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Results tagged with ap.analysis-of-pdes
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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4
votes
Accepted
Classification of a certain System of Linear First Order PDEs whose characteristic polynomia...
Your system of PDE's appears to be of real principal type, as defined by Hormander. Hormander studied the singularities of distributional solutions to such a PDE and how they propagate. This in turn l …
- 27.5k
3
votes
Quasi-linear System of First Order P.D.E.s of "Mixed" type
I gave an answer to your earlier question, but although it was technically correct, I see now that since you have only two independent variables my answer wasn't really the right one. Let me try one m …
2
votes
Accepted
Convergence of elliptic operators
It suffices to show that the $L_2$ operator norm of $A_t\circ A_0^{-1} - I = (A_t - A_0)\circ A_0^{-1}$ is small if $t$ is sufficiently small. To do this, it suffices to show that the operator norm of …
- 27.5k
4
votes
Accepted
meromorphic family of pseudo-differential operators
If you set $A(y) = S(y) - S(z)$, your question is equivalent to asking if $A(z)$ is a holomorphic family of pseudo-differential operators such that $A(0) = 0$, then does $\|A(z)u\|_\infty \rightarrow …
- 27.5k
5
votes
Accepted
Removable Singularities for Elliptic Equations
The proof I had in mind was actually simpler and appears to work only in dimension greater than 2. Here is a sketch for a second order self-adjoint elliptic operator $Pu = \partial_i(a^{ij}\partial_ju …
- 27.5k
4
votes
Solutions to a Monge-Ampère equation on the simplex
This is not an answer, but...
I don't know of any previous work on this, but it appears to be well worth studying. Techniques inspired by convex geometry, like those used to solve the Minkowski probl …
- 27.5k
18
votes
Why can't there be a general theory of nonlinear PDE?
I agree with Craig Evans, but maybe it's too strong to say "never" and "impossible". Still, to date there is nothing even close to a unified approach or theory for nonlinear PDE's. And to me this is n …
15
votes
Why can't there be a general theory of nonlinear PDE?
Some more random thoughts:
The closest thing I've ever seen to a "general theory of nonlinear PDE's" is Gromov's book, Partial Differential Relations. He does many things in there that I don't unders …
5
votes
Accepted
Extremal functions for Gagliardo-Nirenberg inequality
There is a $1$--parameter family of inequalities where the sharp constants and corresponding extremal functions are known. I believe this was first established by Del Pino and Dolbeault. Cordero, Naza …
- 27.5k
7
votes
Accepted
Mathematical difference between entropy and energy
First, note that entropy is well-defined only if $u$ is positive. Using integration by parts, it is in fact true on the flat torus or $\mathbb{R}^n$ (if $u$ is nonnegative and decays in space fast eno …
- 27.5k
2
votes
PDEs involving measures; where to begin?
Almost any book on pseudodifferential operators will discuss how to prove existence, uniqueness, and regularity for the elliptic PDE. I'm not sure about the parabolic PDE.
I learned this stuff from " …
- 27.5k
1
vote
Analyticity of the solutions of PDE
This can't happen if the PDE is either elliptic or parabolic. If the PDE is hyperbolic and you start with initial data that is nowhere analytic, then it seems plausible that the resulting solution is …
- 27.5k
2
votes
PDE with the Jacobian Determinant
This is not really an answer, but I prefer the luxury of the answer box instead of the rather spartan comment box.
You have only one equation for $n$ unknown functions (the components of the vector-v …
- 27.5k
5
votes
PDEs as a tool in other domains in mathematics
The work of Meeks and Yau using minimal surfaces is a beautiful application of nonlinear elliptic PDE's to low-dimensional topology.
4
votes
Accepted
Brenier's theorem
Because there's no reason to ship a whole gaussian en masse to the same target gaussian. It's cheaper to send all the mass that is to one side of the diagonal (line joining (0,0) to (1,1)) to one gaus …
- 27.5k