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Results tagged with ap.analysis-of-pdes
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user 2082
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
1
answer
211
views
Are there applications for the PDE $ - \operatorname{grad} ( \operatorname{div} \vec u ) = \...
As in the title: given a vector field $\vec f$, are there any interesting applications (in physics, biology, or economy, or ...) of the partial differential equation
$ - \operatorname{grad} ( \operato …
- 5,327
18
votes
4
answers
3k
views
Einstein field equations in perspectives from PDE and functional analysis
The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in …
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1
vote
0
answers
102
views
Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product
Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.
Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( …
- 5,327
1
vote
1
answer
751
views
Tensor analysis/Differential forms outside physics
There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, lik …
1
vote
1
answer
246
views
$L^2$-de-Rham complex on Lipschitz domains has smooth harmonic forms?
I would like to know for which choice of boundary conditions the title statement is true.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.
…
- 5,327
28
votes
1
answer
4k
views
Why are viscosity solutions useful solutions?
I refer to definition of viscosity solution in user's guide to viscosity solutions of second order partial differential equations by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.
Viscosit …
- 5,327
59
votes
9
answers
10k
views
Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential opera …
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