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Results tagged with ap.analysis-of-pdes
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user 37103
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
0
votes
Solvability Helmholtz equation
One necessary condition follows from integration by parts:
$$ \int_D f^2=\int_D fLu=\int_{\partial D} (f\partial_nu-u\partial_nf)+\int_D uLf=\int_{\partial D} g\partial_nf. $$
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5
votes
2
answers
320
views
Variational formulation of second order equations of the divergence form
Consider the second order operator
$Lu=\partial_i(a_{ij}\partial_j)u+b_i\partial_iu+cu$.
Can we find a functional $I[u]$ such that $Lu$ is the variation of $I[u]$ with respect to $u$? I have success …
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2
votes
A coarea formula when proving maximum principles for strong solutions in Chapter9 in Gilbarg...
More precisely it is the area formula, see 3.7 of Frank Morgan, Geometric Measure Theory for a sketch of the proof, or 3.2.3 of Federer, Geometric Measure Theory for a complete proof.
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3
votes
Are there closed form solutions available for this equation below?
Now I see. So both your $U$ and $W$ are vector fields on $\mathbb{R}^3$ and your equation is more commonly written as
$$\nabla(\nabla\cdot U)-\nabla^2U=\nabla\times W.$$
By the identity of the vect …
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1
vote
Accepted
Existence and estimates of a solution of a perturbed first order partial differential equation
The PDE is
$$ (\partial_x+s\partial_y+\frac{sx-y}2\partial_z)h=g+B(s)h. $$
The characteristic equation is
$$ \frac{dy}{dx}=s,\quad \frac{dz}{dx}=\frac{sx-y}2,\quad \frac{dh}{dx}=g+B(s)h=g+O(\epsilon)h …
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5
votes
Accepted
Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying fol...
Give yourself "an epsilon of room" and apply the continuity method.
It suffices to check it for $x\in U$.
For one direction, let $\epsilon>0$ and $y\notin U$ such that $d:=d(x,\mathbb{R}^n-U)=|y-x|$ …
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4
votes
Accepted
Lemma 2.11 of Tao's Nonlinear Dispersive Equations
I have now figured out the question, so I'll record it here.
Let $L=ih(\nabla/i)$ be a constant coefficient differential operator, where $h$ is a polynomial. Recall the Bourgain norm is defined as
$ …
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1
vote
0
answers
437
views
$H=W$ for weighted Sobolev spaces
Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights?
Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm
$$\|u\|_{m,p …
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3
votes
1
answer
479
views
Lemma 2.11 of Tao's Nonlinear Dispersive Equations
I'm reading the proof of Lemma 2.11 of that book, for which Tao has an errata showing that the case $b=b'$ is not obvious. But I can't quite understand his explanation on how to show that case. Could …
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8
votes
2
answers
2k
views
Nash's proof of De Giorgi-Nash-Moser theorem
I saw this question, but I think the answer didn't fully address what I want to know about it:
Nash's paper on parabolic equations.
It says almost everything developed later in elliptic and paraboli …
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6
votes
Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyp...
A prototypical example: the wave equation (for sake of simplicity let's work in 1d, but the idea is general):
$$ \partial_t^2u=\partial_x^2u. $$
Let $v=\partial_tu$ and $w=\partial_xu$. Then the abo …
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2
votes
Nash inequality on a compact domain?
Sometimes one has to roll up his sleeves and get his hands dirty in analysis.
So here is the estimate you need:
$$ \|f\|_2^2=(\sum_{|\xi|\le R}+\sum_{|\xi|>R}) |\hat f(\xi)|^2\ll_n R^n\sup_{\xi} |\h …
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5
votes
Existence of a uniformly continuous function $g$ on $\mathbb{R}$ where $f = g$ a.e.?
This is easier when passed to some sort of weak formulation. By Lebesgue differentiation theorem, for almost every x, $\lim_{r\to0} \frac{1}{|B_r|} \int_{x+B_r} f=f(x)$. Replace each f(x) by the left …
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4
votes
1
answer
353
views
Mixed norm estimate for the heat equation
Consider the inhomogeneous linear heat equation
$$\partial_tu-\Delta u=F$$
on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice so …
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11
votes
2
answers
3k
views
What's wrong with the Courant nodal domain theorem?
The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. How …
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