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mollification
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mollification
mollification (mɒlɪfɪˈkeɪʃən) [a. OF. mollificacion (F. mollification), ad. L. mollificātiōn-em f. mollificāre: see mollify v. and -ation.] The action of the verb mollify; an appeasing, appeasement, pacification. Also, † something that softens (a substance) or mitigates the harshness of (an action o...
Oxford English Dictionary
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Ceration
Pseudo-Geber's Summa Perfectionis explains that ceration is "the mollification of an hard thing, not fusible unto liquefaction", and stresses the importance
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mollifaction
† molliˈfaction Obs. rare—1. [f. mollify v.: see -faction.] = mollification.1822–34 Good's Study Med. (ed. 4) III. 460 There is a considerable difference in explaining upon the same principle the mollifaction of the diseased area.
Oxford English Dictionary
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Aubin–Lions lemma
Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a Galerkin method or by mollification of the
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Mollification is Lipschitz Assume that $f$ is $L$-Lipschitz on $\mathbb{R}^n$ And consider a mollifier $$ \eta \geq 0,\ {\rm supp}\ \eta \subset B_1(O),\ \int_{\mathbb{R}^n} \eta(y)\ dy=1 $$ If $$f_1(z)=\int_{\mathbb...
I don't understand your reason for splitting the space. Here's my solution. Observe \begin{align} |f_1(x)-f_1(y)| \leq \int_{\mathbb{R}^n}\eta(z)|f(x-z)-f(y-z)|\ dz \leq L|x-y|\int_{\mathbb{R}^n}\eta(z)\ dz. =CL|x-y|. \end{align}
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Weyl's lemma (Laplace equation)
Idea of the proof
To prove Weyl's lemma, one convolves the function with an appropriate mollifier and shows that the mollification satisfies Laplace's
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Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $...
Mollification with $\phi_n\in C^\infty_0$ functions will do that, since the support will shrink with increasing $n$.
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La Plata City Hall
Overview
Governor Dardo Rocha's proposal for the establishment of a new capital for the paramount Province of Buenos Aires, useful to the mollification
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Mollification fractional Sobolev function(Convergence) I got a function $u\in W^{\frac{3}{2},2}((0,1),\mathbb{R}^n)$ and a standard mollifier $\eta_\epsilon$. It follows that the mollification $u_\epsilon$ converges...
I found a nice reference, they solved the problem in a more universal setting < Lemma 11
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Why is the mollification $\frac{1}{r^n}\int_\Omega\varrho\left(\frac{|x_0-x|}{r}\right)f(x)\;dx$ of an integrable $f$ infinitely differentiable? Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $\overline{B}_...
The mollification is the convolution product of a $C^{\infty}_c$ function $\varrho$ and a function $f$ integrable:
$$f_r = \varrho \star f$$
And it suffice
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A problem about mollification The problem is : Given $M > 0$ a constant, show that exists $\phi \in C^{\infty}(R)$ with the following properties: i) $\phi(x) = x , \forall x \in [-M,M] $ ii) $ 0 \leq\varphi^{'}(x) ...
Fix some $\epsilon>0$ and define for $x\in\mathbb{R}$: $h_1(x)=x$, $h_2(x)=M+\epsilon$ and $h_3(x)=-M-\epsilon$. Let $U_1=(-\epsilon-M,M+\epsilon)$, $U_2=(M,\infty)$ and $U_3=(-\infty,-M)$. Let $\\{\phi_i,U_i\\}_{i=1}^3$ be a partition of unit associated with $U_i$, i.e. I - $\phi_i\in C^{\infty}(\m...
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The convergence of re-scaled function with mollification Let $u\in L^p_{\operatorname{loc}}(\mathbb R^N)$. Let $\Omega\subset \mathbb R^N$ be open bounded. Let $\eta_\epsilon$ be the standard mollification function an...
Similar to the hint in the comment, define $$ L^p (\Bbb {R}^n) \to L^p (\Bbb {R}^n), f \mapsto ( x\mapsto f ((1+\epsilon)x) $$ and show (exercise) that there is $C>0$ with $$ \|T_\epsilon f \|_p \leq C \cdot \|f\|_p \quad \quad (\ast) $$ for all $\epsilon \leq 1$ and all $ f \in L^p (\Bbb {R}^n)$. N...
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Sequence of smooth functions approximating a 2d cylinder step function Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The function...
You might want to try any of formulae 17 to 27 in the MathWorld page for the unit step function, since the function you're interested in is expressible as $$1-H(x^2+y^2-1)$$
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$\nabla u=0~ a.e.\Rightarrow u=\text{constant}~ a.e. $ on Riemannian manifold Let $M$ be a Riemannian manifold and $\Omega\subset M$ be a connected open set. Is it true that for $u\in W^{1, 2}(\Omega)$ $$\nabla u=0~ a...
It seems to me obviously true: from the standard arguments in Euclidean space you have that $u$ is constant a.e. on the connected components of the intersection of $\Omega$ with any chart. That constant must be the same on any pair of overlapping chart-components, and hence, on any pair of chart-com...
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