Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,466 questions
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Finer properties of a sequence of harmonic functions
This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...
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Weakened conditions on the smoothness of the domain in the regularity and a priori estimate of Agmon, Douglis, and Nirenberg for elliptic systems
I have read in a couple of places (e.g. An Introduction to PDEs by Renardy and Rogers, p.309) that the smoothness hypotheses on the domain in the a priori estimate of Agmon, Douglis, and Nirenberg for ...
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vibrations of higher dimensional drums
The solutions of the wave equation for a circular drum with fixed boundary are well known. What do the solutions look like for a spherical drum in three spatial dimensions? What about higher ...
- 479
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Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius
I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
- 101
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Asymptotics of quasilinear elliptic equations with Dirac right hand side
On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form
$$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha \...
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Uniqueness for solution of a d-dbar system related to Davey-Stewartson Solitons
This question concerns a system of equations that arise in the study of one-soliton solutions to the Davey-Stewartson equation.
In what follows, $f(z)$ denotes a function which depends smoothly (but ...
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When can a perturbation be treated as a regular perturbation?
I am working with cauchy problem of the form
$$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$
where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...
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Transformation from domains to half-spaces
In a paper I read, an elliptic boundary value problem
on a bounded domain D x (0,T) is solved by first transforming
it in a set of equations on half-spaces R^n and then applying
partial Fourier ...
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decay for spatially discrete parabolic equations with non-constant non-self-adjoint right hand side
Consider the following uniformly parabolic lattice differential equation
$ \begin{array}{ccc} \dot{u}_{n,m} & = & \alpha_{n,m}(u_{n+1,m} - u_{n,m}) + \beta_{n,m}(u_{n-1,m}-u_{n,m}) \\ & &...
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Generalizations of group algebras for arbitrary manifolds?
In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take ...
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Regularity of reflection coefficients (or more generally the scattering transform)
Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$.
We define the ...
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elliptic system; bounds on $v$ when $u$ is small
I am interested in the following system
$-\Delta u = f(u,v) $
$-\Delta v = g(u,v)$ in $ \Omega$ a bounded domain in $ R^N$ with $ u=v=0$ on the boundary.
The solutions are smooth and positive. ...
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Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
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Comparison Principle for Courant Nodal Domain Theorem
Recall that Courant's Nodal domain gives an upper bound on the number of nodal domains of the $k$-th eigenfunction. I was wondering if it is possible to deduce some relation between the number of ...
- 537
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Finding a particular solution to a Lax pair on $[0, \infty)$ to solve $q_t + q_{xxx} + u(x)q = 0$ using the Fokas Method
I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here).
This post is a follow up to my first that ...
- 547
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On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
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