# 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.

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### Any function in Sobolev $H^{s}(\mathbb{R}^n)$ space is continuous and bounded if $s>n/2$

I'm studying this book M.E. Taylor, and M.E. Taylor, Partial Differential Equations. 1: Basic Theory, Corr. 2. print (Springer, New York Heidelberg, 1997), more specifically, the proposition 1.3. ...
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### Convexity of Fisher Information [duplicate]

I came across with this functional called Fisher Information: $$\int_0^1 \int_{\Omega}|\nabla \log (\bar{\rho})|^2 d \rho,$$ where $\bar\rho$ is the density $\frac{d \rho}{d \mu}=\bar{\rho}$ with ...
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### Convexity of Dirichlet Integral Functional

The following is not homework, it is just personal study. Let $u \in H_0^1(\Omega)$ and consider the map $u \mapsto \int_{\Omega}|\nabla u|^2 dx.$ I want to show that this map is convex. I considered ...
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### Understanding spaces of negative regularity

Let $C^k(\mathbb{R}^n$) be the space of functions with $k$ continuous derivatives, and $H^s(\mathbb{R}^n)$ the Sobolev space $W^{2,s}$. Their dual spaces are commonly denoted as $C^{-k}$ or $H^{-s}$. ...
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### Regularity of continuous approximation on time discretization

I have been trying to understand how to deal with continuous approximations when dealing with time discretizations, consider the abstract PDE $$\partial_t u + A(u) = f(u)$$ and then instead ...
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### convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
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### integration by parts in Hilbert space

I want to prove that $u,v∈H^1 (R^n )$ we have : $∫_{R^n} \frac{∂u}{∂x_i} vdx=-∫_{R^n} \frac{∂v}{∂x_i} udx$ We know that $C_c^∞$ $(R^n )$ is dense in $H^1$ $(R^n)$, this means that $∃ u_n∈C_c^∞ (R^n )$...
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### Determining weak solution for Dirichlet problem

Let $D$ be the unit disk in the plane and let $\Omega= D\setminus\{0\}$. The Dirichlet problem \begin{cases} Δu = 1 & \text{in } \Omega \newline u=0 & \text{on } \partial \Omega \end{cases} ...
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### Eliminating Neumann boundary condition for elliptic PDE

In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated: I am now wondering if this also works with Neumann boundary ...
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### Equivalence of Sobolev spaces Definitions on Riemannian manifold via $k$-Covariant Derivative and $k$-Gradient Derivative

I am studying Sobolev spaces on Riemannian manifolds and have encountered two different definitions in the literature. In the references: E. Hebey. Nonlinear analysis on manifolds: Sobolev spaces and ...
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### A substitution in an integral of Bochner functions

I have given functions $f \in L^2(0,T;L^2(U))$ and $g \in L^2(0,T;H^1(U))$ and a function $H \in C^1([0,T]\times U)$ such that $$1 \leq H(t,x) \leq 2$$ for a.e. $(t,x)$. Here, $U=\partial\Omega$ is ...
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### Regularity of an elliptical problem

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with boundary of class $C^{\infty}$, with $N \geq 3$. Moreover, consider the uniformly elliptic operator $$\mathcal{L}v := -\Delta v - \lambda v.$$...
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### Schwartz theorem

Can i relax the conditions on f of the Schwartz theorem in real analysis? That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the ...
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### Continuous representative of Sobolev function on triangle set

According to Morrey`s inequality $$||u||_{C^{0,\gamma}(U)}\leq C\cdot ||u||_{W^{1,p}(U)}$$there is a continuous representative for every $u\in W^{1,p}(U)$. $U$ is supposed to be a bounded domain with ...
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### Euclidean space is a Lipschitz domain?

I'm reading the concepto of Lipschitz domain. For example, in Question. Is $\mathbb{R}^n$ a Lipschitz domain? i read that, the boundary of $\mathbb{R}^n$ is the empty set. Is empty set a Lipchitz ...
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