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1 answer
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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 $\partial\Omega$. ...
Gio712's user avatar
  • 253
1 vote
0 answers
80 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 ...
user avatar
2 votes
0 answers
320 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 ...
CooLee's user avatar
  • 375
2 votes
1 answer
1k 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 $H^s_x(H^s_y)$, ...
anonymous's user avatar
4 votes
1 answer
204 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 $\Omega$ ...
shuhalo's user avatar
  • 5,327
8 votes
3 answers
1k 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 ...
Pit's user avatar
  • 81
0 votes
0 answers
187 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 12}|(u,\varphi_k)_{L^...
C_Al's user avatar
  • 251
2 votes
2 answers
342 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 \mathcal{D}(...
anonymous's user avatar
3 votes
0 answers
72 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 $$\...
EasyStarter's user avatar
2 votes
2 answers
402 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 \rVert_{L^2(M)...
Upin's user avatar
  • 29
1 vote
0 answers
82 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 L^\infty(I,H^1(M))\...
Alan's user avatar
  • 1,594
2 votes
0 answers
102 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 \{...
MichaelL's user avatar
1 vote
0 answers
191 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 ...
AKA's user avatar
  • 21
2 votes
1 answer
108 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 $\Omega \subset \mathbb{R}^{N}$ ...
user109584's user avatar
1 vote
1 answer
110 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 ...
user109584's user avatar
2 votes
1 answer
340 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 v(...
ChristopherB's user avatar
2 votes
0 answers
119 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 ...
ChristopherB's user avatar
2 votes
0 answers
553 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 ...
user44651's user avatar
  • 1,069
0 votes
1 answer
264 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 ...
Alan's user avatar
  • 1,594
3 votes
0 answers
582 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 ...
user35593's user avatar
  • 2,286
0 votes
1 answer
332 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 $...
XinWang's user avatar
5 votes
1 answer
2k 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 ...
Martin's user avatar
  • 53
1 vote
1 answer
514 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 \...
Charpe's user avatar
  • 35
1 vote
0 answers
116 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 $C_1 \leq a(r) \leq C_2$ ...
C_Al's user avatar
  • 251
4 votes
1 answer
1k 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 ...
Peter's user avatar
  • 437
2 votes
0 answers
535 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 ...
MainW's user avatar
  • 21
1 vote
0 answers
120 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 ...
Recreate's user avatar
3 votes
2 answers
669 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 ...
C_Al's user avatar
  • 251
18 votes
1 answer
2k 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 = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
JakeA's user avatar
  • 201
1 vote
1 answer
432 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 ...
Paul-Benjamin's user avatar
1 vote
0 answers
215 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 = 0\quad\...
jamesC's user avatar
  • 49
3 votes
1 answer
171 views

If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz?

Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$. Suppose $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of ...
ewl's user avatar
  • 53
2 votes
2 answers
754 views

About a completion of a Sobolev space

Let $\Omega$ be a bounded smooth domain and define $\mathcal{C} = \Omega \times (0,\infty)$. Below, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$. The map $\...
jamesC's user avatar
  • 65
0 votes
2 answers
218 views

Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?

Should I expect the following Friedrichs/Poincare inequality to hold for $u \in C^\infty(S_n \times (0,\infty))$ with $u(x,0) = 0$: $$\int_{S_n \times (0,\infty)}|u|^2 \leq C\int_{S_n \times (0,\infty)...
KitA's user avatar
  • 17
3 votes
0 answers
411 views

Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
Matthias Ludewig's user avatar
5 votes
1 answer
369 views

Calculus of variation

This is probably simple but I'm stuck somewhere. I am trying to solve the calculus of variation problem that arise in an applied field: $$\min_{f \in C^1} \int^1_0 \int^1_0 (x-y)^2f(x,y)dxdy$$ $$\text{...
skillfeedback's user avatar
2 votes
0 answers
382 views

Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)

Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$. I am ...
ewl's user avatar
  • 53
3 votes
0 answers
185 views

Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse. Does there ...
DeleMax's user avatar
  • 93
1 vote
1 answer
264 views

Image of (right) inverse trace map $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$ dense in $H^1(\Omega)$?

Let $\gamma\colon H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ be the linear trace map which has a right continuous inverse $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$. Is the image of $...
DeleMax's user avatar
  • 93
3 votes
1 answer
266 views

$\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$. Does the following inequality (or something similar hold) for ...
Marcus_Thel's user avatar
2 votes
1 answer
307 views

Question regarding to approximate continuity

Given $u\in BV(R^N)$, we say $u$ is approximate continuous at $x$ and the approximation limit is $l\in R$ if $$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$ for all $...
JumpJump's user avatar
  • 679
5 votes
2 answers
1k views

Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
Matthias Ludewig's user avatar
2 votes
0 answers
448 views

Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point. I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...
user62319's user avatar
3 votes
0 answers
224 views

Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in W^{1,p}(...
jamesC's user avatar
  • 65
8 votes
1 answer
392 views

Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\...
Robert's user avatar
  • 171
2 votes
1 answer
2k views

Sobolev Space, "characteristic function" for the weak derivative

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product $$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$ for $u\in H_0^1$ and $\mu(...
Xiao's user avatar
  • 485
3 votes
1 answer
880 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ D_{j}f(x):=...
GFF41's user avatar
  • 31
2 votes
1 answer
192 views

Nonlocal Stefan problems

Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality? A search on Google and MathSciNet give me only a handful of results which greatly ...
LDace's user avatar
  • 43
2 votes
0 answers
142 views

Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$. Consider the system, with $u^\epsilon, v^\epsilon \...
LDace's user avatar
  • 43
3 votes
1 answer
752 views

A distributional normal derivative for functions in $H^1(\Omega)$

Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this. For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
riem's user avatar
  • 266