Questions tagged [elliptic-pde]
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1,260 questions
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Asymptotic behavior of the Green function of $(-\Delta_{g_{\mathbb{S}^3}}+1)^{1/2}$ on $\mathbb{S}^3$
Let $A=\left(-\Delta_{g_{\mathbb{S}^3}}+1\right)^{1 / 2}$. I expect the Green function on $(\mathbb{S}^3,g_{\mathbb{S}^3})$ has the asymptotic behavior $G(x, y) \sim c d(x, y)^{-2}$ as $x \rightarrow ...
- 569
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Morse index of mountain pass solution for fractional Laplacian
Suppose $u\in H^s_{0}(\Omega)$ is a classical solution of
\begin{equation}\begin{cases}
(-\Delta)^s u = f(u) & \text{in }\Omega,\\
u=0 & \text{in }\Omega^c.
\end{cases}\end{equation}
with $N\...
- 309
3
votes
1
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118
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Simplicity of elliptic eigenvalue problem
I had asked a (related question) a few months ago - but now I have a different question in the same setting:
Consider the generalized eigenvalue problem:
$$ [- \nabla \cdot (D(\mathbf{x}) \nabla) + \...
- 101
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66
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Abstract higher-order parabolic problems (reference request)
I'm currently reading "Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems" by H. Amann and I have some doubts on how this work may be generalized to higher ...
- 145
3
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86
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Regularity of the Leray Projection on the diffeomorphism group
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed $C^\infty$ Riemannian manifold with $\mathrm{dim}(M) = m$ and let $\mathrm{vol}$ denote the renormalized volume measure on $M$. Let also $s > m/2$...
- 195
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30
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Spectral theory for the boundary operator $P_{g}^{3,b}$ arising from the Paneitz operator on the 4-ball
I am working with the following eigenvalue problem on the 4–ball $(\mathbb{B}^4,g)$, associated with the Paneitz operator $(P_g^4)$ and its boundary operator $P_g^{3,b}$ on $(\mathbb{S}^3,\hat g)$:
\...
- 569
2
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1
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242
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Nontrivial solutions of an elliptic pde
Consider the following PDE:
$$
-\Delta u + \alpha u + \beta (x \cdot \nabla) u = 0.
$$
Is there any nonzero weak solution of this equation on $\Bbb R^n$, in $H^1$ or other function spaces, for some ...
- 873
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119
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how to prove this estimate(gradient estimate)
$Lu=u_{xx}+u_{yy}-\frac{1}{x}u_{x}$,I want to know how to prove the following estimate
$\vert\nabla u\vert\leq\frac{c}{s\vert\partial B_{s}\vert}\int_{\partial B_{s}(X)}u$
(remark:By representing $w(X'...
- 11
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2
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304
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A system of 1st order non-linear PDEs
From a problem in mechanics there comes the PDE system for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$,
\begin{align}
\tau_u&=A(\gamma;u,v)\:\tau^2+F(\gamma;u,v)\:\gamma_u\\
\...
- 176
3
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128
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Any direct and concise way to prove that $ \Delta u+(\theta, d u)_g=f $ is the E–L equation of some functional if and only if $\theta$ is exact
I wonder if there is any direct and concise way to prove that
On Riemannian manifold, $ \Delta u+(\theta, d u)_g=f $ is the E–L
equation of some functional if and only if $\theta$ is exact ($\theta$
...
- 1,041
9
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answers
257
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Additive decomposition of Green's function for elliptic Dirichlet problems
Let $\mathcal{D}\subset\mathbb{R}^n$ be a bounded Euclidean domain with Lipschitz (possibly smoother) boundary and consider an Elliptic Dirichlet problem of the form
\begin{align}
\mathcal{L} u +\...
- 4,852
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48
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Existence for second order elliptic PDE with Robin boundary conditions reference
The paper I am reading is considering the following PDE:
$$
- \nabla \cdot \nabla \rho + (1 - 2 \bar{\rho}) \nabla \rho \cdot u =0
$$
on bounded $\Omega \subset \mathbb{R}^n$ ($n=1,2,3$) with $C^2$ ...
0
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1
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212
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The Leray self-similar solution to Navier-Stokes
The self-similar solution of Navier-Stokes has the form as follows (the $u=(t-T)^{\lambda}U(y)$) has only form of $\lambda = -1/2$.
The work of V. SVERAK et. al. above shows that if $U\in H^1(R^3)$ ...
- 873
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59
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$L^\infty$-$L^1$-bounds for subsolutions of certain elliptic PDE's
Let $\Phi:\mathbb R \times \mathbb T^2\to [0,+\infty)$, $\Phi(s,x,y)\in W^{1,1}_{\mathrm{loc}}$, satisfying $$-\Delta \Phi \leq a (1+\Phi)$$ for some $a>0$, and such that $\Phi(s,\cdot) \to 0$ for $...
- 21
3
votes
1
answer
187
views
Existence of harmonic coordinates on Lipschitz manifolds
Suppose that we have a Riemannian manifold $M$ whose metric is only Lipschitz. Does $M$ admit harmonic coordinates?
- 7,404