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Results tagged with ap.analysis-of-pdes
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user 66623
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
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0
answers
184
views
Poisson equation on exterior of a ball
Let $ B_1^c$ denote the compliment of the unit ball centered at the origin in $ R^N$ where $N \ge 3$. I am interested in $ -\Delta u(x)=f(x)$ in $ B_1^c$ with $ u=0$ on $ \partial B_1^c$. In partic …
- 1,385
1
vote
0
answers
89
views
Two dimensional embedding
Take $ \alpha\gt0$ and consider $\Omega=\{ x \in R^2: x_1^2+x_2^2 \lt 1, x_i\gt 0 \}$ (first quadrant of unit ball in plane). I am interested in optimal (so I am looking for the range of $p$) embedd …
- 1,385
4
votes
0
answers
95
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Biharmonic operator and maximum principle (PPP)
I have a question related to the Positivity Preserving Principle (PPP) for $ \Delta^2$ and related topics. Recall if $u$ solves
$$\Delta^2 u = f(x) \mbox{ in } \Omega, \quad u=\partial_\nu u =0 \m …
- 1,385
2
votes
0
answers
75
views
elliptic pde question
Suppose $\Omega$ is a smooth bounded domain in $ R^N$ and suppose we have
$$-\Delta u(x) = (u(x)_+)^3 \qquad \mbox{ in } D$$
and $$ -\Delta u(x) = (u(x)_-)^3 \qquad \mbox{ in } \Omega \backslash \ov …
- 1,385
1
vote
0
answers
47
views
elliptic pde with supercritical advection term
Let $B$ denote the unit ball in $ R^N$ centered at the origin and consider
$$ -\Delta u(x) + \frac{ x \cdot \nabla u(x)}{|x|^\alpha} = f(x) \quad \mbox{ in } B$$ with $u=0$ on $ \partial B$. (or i …
- 1,385
0
votes
1
answer
159
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boundary integral estimates for elliptic pde
Consider smooth positive solutions $u_m$ of
$$-\Delta u_m(x) = u_m(x)^p \quad \mbox{ in } \Omega$$ with $u_m=0$ on $ \partial \Omega$. My interest is in obtaining some sort of global integral estim …
- 1,385
2
votes
1
answer
99
views
Maximum principle for an elliptic system
Suppose $\Omega$ is a bounded smooth domain in $ R^N$ and $ a,b$ are bounded smooth vector fields in $ \Omega$. Suppose we have
$$ -\Delta u(x) + a(x) \cdot \nabla v(x) = f(x) \ge 0 \quad \mbox{ in …
- 1,385
3
votes
0
answers
75
views
solutions of a pde smooth with respect to a parameter
I have a generic type question, but I will pose it with a specific example. Suppose $1<p<\frac{N+2}{N-2}$ and $B_1$ is the unit ball in $ R^N$ for $N \ge 3$. Consider the pde
$$-\Delta u(x) = t …
- 1,385
2
votes
1
answer
165
views
Simple elliptic pde problem
I have a question that is clearly not research level, but it's confusing me so I will ask anyway.
There must be some little logic flaw I am missing. Take $\Omega$ a bounded smooth domain in $\mathbb …
- 1,385
2
votes
0
answers
50
views
Rescaling an elliptic system
I have a basic question regarding rescaling an elliptic system when trying to get apriori estimates.
Consider an elliptic system say of the form
$$-\Delta u(x) = u^{p_1} v^{q_1}, \quad -\Delta v = u^{ …
- 1,385
2
votes
0
answers
86
views
Elliptic pde and boundary layer estimates
I have a question that is related to finding some boundary layer estimates. I am sure there is a general method for this but I don't know it.
I will pose the problem in two dimensions (but here there …
- 1,385
3
votes
1
answer
133
views
Positive first eigenvalue; operator satisfies maximum principle
I am attempting to understand a paper. They have $u$ is a stable smooth solution of $ -\Delta u = f(u)$ in $ \Omega$ with $ u=0$ on $ \partial \Omega$ where $\Omega$ is a bounded domain in Euclidean …
- 1,385
1
vote
0
answers
47
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Odd extension of radial solutions of elliptic pde
I have a question about trying 'odd' like extension to obtain some sign changing solutions of an elliptic equation.
Lets first consider the 1 dimensional problem. To solve a sign changing solution of …
- 1,385
4
votes
0
answers
105
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Gradient bounds on a solution of a linear elliptic problem
Take $\Omega$ to be a bounded domain in $N$ dimensional Euclidean space with smooth boundary and we assume $\Omega$ contains the origin. I am interested is the following equation
$$ \Delta \phi(x) …
- 1,385
3
votes
0
answers
63
views
Liouville theorem for linear equation
Let $\Omega$ denote a an exterior domain in $R^N$ with smooth boundary.
I am interested in Liouville Theorems related to smooth solutions of
$$\Delta \phi(x) + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x| …
- 1,385