# Tagged Questions

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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### Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
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Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\... 2answers 2k views ### Dual space of H^1 It holds that W^{1,2}=H^1 \subset L^2 \subset H^{-1}. This is clear since for every v \in H^1(U), u \mapsto (u,v)_{H^1} is an element of H^{-1}. Moreover for every v \in L^2(U), u \mapsto (... 3answers 607 views ### Are polynomials dense in Gaussian Sobolev space? Let \mu be standard Gaussian measure on \mathbb{R}^n, i.e. d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx, and define the Gaussian Sobolev space H^1(\mu) to be the completion of C_c^\infty(\mathbb{R}^n)... 1answer 201 views ### Pointwise estimate for a sequence of mollified functions In the answer to Characterisation of one-dimensional Sobolev space Tomás wrote ... let \eta_\delta be the standard mollifier sequence. Let u_\delta=\eta_\delta\star u and note that for any c\... 2answers 495 views ### Elliptic regularity in Sobolev spaces of negative order I am having some trouble with Sobolev spaces of negative order. More precisely I am considering the space W^{-1,p}(\mathbb{R}^2), considered as 'the' dual space of W^{1,q}(\mathbb{R}^2). Question ... 1answer 300 views ### Intuition behind losing half a derivative via the trace operator This is an informal question, but here goes: For a function f \in H^s(\Omega) (s > 1/2), there is a well-defined operator (the trace) T such that Tf = f\vert_{\partial \Omega} if f \in C^\... 2answers 1k views ### Some basics of Sobolev spaces Let W^{m,p}(\Omega) = \{ f \in L^p(\Omega): D^\alpha f \in L^p(\Omega) \text{ for multi-indices } |\alpha| \leq m\}, where D denotes the weak derivative. Let W_0^{m,p} denote the closure of C_c^... 1answer 867 views ### Why no trace operator in L^2? We have trace operator which allows us to define boundary values of an H^1 function. This is because of the fact that C^\infty is dense in H^1 under the H^1 norm, I believe. I'm sure either ... 2answers 145 views ### Properties of function f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R} with 0 < \alpha < 1. Consider the function$$f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R},$$with 0 < \alpha < 1. How do I see that f \in W^{1, p}(\mathbb{R}) for all p \in [1/\alpha, \... 2answers 619 views ### Extension and trace operators for Sobolev spaces Given that \Omega \subset \mathbb{R}^{n} is an open, convex, Lipschitz bounded set. Let O \subset \Omega be open bounded set then consider.$$u_{m} \rightharpoonup^{*} u \text{ in } W^{1,\infty}_{...
Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$u_t + \Delta^2 u = 0$$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...