Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

3,530 questions
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Existance of $\phi \in L^2$ such as $L(\vec v)=\int_{\Omega}\phi \;\text{div} \vec v \;dx$

[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere] Let $\Omega \in \mathbb{R}^N$ an open bounded connected set such as $\partial \Omega$ is $\mathcal{C}^1$. And ...
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what is $W^{-1,2}$?

I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet. Let $\Omega$ be a simply connected domain in $\mathbb R^n$...
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$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$). So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous ...
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Different topologies in Sobolev space $W^{1,p}$

In paper , L.Ambrosio talks about the space $W^{1,p}(\Omega),\ 1\leq p<+\infty$ endowed with four different topologies: The strong topology, denoted by $W^{1,p}(\Omega)$. The weak topology, ...
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Confirmation of convergence in subdomain

Let $\{u_{n}\}_{n\in\mathbb{N}} \subset L^{p}(\Omega)$ be a sequence of function such that $u_{n} \to u$ in $L^{p}(\Omega)$ for $\Omega\subset \mathbb{R}^{N}$ bounded domain. I want to show that for ...
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Computing Euler Lagrange Equation for a Certain Functional

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$J(v) = \int_\Omega |v(x)|^p\mathrm{d}x$$ where ...
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Mean Value Property for Linear Partial Differential Equation

Given any linear second order partial differential equation. I would like to know the steps to follow in order to obtain the mean value property of the equation. For example, I was studying a book by ...
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$H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$?

In a paper I see that the authors used $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary. I think that this imbedding holds ...
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An inequality in Sobolev spaces

Suppose that $f\in C^{\left[\frac{n}{2}\right]+1+s}$ and $v(x,t)\in H^{\left[\frac{n}{2}\right]+1+s}$ differentiable. Then $f(v(x,t)) \in H^{\left[\frac{n}{2}\right]+1+s}$, is differentiable and ...
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Convergence in $W^{1,\infty}$ implying convergence in $W^{-1,\infty}$.

I'm looking for references or proofs of the following facts: Weak star convergence in $L^\infty$ implies strong convergence in $W^{-1,\infty}_{\text{loc}}$, and Strong convergence in $L^\infty$ + ...
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Sobolev spaces embedding on $R^n$ [closed]

It's true this Sobolev spaces embedding? if $k < s$ then $H^{s}(R^n)\hookrightarrow H^{k}(R^n)$ ? Thank you.
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Morrey's Inequality in 1D

Morrey's Inequality in 1D for $p=2$: There exists a constant $C$ such that $||u||_{C^{0,1/2}(\mathbb{R})} \leqslant C ||u||_{H^{1}(\mathbb{R})}$ for all $u \in C^{1}(\mathbb{R})$. Of course, for ...
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A continuity result for functions on a Sobolev space

Let $$W^{1,p}_T = \{u \in W^{1,p}([0,T];\mathbb{R}^N) \mid u(0) = u(T)\},$$ where $W^{1,p}([0,T],\mathbb{R}^n)$ is the usual Sobolev space of functions from $[0,T]$ to $\mathbb{R}^N$. Let F:[0,T] \...
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global approximation by smooth functions theorem

how the yellow shades comes and what is the use of this theorem and how can we say that $supp(U^i) \subseteq supp(W_i)$
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$f\in C_c^{\infty}(U) \implies D^{\alpha}f\in C_c^{\infty}(U)$, $U$ is open in $\mathbb{R}^n$ and $\alpha$ is multi-index

If $f\in C_c^{\infty}(U)$ then $D^{\alpha}f\in C_c^{\infty}(U)$ here $U$ is open in $\mathbb{R}^n$ and $\alpha$ is multi-index what I know is since $f\in C_c^{\infty}(U)$ then $f$ is infinitely ...
Given a tempered distribution $T$, in order to show that it is a Schwartz function, does it suffice to prove for any $f$ Schwartz, $T(f) = \int g f$ for some $g$ Schwartz? Now if $T$ is a tempered ...
In chapter 4 of "Measure Theory and Fine Properties of Functions" by Evans and Gariepy, part (ii) of theorem 4.4 states: If $f \in W^{1,p}(U)$ and $F \in C^{1}(\mathbb{R})$, \$F' \in L^{\infty}(\...