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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2answers
37 views

estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
2
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1answer
48 views

A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

Let $f_n \to f$ on compact subsets of the real line. If $u_m \rightharpoonup u$ in $L^2(0,T;H^1) \cap L^p(0,T;L^p)$ and $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in ...
1
vote
1answer
42 views

Small question about strong convergence

I have a small question I have that $ \lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...
-2
votes
1answer
71 views

eigenvalues to Laplacian

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 ...
0
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1answer
47 views

Hölder-Zygmund Spaces on compact sets and for integer smoothness parameters

I know from Triebl, Theory of Function Spaces II, that for $\alpha \notin \mathbb{N}$ Hölder-Zygmund Spaces on $\mathbb{R}$ are equal to the classical Hölder Spaces. However, I have two questions ...
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0answers
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 ...
1
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1answer
78 views

Help please eigenvalue of Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2\left(\Omega \right)$. Let $\left(\lambda_n\right)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and ...
3
votes
1answer
34 views

Smooth function composed with sobolev function vanishes at 0

Let $\Omega$ be a bounded domain with sufficiently smooth boundary. Let $u \in W^{1, 2}_{0}(\Omega)$ and $F \in C^{\infty}(\mathbb{R} \rightarrow \mathbb{R})$ such that $F(u(x)) = 0$ for almost every ...
0
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0answers
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 ...
1
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1answer
34 views

Operator compact on $H^1 (0,\pi)$

Consider the operator $K\colon H^1(0,\pi)\to H^1(0,\pi)$ defined by duality (Riesz. Theorem) as $$ \langle K\phi,\psi\rangle = \int_{0}^{\pi}{\phi(x)\psi(x)\,dx} $$ for all $\psi \in H^1(0,\pi)$, ...
0
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0answers
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 ...
2
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1answer
67 views

Characterization of Sobolev Space

I have just started learning about Sobolev spaces. So this might be trivial. I am working through the book "Partial Differential Equations" by Lawrence Evans, it came highly recommended. Taking ...
2
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2answers
57 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
1
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0answers
27 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
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2answers
26 views

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
1
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0answers
50 views

eignvalues of Laplacian operator and distributions

Let $\Omega$ be open and bounded in $\mathbb{R}^n$ and $I$ an interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigenvalues ...
1
vote
2answers
200 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
2
votes
3answers
87 views

Is a function in $L^2$ which second derivative is in $L^2$ in $H^2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary. Assume that $f\in L^2(\Omega)$ and $f^{\prime\prime}\in L^2(\Omega)$. Does one have $f\in H^2(\Omega)$? Useless comments: ...
0
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0answers
35 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 ...
2
votes
1answer
31 views

Approximate an $L^2$ function from “inside”

Consider a bounded domain $\Omega \subset \mathbb R^d$ and a function $f \in L^2(\Omega)$. Now $f$ can be approximated through a sequence of functions $f_n \in H^1(\Omega)$ (or even ...
0
votes
1answer
27 views

Question on Gagliardo-Nirenberg.

On page 679 of this paper, the authors claim they can get a special case of Gagliardo-Nirenberg with a constant of 1/2. They prove this using functions in $C_0^\infty(\mathbb{R}^2)$, for which the ...
3
votes
1answer
57 views

Elliptic PDEs in Banach space

The standard textbooks discuss weak solutions and regularities in Hilbert spaces $W^{k,2}.$ I could not find a good reference on the theory based on Banach spaces $W^{k,p}.$ It would be good to point ...
5
votes
1answer
158 views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
1
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1answer
72 views

$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some boundary conditions. Consider the weak form ...
1
vote
2answers
56 views

Why does Boundary $H^2$ regularity fail for trace non-zero functions?

In Evans, Section 6.3, Theorem 4; we know that if $\Omega$ is a bounded region in $\mathbb{R}^n$ with smooth boundary (say), and $u\in H_0^1(\Omega)\cap H^2(\Omega)$ then we have the bound ...
1
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1answer
35 views

Completeness of Sobolev space constructed from seminorm

Define $W^{p,r}(\mathbb{R}^d):=\{f\in L^p(\mathbb{R}^d) : D^\alpha f\in L^p(\mathbb{R}^d), \forall 0<|\alpha|\le r\}$ where $1\le p\le\infty$. Let the seminorm on $W^{p,r}(\mathbb{R}^d)$ be ...
1
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1answer
23 views

Approximating with a sequence convergent in multiple Sobolev Norms.

I'm stuck proving a Gagliardo-Nirenberg Interpolation-type inequality. Typically authors prove the inequality for functions of their favorite regularity and try a density argument. This often ...
1
vote
1answer
64 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
0
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1answer
24 views

Looking for a “trace inequality” of normal derivative in $L^1$

Let $U = \Omega \times (0,\infty)$ with $\partial U = \Omega \times \{0\}$. Let $w \in H^1(U)$ with $(w|_{\partial U})^{\frac 1 m} \in L^1(\partial U)$. I am looking for a trace inequality of the ...
0
votes
1answer
30 views

If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.

I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is "Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, ...
1
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1answer
52 views

equation in Sobolev space

i have this exercice: Let $f\in L^2(\mathbb{R}^n)$. 1- Prouve that the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admeit a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prouve the ...
4
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2answers
64 views

Bounded data means bounded solution to parabolic PDE

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider $$u_t - \Delta u = f$$ $$u|_{\partial\Omega} = 0$$ $$u(0) = u_0$$ or more generally replace $\Delta$ with a suitable ...
1
vote
1answer
75 views

Hardy-Littlewood-Sobolev fractional integration inequality fails at endpoints

Here's a version of the theorem: If $1 < p, r < \infty$ and $ 0 < \alpha < n $ be such that $ \frac{1}{p} + \frac{ \alpha }{ n} = \frac{1}{r} + 1 $. Then for any $ f \in L^p ( \mathbb R ^n ...
0
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1answer
43 views

Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
3
votes
0answers
46 views

Missing step in Evan's PDE?

In the attached proof, I am having trouble justifying the presence of "$x_n^{p-1}$" in eh line immediately below "Thus" and "Letting $m\to\infty$". I understand he uses Minkowski's inequality, and ...
3
votes
0answers
43 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
0
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0answers
26 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 ...
2
votes
1answer
44 views

Discontinuous function in $W^{1, 1}(\mathbb{R}^{2})$

What's an example of a bounded function in $W^{1, 1}(\mathbb{R}^{2})$ which is discontinuous? Can this function be discontinuous on a set of positive measure?
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0answers
54 views

Functions of fractional-order Sobolev spaces

In fracture mechanics one might end up dealing with functions such as $$w(r,\theta) = \sqrt{r} \sin \frac \theta 2,$$ which is defined, for example, on a cracked unit circle, $\Omega = ...
1
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1answer
40 views

Proposed proof for Sobolev space result

I have the following result which seems that it must be true, but I would like to prove it: This is my proposed proof. If $U \subset \mathbb{R}^{n}$. Given $u \in W^{1,p}(U)$, where $u$ has compact ...
0
votes
1answer
25 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
10
votes
2answers
188 views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
5
votes
1answer
173 views

Stokes theorem and Sobolev spaces.

I am interested under which regularity condition is Stokes' theorem is still valid. For concreteness I am interested in the following problem Let's consider a domain $\Omega$ in $\mathbb{R}^{3}$ ...
0
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0answers
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 ...
0
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0answers
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}, ...
2
votes
1answer
67 views

Proving $u\mapsto |u|^2u$ is Lipschitz on bounded subsets of $H^2(\Omega)\cap H_0^1(\Omega).$

I'm reading a paper and am stumped verifying two details. Let $\Omega$ be a bounded region in $\mathbb{R}^2$ with smooth boundary. I'd like to show that the map $u\mapsto |u|^2u$ is a map from ...
2
votes
2answers
73 views

Variational methods : Why i can't apply this theorem?

Consider the following problem: Find a weak solution for $$u'' + u = \sin t ,\,\, 0 < t < \pi$$ $$u(0) = u(\pi)=0 $$ the corresponding functional for the problem is $\varphi(u) = ...
0
votes
0answers
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 ...
1
vote
0answers
61 views

Extension of function from $W^{1,p}(\Omega - \{x\})$ to $W^{1,p}(\Omega)$

Suppose $\Omega\subset \mathbb{R}^n$ is open, $p\geq 1$, $n\geq 2$ and $u \in W^{1,p}(\Omega-\{x\})$. Show that $u$ extends to a function in $W^{1,p}(\Omega)$. So far I have; clearly $u$ and each ...
2
votes
0answers
49 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...