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Results tagged with ap.analysis-of-pdes
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user 103792
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
9
votes
1
answer
586
views
When does the eikonal equation $\lvert Du \rvert^2 = f$ admit a local solution?
Let $f$ be a smooth function defined on the unit disc $D \subset \mathbf{R}^2$ with
\begin{equation}
f \geq 0 \text{ in $D$ and } f(0) = 0.
\end{equation}
This is allowed to have a degenerate minimum …
- 5,048
6
votes
0
answers
118
views
Entire solutions of the Ginzburg-Landau equation in the plane
Question. Are all solutions $u: \mathbf{R}^2 \to \mathbf{C}$ of the Ginzburg-Landau equation (1) radially symmetric? What if one imposes additionally that $\int_{\mathbf{R}^2} ( 1 - \lvert u \rvert^2) …
- 5,048
6
votes
0
answers
318
views
Are the two-valued homogeneous harmonic functions classified?
Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$?
For reference, multi-valued functions are familiar objects in comple …
- 5,048
5
votes
0
answers
165
views
Singularities of phase interfaces in closed surfaces
Let $(\Sigma,g)$ be a compact surface without boundary. Given $\epsilon > 0$, the $\epsilon$-Allen-Cahn equation is the semilinear elliptic PDE $\epsilon \Delta_g u - \epsilon^{-1} W'(u) = 0$, with un …
- 5,048
4
votes
0
answers
148
views
What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?
As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ sho …
- 5,048
4
votes
1
answer
336
views
Is there a harmonic function with just one singular point?
Let $D \subset \mathbf{R}^2$ be the unit disc, and $L > 0$. Let $u: D \times (-L,L) \to \mathbf{R}$ satisfy
\begin{equation}
\begin{cases}
\Delta u = 0 \quad \text{ on $D \times (-L,L)$ } \\
\frac{\p …
- 5,048
3
votes
1
answer
343
views
Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation?
I should preface this question by saying that I strongly suspect the answer is negative, but I couldn't find the counterexample myself.
Say we are working on the unit disc $D \subset \mathbf{R}^n$, wh …
- 5,048
3
votes
1
answer
107
views
A harmonic function degenerate in one direction
Question. Let $u: B^3 \to \mathbf{R}$ be a harmonic function with $u(0) = 0$, $Du(0) = 0$, where its homogeneous harmonic blow-up is a polynomial $p = p(x,y)$ in two variables, so independent of $z$; …
- 5,048
3
votes
0
answers
116
views
Approximation of viscosity subsolution
Let $u: \Omega \to \mathbf{R}$ be a $C^{0,\alpha}$ function, with $\alpha \in (0,1]$, defined on a bounded, open domain $\Omega$. Suppose that $u$ is a viscosity subsolution of the equation $\Delta U …
- 5,048
3
votes
0
answers
56
views
Intersection of $n$-dimensional minimal surfaces with two-dimensional planes
Let $M^n \subset \mathbf{R}^{n+k}$ be a smoothly embedded minimal surface. When the dimension is $n = 2$ and the codimension is $k = 1$ the intersection of $M$ with planes is well understood. If $M$ a …
- 5,048
3
votes
1
answer
145
views
'Degenerate' tangent point of a minimal graph
Let $u: D_1 \to \mathbf{R}$ be a smooth function defined on the unit disk $D_1 \subset \mathbf{R}^2$ which describes the minimal graph $G$. Suppose that at the origin $G$ is tangent to the horizontal …
- 5,048
3
votes
2
answers
381
views
Heating a long cylinder: steady states
Consider a long cylinder $C = D \times (-L,L) \subset \mathbf{R}^3$, with heat applied to its horizontal boundary according to $\varphi$ and perfectly insulated ends. The steady state $u: C \to \mathb …
- 5,048
2
votes
0
answers
148
views
Extensions of minimal hypersurfaces
Let $B \subset \mathbf{R}^{n+1}$ be the unit ball, and $M \subset B$ be a minimal hypersurface. By this we mean that $M$ is an embedded $n$-dimensional submanifold with vanishing mean curvature. We al …
- 5,048
2
votes
0
answers
207
views
Can you compute one eigenspace without computing them all?
Maybe the simplest non-trivial settings in which the spectrum of the Laplacian be can be computed is on the round sphere $\mathbf{S}^n$, and for products of manifolds. I want to use the two as example …
- 5,048
2
votes
1
answer
184
views
Reference for harmonic functions in cylinders
Question. What is a good reference (textbook, article etc.) to learn more about harmonic functions on finite (and infinite) cylinders?
I am trying to gain a better understanding of the behavior of har …
- 5,048