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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197 views

A continuous embedding.

If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
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42 views

One of conjectures of De Giorgi

conjecture: If $\exp(tw), \exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight. $w$ is regular if weighted Sobolev space $W^l_p(\Omega,w)$ is equal to the ...
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144 views

A weak convergence in Sobolev space.

Let $u \in C^0([0,T], H^{s-1}(\Bbb R^n)). $ Let $\{t_n \} \subset [0,T]$ such that $\lim_{n \to \infty} t_n = t_0$. Let $ u(t_n ) \to u(t_0) $ in the Sobolev space $H^{s-1} ( \Bbb R^n )$ for $s = ...
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65 views

Proving $ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) $ by using the Sobolev inequality.

Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ...
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66 views

details in a proof

I saw this statements in a proof, but I would like to see the details. Let $U \subset \mathbb{R}^{n}$ and $u$ sperharmonic function in the sense that \begin{equation} \int_{D} \langle \nabla u(x) ...
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136 views

Is $C_0^\infty $ dense in $W^{m,p}$?

Is $C_0^\infty $ dense in $W^{m,p}$? Here $C_0^\infty$ = $C_c ^ \infty$ : $C^\infty$ with compact supports, and $W^{m,p}$ : Sobolev spaces.
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167 views

A question about weak lower semicontinuity

Let $\Omega \subset \mathbb{R^{n}}$ be a bounded domain and $u , u_j \in H^{1}(\Omega)$ such that $u_j \rightharpoonup u$ in $H^{1}(\Omega)$ $$ F_{1}(u) = \int_{\{ u > 0\}} \dfrac{1}{2}\langle A_1 ...
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187 views

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + ...
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97 views

Eigenfunction associated to the smallest eigenvalue of an elliptic operator

Let $(T_n)$ be a sequence of elliptic operators defined in $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$, with $\Omega$ being a bounded domain with smooth boundary. All of them have a smallest ...
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64 views

Show properties of elements of $\mathcal{H}^2$

I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
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20 views

Traces of Sobolev functions in an unbounded domain

I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev ...
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16 views

associating a function v \in H^{k+1}(K) with a polynomial function

for all $v\in H^{k+1}(K)$ we can associate a polynomial $p\in P_k$ (space of polynomial functions with degree $\leq k$), defined by $\forall \alpha \in N^n$, with $|\alpha|\leq k,\ \int_K ...
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30 views

When is the second derivative of a $H^{1}$ function in $L^{2}$?

Is there a characterisation of all functions $\phi\in H^{1}$ such that for given functions$\{g^{ij}\in L^{\infty}\}$ then $\sum_{ij} g^{ij}\phi,_{ij}\in L^{2}$ where $\phi,_{ij}=\frac{\partial ...
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51 views

Characterization of $H^{-1}(U)$

I am trying to understand the proof of characterization of $H^{-1}(U)$ as in Evan's Partial Differential equations. My questions is different to that of this link. In the theorem regarding the same, ...
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23 views

Find $u:[0,T]\to H^2$ such that $u(0)=u_0\in H^2$ and $u_t(0)=u_1\in H^1$.

Let $u_0\in H^2$ and $u_1\in H^1$. If we define $$ \begin{align*}u:[0,T]&\longrightarrow L^2\\ t&\longmapsto u_0+\int_0^tu_1\;ds \end{align*}$$ then $u(0)=u_0$. Furthermore, the weak ...
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48 views

About the image of the trace operator for Sobolev spaces.

Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain. Once every convex function is locally Lipschitz, we have that $\partial\Omega$ is Lipschitz, therefore, the trace operator $T: ...
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26 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
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28 views

Fractional order Sobolev space over manifold

I stuck with a problem: If $\Omega\subset\mathbb{R}^{n}$ is open and bounded with a sufficiently smooth boundary $\Gamma$. Then how to prove dual space of $H^{1/2}(\Gamma)$ is $H^{-1/2}(\Gamma)$ ? ...
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36 views

When can we interchange summation with $L^2$ inner product?

(This question concerns a step in the solution given to Eignvalues of Laplacian operator and Sobolev spaces.) Why can we interchange the sum and the $L^2$ inner product in the following? $$(\sum_n ...
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46 views

Question with tried Eigenvalues of Laplacian operator and Sobolev spaces III.

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
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21 views

eignvalues of laplacian operator and Sobolev spaces -II

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, Let $F=(F_t) \in C^0(I,L^2(\Omega))$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od the ...
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37 views

Question about functions in Sobolev space.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. If I consider a function $g:\mathbb{R}\rightarrow\mathbb{R}$ which has the following properties: $$ |g(x)|\leq M \qquad |g(x)-g(y)|\leq ...
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27 views

Dual of a Sobolev space (ex)

Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$. 1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$? 2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$? For the first ...
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26 views

injection in H^{-1}

let $\Omega$ an open on $\mathbb{R}^n$. if $f \in H^{-1}(\Omega)$ and $g \in L^1(\Omega)$. Who is the Sobolev space $V$ who can contains $f-g$ such as $V$ is injected on $H^{-1}(\Omega)$? Thanks for ...
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13 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
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28 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
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38 views

Convergence in $L^2$ of difference quotients to derivative of function in $H^1$

Is it true that if $u\in H^1({\mathbb R})$, then $(u(x+h)-u(x))/h$ converges to $u'(x)$ in $L^2({\mathbb R})$, as $h\to 0$? It's hard for me to get a handle on this, since $u'$ doesn't have to be ...
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8 views

How to check whether f belongs to H^(\beta+t)

In the Lemma 3 of the paper"Wendland H. Multi scale analysis in Sobolev spaces on bounded domains" How to check whether f belongs to H^(\beta+t)? Thank you!
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62 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
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17 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
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35 views

Trouble understanding proof of the Poincare Inequality

In the proof of the Poincare Inequality in this article on page 4 the domain of integration is divided into two parts, $v< 0$ and $v>0$. I think this has to do with the absolute norm appearing ...
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35 views

Boundary value problem for two functions

The question is: Let $\mathcal{H}=H_0^1(\Omega)\times H^1(\Omega)$ and consider the solution $(u,v)\in\mathcal{H}$ to the differential problem \begin{equation} -\Delta u=f+a(v-u)\quad\text{in }\Omega ...
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26 views

Why do the following regularisations of a function in Sobolev space exist?

Suppose $v_1, v_2$ satisfy $\mu \leq v_1(x,t), v_2(x,t) \leq M$ a.e. in $Q:=\Omega\times(0,T)$ and $$(v_1, \eta_1) \quad\text{and} \quad (v_2, \eta_2) \in L^2(0,T;H^1(\Omega)) \cap L^2(Q).$$ Define ...
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96 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
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26 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
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31 views

inequality for linear functions in sobolev space

Ist the following Statement true for $f$ and $g$ linear? $\vert fg \vert_{H^2} \leq C \Vert f \Vert_{H^1} \Vert g \Vert_{H^1}$, where $\vert \cdot \vert_{H^2}$ denotes the seminorm. My Idea: It is ...
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52 views

Sobolev spaces in polar coordinates

I need some properties about Sobolev spaces in polar coordinates. To be precise, let $U = \{(x,y)\in\mathbb R^2 : x^2 + y^2 < R\}$ be an open disc and let $H_0^1(U)$ be the usual Sobolev space ...
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24 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
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23 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
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34 views

Simple question about the Gagliardo Niremberg interpolation inequality

Consider the Gagliardo Niremberg interpolation inequality : (Gagliardo Niremberg interpolation inequality)Let $q,r$ be any numbers satisfying $1 \leq q, r \leq \infty $ and let $j,m$ be any ...
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26 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 $$W_T = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*) \text{ and } u(0)=u(T)\}.$$ ...
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42 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 ...
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41 views

Laplace equation with Dirichlet homogeneous BC

I'm studying the Laplace equation with homogeneous Dirchlet boundary conditions that is, $$ -\triangle u + u = f \quad \text{in } \Omega \quad u=0 \quad \text{on } \Gamma $$ With $\Omega$ an open ...
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66 views

Estimates for linear finite element nodal basis functions

Let $\Omega\subset\mathbb R^2$ be a domain with $\operatorname{diam} \Omega=H$ and $\mathcal T^h$ be shape-regular triangulation of $\Omega$ with triangular $\mathcal P^1$-elements. (That means we are ...
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16 views

Bessel Potential spaces

Let $\Omega_1,\Omega_2 \subset \mathbb{R}$ be bounded. The mapping $F: \Omega_1 \rightarrow \Omega_2$ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||(DF(x))^{-1}||$ ...
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32 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
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22 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
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48 views

A Sobolev norm inequality

Assume $\mu\in H^{1}$ is compactly supported in $\Omega$ and $\Delta_{g}\mu\in H^{m}$ for some $m\ge 0$. Then if $K\subset \Omega$, there is a constant $C=C(K,m)$ such that $$|\mu|_{H^{m+2}(K)}\le ...
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29 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
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55 views

Chainrule: “piecewise smooth” and Sobolev functions

since we had a lengthy discussion on my last question (Generalized chainrule for Sobolev functions with a cut-off I didn't find an answer to it yet, I'm going to post a related more specific question ...