Questions tagged [sobolev-spaces]
For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.
5,842 questions
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Step in Brezis' proof of chain rule for Sobolev spaces
I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and ...
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Extension operator for ball domain in BV spaces [closed]
Does there exist an extension operator: $\mathbb{E}: BV\left(\partial B(x_0, R); \mathbb{R}^N\right)\mapsto BV\left( B(x_0, R); \mathbb{R}^N\right)$ ?
Hopefully one of you can help! Thanks.
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Extension operator in BV spaces [closed]
Does there exist an extension operator: $\mathbb{E}: BV\left(\partial B(x_0, R); \mathbb{R}^N\right)\mapsto BV\left( B(x_0, R); \mathbb{R}^N\right)$ ?
Hopefully one of you can help! Thanks.
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How to construct a step function to approximate a quadratic integrable function？ [closed]
Assume $f\in L^2(0,T; V)$, I approximate $f$ by constructing a sequence step functions a sequence of step functions $f^n:=\sum_{i=1}^{n}f(t_i)\mathbb{I}_{[t_{i-1}, t_i)}$, where $0=t_0<t_1<\...
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Is $C^\infty(U)\cap L^\infty(U)\cap W^{k,p}(U)$ dense in $W^{k,p}(U)\cap L^\infty(U)$?
I'm interested in whether smooth bounded functions are dense in Sobolev spaces. Specifically, letting $U\subset \Bbb R^n$ be open and bounded, is $C^\infty(U)\cap L^\infty(U)\cap W^{k,p}(U)$ dense in $...
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Reference request on Sobolev inequality on compact manifolds.
What is a reference of the fact that on connected, positively curved, compact Riemannian manifolds (such as the sphere) $M$ with dimension $d$, the following inequality
$$\|f\|_{L^\infty(M)} \lesssim \...
- 2,411
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Spectral decomposition of the Neumann-Laplacian
I'm currently trying to find a spectral decomposition for the Laplace Operator
such that the eigenfunctions are a part of $H^2_N$ := $\{u \in H^2 \,|\,\frac{\partial u}{\partial \nu} = 0 \,\,\text{on} ...
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Function in Hilbert space with bounded derivatives
We have a function $f \in H^2(\mathbb{R}^2)$ and $f \in C^{2,\alpha}(K)$ for any compact set $K \subset \mathbb{R}^2$ and for any $\alpha \in (0,1).$ Moreover, we have that $\|\Delta f\|_{L^{\infty}(\...
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Equivalency of a mixed Sobolev norm
I was working on a problem and the following question arose. Consider the norm
$$ \| (-\Delta)^\gamma (1-\Delta)^{- \gamma /2} f\|_{L^2}.$$
It appears to combine features of the usual inhomogeneous ...
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If $\nabla f \in L^p(D)$ for some $f \in L^1(D)$, does it follow that $f \in L^p(D)$?
Suppose that $D \subseteq \mathbb R^n$ is a bounded domain.
Suppose that $f \in L^1(D)$ is an integrable function such that $\nabla f \in L^p(D)$, for some $p > 1$.
Does it follow that $f \in L^p(D)...
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Optimal constant in Sobolev embedding $H^1(0,s) \subset L^\infty$
The paper by Marti Evaluation of the Least Constant in Sobolev's Inequality (Marti) for $H^1(0,s)$ has a sketch of a simple fundamental theorem of calculus computation to obtain a less optimal ...
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The Paper: "On a price formation free boundary model by Lasry and Lions"
I am reading the following paper "On a price formation free boundary model by Lasry and Lions".
https://www-sciencedirect-com.ezp1.villanova.edu/science/article/pii/S1631073X11001488
The ...
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Reproducing Kernel Hilbert Spaces with initial / boundary conditions
Let $\Omega \subset \mathbb{R}^n$ be a domain, $I := [0,T)$ an interval and $s,r\in\mathbb{R}, r\geq 1$. The space $H^s_0(\Omega)$ denotes the standard zero-trace Sobolev space of $s$-times weakly ...
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Evan's PDE - Sobolev Spaces, Chapter 5 Problem 10
Integrate by parts to prove,
\begin{align*}
\int_U |Du|^p dx \le C \left(\int_U u^p dx\right)^{1/2}\left(\int_U |D^2u|^pdx\right)^{1/2}
\end{align*}
for $2 \le p < \infty$ and all $u \...
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First eigenfunction of Laplace-Beltrami Operator
In the book of Jost (Riemannian Geometry and Geometric Analysis), in section 3.2, he is finding the eigenfunctions of the Laplace-Beltrami operator. He defines the first eigenvalue as the Rayleigh ...
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