Questions tagged [sobolev-spaces]
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
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Sobolev space integral
We know that if $ u \in W^{2,4}( \mathbb{R}^8), $ then $$ \int_{\mathbb{R}^8} \Delta u \cdot u dx = - \int_{\mathbb{R}^8} \left| \nabla u \right|^2 dx \leq 0. $$
I wonder if the following integral $$ \...
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Regularity inequality in $ W^{2,p}(\mathbb{R}^N) $
Let $ N \geq 3 $ and $ 1 < p < + \infty $ is rational number. Let $ f \in L^p( \mathbb{R}^N) $ and $ u \in W^{2,p}( \mathbb{R}^N) $ such that $ - \Delta u + u = f $ in the distributions sense ...
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Derivation in the sense of distributions
I consider the Bessel kernel $$ K(x) = \frac{1}{(4 \pi)^{\frac{N}{2}}} \int_0^{+ \infty} e^{-t} e^{- \frac{\left|x\right|^2}{4t}} t^{- \frac{N}{2}} dt,\ x \in \mathbb{R}^N, x \neq0. $$ We can assume ...
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Poincaré inequality on domains bounded in one direction via compactness
Let $\Omega \subset \mathbb{R}^n$ be an open set which is bounded in one direction, i.e. there exist a unit vector $e \in \mathbb{R}^n$ and constants $a<b$ such that
$$
a < x\cdot e < b \...
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Smoothness of the differential on the group of diffeomorphisms over a compact Riemannian manifold
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed Riemannian manifold (compact without boundary) and consider $\mathcal{D}^s$ the group of diffeomorphisms from $M$ to $M$ of class $H^s$ ($s$-th ...
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About the first variation of total variation of BV functions
Let $u v\in W^{1,1}(\Omega)$ and consider the gradient functional
$$
J(u) = \int_\Omega |\nabla u(x)| \, dx
$$
and the perturbation $u_\varepsilon = u + \varepsilon v$, with $\varepsilon > 0$.
On ...
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Is the restriction of a Sobolev function to some full-measure set continuous?
Motivation: As a consequence of this question, every function $u$ in $W^{1,2}(\mathbb{R}^n,\mathbb{R})$ has a representative $u^*$ such that there is a null set $E\subset \mathbb{R}^n$ with the ...
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Classical PDE theorems on manifolds with boundary
At first, let $M$ be a smooth, non-compact, complete (geodesic balls are precompact) and connected Riemannian manifold with boundary. We define the weak gradient of a function $u \in L^2_{\mathrm{loc}}...
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An interpolation scale for uniformly local Sobolev spaces?
Are there any known results about the interpolation of uniformly local Sobolev spaces on the real line? For $s \geq 0$, the space $H_{\mathrm{u,l}}^s(\mathbb{R})$ is defined as follows,
$$
H^s_{\...
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Regularity of the gradient for elliptic equations set in a bounded open set
Let $\Omega$ be a bounded open set (no regularity assumptions), and let $f \in L^{\infty}(\Omega)$. It is well-known that there exists a unique solution $u \in H_0^1(\Omega)$ of $-\Delta u = f$. In ...
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What is the curl of inverse Dirichlet laplacian?
Let $\Omega$ be bounded smooth simply connected. Let $\Delta^{-1}_D$ be the inverse of the Dirichlet problem.
I was wondering what ${\rm curl}(\Delta^{-1}_D({\rm grad}(p))$ is for $p\in L^2$.
Firstly ...
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Necessary and sufficient conditions for coefficients of elliptic operator to obtain interior regularity
In the comments section in this other MSE question concerning a certain calculation left to the reader in Evans's Partial Differential Equations, @peek-a-boo and I were discussing the requirements on ...
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Functions with compactly supported Fourier transform
The following is a repeat from the following question on MathStack Exchange. I am just hoping to have more success here.
This is a follow-up from that question. The question is this: I want to ...
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A generalized comparison principle - is it true?
Consider $v_1,v_2\in H^1([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ (Bochner spaces) be two weak solutions of the following doubly-nonlinear parabolic problems
$$\begin{cases}\dfrac{\partial v_2^2}{\...
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About weak solutions of PDE's - integral inequality
Working with weak solutions of PDE's we often can deduce a inequality like the following one:
$$
\int_{\Omega}w(x)\phi(x)\ dx+\int_{\Omega}\xi(x)\cdot\nabla\phi(x)\ dx\geq 0,\quad \forall\ \phi\in H^1(...
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