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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.
2
votes
1
answer
205
views
Does unique continuation also hold for derivatives of solutions?
Let $D \subset \mathbf{R}^n$ be the unit ball, and $u \in C^2(D)$ be a solution of the linear elliptic PDE
\begin{equation}
a^{ij} D_{ij} u + b^i D_i u + cu = 0 \quad \text{in $D$},
\end{equation}
whe …
- 5,048
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
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
1
answer
105
views
'Dirichlet problem' along axis for harmonic functions
Question. Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the axi …
- 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
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
2
votes
1
answer
96
views
The attractive 'force' between phase interfaces in the Allen-Cahn model
The heuristic explanation of the behavior of phase transition in the Allen–Cahn model describes two 'forces' at play:
the curvature of the phase interfaces—they each 'want to' minimize length;
and an …
- 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
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
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
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
2
votes
1
answer
248
views
Existence of divergence-free unit vector field in conformally rescaled euclidean metric
Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div …
- 5,048
2
votes
0
answers
53
views
Has the nodal map been studied?
Let $D \subset \mathbf{R}^n$ be the unit disc, and $\alpha \in (0,1)$. Let $f \in C^{0,\alpha}(\partial D)$, and $u \in C^{2,\alpha}(D)$ be the harmonic function with $u = f$ on the boundary.
Define t …
- 5,048
0
votes
1
answer
187
views
Distance function to mean curvature flow
In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the rea …
- 5,048