closed $C^k$ hypersurface. If $f \in C_c^\infty((0,T);H^1(\Gamma))$ is $f$ uniformly bounded on $[0,T]\times \Gamma$? Or even does it hold that $|f(t)| \leq C_t$ for ...