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5
votes
0answers
44 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
2
votes
0answers
51 views

Sobolev embedding on warped product

Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...
0
votes
0answers
32 views

interpolation between Bochner spaces

Is there a reference for the interpolation result stating the existence of an embedding \begin{equation} L^2(I;W^{2,p}(\Omega)) \cap H^1(I;L^p(\Omega)) \hookrightarrow ...
0
votes
0answers
47 views

Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
0
votes
0answers
17 views

Compact embedding for Bochner spaces

I was reading some topics about Bochner integral and this question arises: Assume we have a sequence $u_n \in H^s(0,2\pi,L^2(\mathbb{T}))$ converging weakly to $v$ for all $s \in (\frac12, \frac58)$, ...
2
votes
0answers
20 views

Finding an explicit constant in finite element error estimates

Background: In a finite element approximation to the solution of a linear PDEs, estimates on the order of convergence of the approximation to the solution rely on a theorem of Bramble and Hilbert ...
0
votes
0answers
66 views

Sobolev trace theorem

Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$, where $\Omega$ is knows as a bounded domain with smooth boundary $\partial D$. We choose any subdomain $D\subset Q$ with smooth boundary $\partial ...
3
votes
1answer
114 views

Mixed (anisotropic) Sobolev spaces

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...
3
votes
1answer
96 views

Can you pair $H^s(\Omega)$ and $H^{-s}(\Omega)$ on a domain $\Omega$?

Consider the fractional Sobolev spaces on $\mathbb R^n$ $H^s(\mathbb R^n) := \left\{ u \in \mathcal S'(\mathbb R^n) \; : \; ( 1 + |\xi|^2 )^{s/2} \hat u \in L^2(\mathbb R^n) \right\}$. Let $\Omega$ ...
3
votes
1answer
80 views

Horizontal Sobolev space on Carnot group

This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and ...
1
vote
1answer
43 views

Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem. Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
1
vote
1answer
68 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
0
votes
0answers
64 views

About equivalence of two fractional Sobolev/Hilbert spaces

Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space $$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac ...
1
vote
1answer
148 views

compact inclusion of domains of unbounded operators

Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold. Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset ...
3
votes
0answers
54 views

Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$

Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$ Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write ...
3
votes
1answer
97 views

Interpolation Operator Bounded in Sobolev Norm

Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$, ...
1
vote
2answers
112 views

Sobolev trace map: is the fractional seminorm bounded by just the gradient?

Let $M$ be a compact Riemann manifold. Consider the trace map $T:H^1(M) \to H^{\frac 12}(\partial M)$. Is it always the case that $$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u ...
1
vote
0answers
57 views

boundedness of a sequence $ \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ implies that its temporal derivative is bounded as well [closed]

Hi I have the next claim which I would like to find a proof of it. I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in ...
0
votes
0answers
66 views

Sobolev trace of $H^1(\mathcal{M} \times I)$ functions

Let $\mathcal{M}$ be a compact Riemannian manifold and let $I=(0,1)$. I seek a trace theorem saying that functions $u \in H^1(\mathcal{M} \times I)$ have a well-defined trace at $\mathcal{M} \times ...
4
votes
1answer
115 views

Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS

Using Hardy-Littlewood-Sobolev inequality, we can prove that: $$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C ...
1
vote
0answers
102 views

$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of $$u_{tt} + \Delta u =0$$ $$u|_{t=0}= u_0$$ $$u|_{t=T}=0$$ where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
2
votes
1answer
80 views

Follow up question to: Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$

This is a follow up question of the question Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ Let $\Omega \subset \mathbb{R}^{N}$ ...
1
vote
1answer
80 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
3
votes
1answer
293 views

Lemma 2.11 of Tao's Nonlinear Dispersive Equations

I'm reading the proof of Lemma 2.11 of that book, for which Tao has an errata showing that the case $b=b'$ is not obvious. But I can't quite understand his explanation on how to show that case. Could ...
2
votes
1answer
209 views

Does this linear elliptic equation have a weak solution?

Let $Q = \Omega \times (0,C)$ where $\Omega$ is a bounded domain, write $(x,y) \in Q$ for $x \in \Omega$ and $y \in (0,C)$. Is the problem $$\Delta_{(x,y)}v = 0\quad\text{in $Q$}$$ $$\frac{\partial ...
2
votes
0answers
69 views

Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve $$\int_0^\infty\int_\Omega \nabla v\nabla ...
0
votes
0answers
37 views

Uniform bound in Faedo-Galerkin method with time-dependent weight in inner product

Let $v_j$ be an orthonormal basis for $V=H^1(\Omega) \subset L^2(\Omega)$ which is orthogonal in $L^2(\Omega)$. Let $w:[0,T]\times\Omega \to \mathbb{R}$ be a time-dependent weight which is smoooth ...
2
votes
0answers
104 views

Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
0
votes
0answers
80 views

if $u_\epsilon \rightarrow u$ weakly in $L^2$ then also $\partial_t u_\epsilon \rightarrow \partial_t u $ weakly in $L^2$

I am looking for a citation of the above claim, I am not sure under what conditions on the sequence $u_\epsilon$ exactly this applies. $u_\epsilon$ is a sequence of functions that depend on $x,t$ ...
2
votes
1answer
43 views

Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma\in (0,1)$

I am Looking for a theorem that says that the embedding $H^{1-\sigma}(M)\subset C^1(M)$ is compact for $\sigma \in (0,1)$, where $M$ is a compact manifold. Any references are appreciated. PS I am ...
0
votes
0answers
96 views

Banach space dual to $L^\infty(I,H^1(M))$

What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus. Any references that show ...
3
votes
0answers
102 views

Lipschitz continuity of a composition operator

Let $M$ be a compact Riemannian submanifold of $\mathbb{R}^K$, $U\subset \mathbb{R}^K$ an open neighboorhood of $M$ such that the shortest point Projection $P_M\colon U\rightarrow M$ is well-defined ...
1
vote
0answers
73 views

$H=W$ for weighted Sobolev spaces

Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights? Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm ...
0
votes
1answer
159 views

Sobolev's lemma on manifolds

Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying ...
0
votes
0answers
36 views

A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...
4
votes
1answer
162 views

Density of smooth functions in Sobolev space, respecting nonnegative traces

I am looking for a reference for the following result (if it is true, which I would expect): Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be ...
4
votes
1answer
149 views

Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case. Question: Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
0
votes
1answer
143 views

$u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$?

Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u_n\rVert_{L^\infty(0,T;H)} + \lVert u_n \rVert_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u_n ...
1
vote
0answers
57 views

Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain $$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$ $$u|_{\partial\Omega} = 0$$ where $a$ is real-valued and satisfies $C_1 \leq a(r) \leq C_2$ ...
2
votes
1answer
130 views

Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...
1
vote
0answers
102 views

Weak (Sobolev) derivative and the Frechet derivative (chain rule) [closed]

Let $A: H^s(\Omega) \to H^1(\Omega)$ be a bounded linear map. Let $u \in H^s(\Omega)$. Let $f:\mathbb{R} \to \mathbb{R}$ be nonlinear and Lipschitz such that $f(u) \in H^s(\Omega)$. Is it possible to ...
1
vote
0answers
53 views

Fractional Poincare inequality on closed manifold

Let $u \in H^{\frac 12}(M)$ on a compact closed Riemannian manifold. Can someone refer me to a source where the inequality $$\lVert u - \bar u \rVert_{L^{2^*}} \leq C|u|_{H^{\frac 12}}$$ is proved, or ...
3
votes
1answer
102 views

Extension of Sobolev Functions

Let $\,D\subseteq\mathbb{R}^{n-1}$ be a convex bounded domain. Let$A:D\to(0,\infty)$ be a Lipschitz continuous function. Let $\,\Omega\,$ be bounded domain in $\,\mathbb{R}^{n}\,$ of the form ...
1
vote
1answer
112 views

Identifying the weak limit of a gradient (Bochner spaces)

Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f_n \to f$, $f_n(0)=f(0)=0$ and $f_n$ Lipschitz with Lipschitz constant ...
3
votes
0answers
97 views

Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} ...
0
votes
1answer
117 views

Does Trudinger inequality implies this critical Sobolev embedding?

Let $1<p<\infty$, $\mathrm{L}^p_s(\mathbb{R}^n)=J_s(\mathrm{L}^p(\mathbb{R}^n))$, where $J_s=(I-\Delta)^{-\frac{s}{2}}$, or $\mathscr{F}(J_sf)(\xi)=(1+|\xi|^2)^{-\frac{s}{2}}\hat{f}(\xi)$. And ...
0
votes
0answers
48 views

The minimizing problem over a sequence of shrinking balls

Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define $$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$ where $1<q<5$. Hence we know that each ...
1
vote
0answers
136 views

Solution to a PDE with constant data - what is the fault in my proof? [closed]

Let $C=\Omega \times (0,\infty)$. We want to find a solution $v \in H^1(C)$ such that given $u \in H^{\frac 12}(\Omega)$, $$\int_0^\infty\int_\Omega \nabla v \nabla \varphi + v_y\varphi_y = ...
1
vote
0answers
36 views

Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?
0
votes
0answers
74 views

Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$ \begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases} $$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...