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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2
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1answer
35 views

What is the dual of $A\cap B$

I encountered with some elliptic problem which admits a variational formulation in terms of space $X$ and I need to understand its dual. Suppose that $2<p<\infty$, $\Omega\subset {\mathbb R}^d$ ...
2
votes
1answer
36 views

Convergence in $L^p$ plus bounded gradient implies that the limit belongs to $W^{1,p}$?

I have a question with this problem I have found in the latest edition of the book Functional analysis, Sobolev Spaces author Haim Brezis pag 264 Remark 4 Let $(u_n) \subset W^{1,p} $ such that $u_n ...
1
vote
1answer
28 views

Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
1
vote
2answers
67 views

Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
0
votes
1answer
44 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
1
vote
1answer
77 views

On the density of $\mathcal{C}^\infty(\Omega) \cap W^{1,\infty}(\Omega)$ in $W^{1,\infty}(\Omega)$

Let $\Omega$ be an open set of $\mathbb{R}^n$ ($n\ge 1$). We know that the Meyers-Serrin theorem isn't true in $W^{1,\infty}(\Omega)$. But is it true that $\mathcal{C}^\infty(\Omega) \cap ...
2
votes
0answers
49 views

Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset ...
1
vote
1answer
23 views

poincare-sobolev inequality

How can we prove this inequality? For $q=\frac{np}{n-p}$ and $1\leq p<n$, there is a constant $c=c(n,p)$ such that if $u\in W^{1,p}(B_r)$, then ...
0
votes
1answer
51 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
0
votes
1answer
39 views

Chain rule for $(f(u))'$ when $u \in H^1$ and $f$ is only piecewise Lipschitz?

Let $f:\mathbb{R} \to \mathbb{R}$ be a peicewise lipschitz function, eg. $f(x) = \chi_{(0,1)}(x)$. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and let $u \in H^1(\Omega)$. Is the chain rule ...
3
votes
2answers
68 views

If $u=v$ on $A \subset \Omega$, then $\nabla u = \nabla v$ on $A$ too

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $A \subset \Omega$ be measure nonzero. For $u, v \in H^1(\Omega)$, if $u=v$ (a.e) on $A$, how to prove that $\nabla u = \nabla v$ on $A$? ...
1
vote
0answers
37 views

A bound on $\nabla u$ in $L^\infty(0,T;L^2)$; how to make argument rigorous?

Suppose $u \in L^2(0,T;L^2)$, $u_t \in L^2(0,T;H^{-1})$ and $f \in L^\infty((0,T)\times\Omega)$. I have the weak form $$\langle u_t, \varphi \rangle_{H^{-1}, H^1} + \int_\Omega\nabla u \nabla \varphi ...
2
votes
2answers
32 views

Asymptotics of a Sobolev function on unbounded interval

Suppose $f : (0,\infty) \to \mathbb{R}$ is locally $H^1$ and $\int_0^\infty (|f'(t)|^2 + |f(t)|^2) e^{-t} dt$ is finite. Is then $\lim_{t \to \infty} e^{-t} |f(t)|^2 = 0$?
5
votes
3answers
73 views

For which $s\in\mathbb R$, $H^s(\mathbb T)$ is a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
0
votes
1answer
54 views

p--laplacian problem

For $f \in W^{-1,p'}(\Omega)$, $f \geq 0$, we definr $v,$ $v_{\epsilon}$, $g_{\epsilon}$ by $$-\mathrm{div}(|Dv|^{p-2} Dv)=f \quad \mbox{in} \mathcal{D}'(\Omega), \quad v \in W^{1,p}_0(\Omega)$$ with ...
0
votes
1answer
22 views

A bound in Sobolev spaces of negative order

Let's consider the domain $U=[-\pi,\pi]\times[-1,1]$. Assume that we have two functions $f\in H^2$ and $g\in H^{1/2}$. I wonder if the following bound is true: $$ \|f g_{x_1}\|_{H^{-0.5}(U)}\leq ...
0
votes
2answers
23 views

Inner Product Representation of Functional on $H^1_0$

Let $T\colon H^1_0(\mathbb{R})\to \mathbb{R}, \;T(f)=\int f'\phi \;dx$, where $\phi\in L^2$ is fixed. By Hölder, $|T(f)|\leq\|\phi\|_2\|f'\|_2\leq C \|f\|_{H^1_0}$, i.e. $T$ is continuous. Therefore, ...
2
votes
1answer
25 views

$ \int_U |Du|^2\leq (\int_U u^2)^{1/2} (\int_U |D^2 u|^2 )^{1/2}$ for $u\in H^1_0(U)\cap H^2(U)$

This is an exercise in Evans's book PDE. For $u \in C_c^\infty(U)$, we have $$ \int_U |Du|^2\leq (\int_U u^2)^{1/2} (\int_U |D^2 u|^2 )^{1/2}$$ by Holder inequality and ...
2
votes
2answers
40 views

$f\in W^{1,p}((0,1))$ is absolute continuous

This is an exercise in Evans's PDE book $f\in W^{1,p}((0,1))$ is absolute continuous where $1\leq p < \infty $ Try : By definition of sobolev space, $f$ has weak derivative $f'$ So $$\ast\ ...
4
votes
1answer
51 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
0
votes
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
votes
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
41 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
votes
1answer
44 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 ...
0
votes
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
vote
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
33 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
votes
0answers
45 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
vote
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
votes
0answers
20 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
votes
1answer
63 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
votes
2answers
52 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
vote
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
votes
2answers
25 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
vote
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
197 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
85 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
votes
0answers
34 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 ...
4
votes
1answer
103 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
vote
1answer
68 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
53 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
vote
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
vote
1answer
22 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
58 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
votes
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, ...