Questions tagged [boundary-value-problems]
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9 questions
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Neumann problem for the Laplacian with Dirac delta functions
I have encountered a problem while dealing with the adjoint method in potential flow that is also described, in a similar fashion, in (eq. 39) of this paper. The problem is essentially this:
$$\begin{...
- 21
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type of solutions of $-u^{\prime\prime}=\lambda e^{u}$ based on the value of the parameter $\lambda$. (Gelfand problem)
My question comes from the book
Stable solutions of elliptic partial differential equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143. Boca Raton, FL: CRC ...
- 11
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124
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Solve coupled ODEs analytically in the limit of a small parameter
I have the following set of coupled second order non-linear ODEs :
$$ x^2 a''(x) + x a'(x) - \Big(\frac{1}{\epsilon^2}\Big)b^2(x) a(x) = 0 \\
x b''(x) - b'(x) - 2x b(x)a^2(x) = 0$$
with boundary ...
- 11
2
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Intuition behind oscillating eigenvalues for inhomogeneous Sturm-Liouville problem
I was looking at an article where the author finds an example of a solotone effect in a heat transfer context of a 1D layered composite rod rather than the usual metal rod made out of one material (...
- 5,146
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37
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Definition of coercive boundary value problems
In Folland's Introduction to PDE,page 242, he defines what it means for a Dirichlet form to be coercive (a standard definition). Let $X$ be a closed subspace of $H_m(\Omega)$ with $H_m^0(\Omega)\...
- 263
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The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients
Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by
$$Lu = \partial_i(a^{ij}...
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73
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Elliptic regularity for Dirichlet problem
Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$.
Let $P$ be an injective ...
- 577
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$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint.
Consider the problem
$$\Delta u = f \quad\text{in $\...
- 93
1
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1
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Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
- 3,065