Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
Questions tagged [ap.analysis-of-pdes]
4260 questions
83
votes
10 answers
Why can't there be a general theory of nonlinear PDE?
Lawrence Evans wrote in discussing the work of Lions fils that
there is in truth no central core
theory of nonlinear partial
differential equations, nor can there
be. The sources of partial
differential equations are so many -
physical,…
Steve Huntsman
- 15,258
42
votes
3 answers
Do we lose any solutions when applying separation of variables to partial differential equations?
For example, consider the following problem
$$\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2},\hspace{0.5cm} u(x,0)=f(x),\hspace{0.5cm} u(0,t)=0,\hspace{0.5cm} u(L,t)=0$$
Textbooks (e.g., Paul's Online Notes) usually apply…
24
votes
3 answers
Does elliptic regularity guarantee analytic solutions?
Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard Sobolev methods seem useless here, and I can't find any…
Paul Siegel
- 28,772
18
votes
4 answers
unique continuation principle
I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on all of $D$ if it vanishes on an open set in $D$"…
Viktor Bundle
- 1,671
14
votes
1 answer
Integrable solutions to an elliptic PDE on divergence form have a definite sign
Let $f\colon\mathbb{R}^n\to\mathbb{R}^n$ be a smooth, bounded vector field. Further, let $u\colon\mathbb{R}^n\to\mathbb{R}$ satisfy $$-\Delta u=\operatorname{div}(fu).$$
If $u\in L^1(\mathbb{R}^n)$, then $u$ has one sign, i.e., either $u>0$…
Harald Hanche-Olsen
- 9,146
- 3
- 36
- 49
10
votes
4 answers
Solutions to a Monge-Ampère equation on the simplex
Let $\Delta_k$ be the k-simplex and $\mu$ a non-negative measure over $\Delta_k$. I want to know if there exists a function $u : \Delta_k \to \mathbb{R}$ such that $u$ is convex, $u(e_i) = 0$ for all vertices $e_i$ of $\Delta_k$, and $M[u] = \mu$…
Mark Reid
- 325
8
votes
2 answers
Which PDE from physics (and geometry) are supercritical?
I am currently trying to understand the notion of criticality (as discussed, e.g., in Terence Tao's book on nonlinear dispersive equations) from a physical viewpoint. That's why i'm interested in the question which PDE from physics (apart from the…
8
votes
2 answers
Methods for determining domains of influence
Given a hyperbolic PDE, the domain of influence of a spacetime point $x$, say $I_x$ though $x$ could be replaced by any set, can be defined in two ways. Lets call one of them geometric ($I_x^G$) and the other analytical ($I_x^A$). In Lorentzian…
Igor Khavkine
- 20,710
8
votes
2 answers
Proof of L^p Elliptic Regularity
Let $L = \sum_{i,j=1}^n -\frac{\partial}{\partial x^i} (a^{ij}(x)\frac{\partial}{\partial x^j}) + \sum_{i=1}^n b^i(x) \frac{\partial}{\partial x^i} + c(x)$ be a second order elliptic operator with smooth coefficients, $\Omega$ a bounded open domain…
8
votes
3 answers
How to find the associated conservation law from a given symmetry
It is a very well-known fact that any conservation law associated with some given PDE has an associated invariance (by Noether's Theorem). However, it is completely mysterious for me how to compute/derive these conservation laws just by knowing the…
Sharik
- 395
8
votes
0 answers
Simple proof for Fefferman-Phong inequality?
The following inequality was proved firstly by Fefferman in the paper: The uncertainty principle, Bulletin of the AMS, 1983. Then it was improved by several authors. The proofs presented in these papers are quite involved. I wonder do we have a…
Dj kahle
- 91
8
votes
1 answer
Space of solutions of nonlinear Helmholtz equation on a torus
On a unit torus $T^n$ (or equivalently, on $\mathbb{R}^n$ with periodic boundary conditions), the linear Helmholtz equation:
$\nabla^2 \phi + k^2 \phi=0$
will have no non-trivial solutions for generic values of $k$, while for special values of $k$…
Greg Egan
- 2,852
7
votes
2 answers
the inverse for the trace theorem
The trace theorem says that the restriction of a $W^{1,p}(\Omega)$ function $u$, $Tu$ belongs to $W^{1-1/p,p}(\partial\Omega)$ if $\Omega$ satisfies some smooth condition, for example, $\Omega$ is convex.
Now my question is the inverse of the…
huilian jia
- 71
7
votes
1 answer
Heat equation bounds
I am interested in the following damped heat equation on $\mathbf{R}$, $u_t = u_{xx} - 1_{x \in [-1,1]} u$ with initial data $u(0,x) = \delta(x-x_0)$ for some $x_0 \in \mathbf{R}$.
In particular I am interested in obtaining non-trivial bounds on…
Matt Cooper
- 143
7
votes
2 answers
Uniqueness of weak solution L[u]=0
Suppose L is a partial differential operator of arbitrary order with constant coefficients.
If u is in $L^p(\mathbb{R}^n)$ and Lu=0 in distributions, is it necessarily the case that u=0? Does the answer depend on p?
Also, if u is a compactly…
Marie
- 73