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Results tagged with elliptic-pde
Search options questions only
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user 241460
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1
vote
1
answer
97
views
Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>2 $
Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>2 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.
Since …
1
vote
0
answers
39
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are nonemp …
2
votes
0
answers
77
views
$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega, …
1
vote
0
answers
490
views
How to deal with the boundary estimate for the Schauder estimates of laplacian equations?
Recently, I am learning Schauder estimates for elliptic equations and I come across a proposition as follows
Let $ \alpha\in (0,1) $ and $ \Omega $ be a bounded $ C^2(\Omega) $ domain on $ \mathbb{R} …
0
votes
1
answer
99
views
Limit of minimizers of a class of functionals
Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ u_0 …
1
vote
1
answer
153
views
How to show that $ u $ is vanishing in $ \mathbb{R}^3\setminus B_1 $?
I come across an interesting question.
Let $ B_r=\{x\in\mathbb{R}^3:|x|\leq r\} $ be the ball in $ \mathbb{R}^3 $ with radius $ r $. Assume that $ u \in C(\mathbb{R}^3\setminus B_1) $ satisfies
$$
\D …
5
votes
2
answers
1k
views
How to prove the second Korn inequality?
$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\
\begin{eqnarray}
\sqrt{2}\left\|\triangledown u\right\| …
3
votes
1
answer
228
views
Schauder estimates with boundary conditions
For the elliptic equation with non-divergence form
$$
\sum_{i,j=1}^na_{ij}(x)\partial_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1),
$$
where $ \{a_{ij}(x)\} $ is a matr …
8
votes
2
answers
2k
views
Why don't we study hyperbolic equations as elliptic and parabolic equations?
In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small bal …
-1
votes
1
answer
76
views
Applications and motivations of resolvent for elliptic operator
Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
…
4
votes
1
answer
320
views
The behavior of $ \nabla u $ on the boundary for Poisson equations
Let $ \Omega $ be a bounded domain with smooth boundary. Consider the Poisson equation
\begin{eqnarray}
-\Delta u&=&f\text{ in }\Omega\\
u&=&0\text{ on }\partial\Omega
\end{eqnarray}
where $ f\in C_0^ …
1
vote
0
answers
71
views
Elliptic systems with two dimensions
Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\alpha, …
1
vote
0
answers
80
views
Boundary estimates for elliptic systems
Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\alpha, …
1
vote
0
answers
53
views
What is the the "method of ascending" in the study of elliptic systems in dimension two?
I have read a paper of Z. Shen [1]. In the paper the author mentioned we can deal with two-dimensional elliptic systems by adding a dummy variable (the method of ascending) and use the results on the …
2
votes
0
answers
312
views
Caccioppoli inequality in $ \mathbb{R}^2 $
Let $ \Omega\subset\mathbb{R}^d $ be a Lipschitz domain and $ A=(a_{ij}(y)):\Omega\to\mathbb{R}^{d\times d} $ be a matrix valued function with uniformly elliptic conditions i.e. $ \lambda|\xi|^2\leq a …