4 votes
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how do you calculate a derivative with respect to another derivative

The Lagrangian is just a function of two variables (or more depending on how general you want to be). For example, $L:\Bbb{R}^2\to\Bbb{R}$, defined as say $L(\xi,\eta)=\frac{m}{2}\eta^2-\frac{k}{2}\xi^...
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3 votes
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Why does $\lVert u\rVert_{W^{k+2,p}}\leq C\lVert\Delta u\rVert_{W^{k,p}}$ imply injectivity of $\Delta$?

By linearity of the Laplace operator $\Delta$ $$\|u-v\|\leq C\|\Delta(u-v)\|=C\|\Delta u-\Delta v\|=C\|0\|=0$$ and by positive definiteness of the norm this implies $u-v=0\iff u=v$ (a.e).
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2 votes
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Integration by parts involving $\Delta^\frac{-1}{2}$

For the sake of simlicity, let $f,g$ be Schwartz functions in $\mathbb R^n$. The first equality $$\nabla (f \Delta^{-1/2}g)= (\Delta^{-1/2}g) \nabla f +f \nabla \Delta^{-1/2}g $$ is true, but it is ...
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2 votes

Solving a 3rd order differential equation with a non constant coefficient

Since you have derivatives of $h$ of orders $1$, $2$, and $3$ (but not $0$, namely $h$ itself), we can make the substitution $y = \frac{dh}{dR}$, resulting in the second-order linear equation in $y$: $...
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2 votes
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Eigenfunction expansion of a heat equation with Legendre polynomials

The equation for $\Phi$ is actually Mathieu's equation. The standard form of the equation is $$ \frac{d^2 y}{dx^2} + (a - 2 q \cos(2x)) y = 0. $$ If we take the substitution $v = 2 x$, the resulting ...
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1 vote
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How to pass partial derivatives with respect to t to the integral - Heat Equation in $\mathbb{R}^{n}$

By the Leibniz integral rule, \begin{multline} \partial_t u = \lim_{s->t^-}\left[\int_{\mathbb R^n}K(x-y, t-s)f(y,s)dy\right] + \int_0^t\int_{\mathbb R^n} \partial_tK(x-y,t-s)f(y,s)dyds \\ = \...
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1 vote
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question about the proof for the Key Lemma for Alexandroff maximum principle

I'll try to speak to your stated questions 1 and 2, assuming you are okay with $D\chi_{\epsilon}$ being negative definite in $\Gamma^{+}$, I've also taken the liberty to include a calculation at the ...
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1 vote

notation when changing variables in Partial Differential Equation

Defining $g$ as you have done is indeed the right thing to do, but for the sake of everyone's sanity, might I suggest that the function $h$ be defined as in the order $h(t,r,C)$ (so compose your $h$ ...
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