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The linearization intended by Bethuel et al. is around $\tilde{v}=1$, so around $f=0$, $g=0$, so the linearized equation is $0=L_0 \left( \begin{array}{l} f \\\ g \end{array} \right)$ with $L_0 = \le …

The case $p=2$ is the nonlinear Schrödinger equation, more generally written as $$Eu=-\Delta u+\kappa|u|^2 u,$$ with coefficients $E,\kappa\in\mathbb R$. It describes the propagation of light in nonli …

The gradient flow $$\frac{\partial u}{\partial t}=\epsilon^2\Delta u+f(u),\;\;\text{with}\;\;f(u)=u(1-u^2),$$ of the Allen-Cahn functional has no inertia (there is no second order derivative in time), …

Here is one study Overdetermined elliptic problems in physics. It is proven in $\mathbb{R}^2$ that the Poisson equation $\Delta u=-{\rm constant}$ in $\Omega$, with boundary conditions $u=0$, $\partia …

The solution to $$(-\Delta)^s u(x)=1,\;\;-1<x<1, $$ with $u(x)=0$ for $|x|\geq 1$ is $$ u(x)=\frac{\sqrt{\pi}(1-x^2)^s }{4^{s}\Gamma(\tfrac{1}{2}+s)\Gamma(1+s)},$$ as follows from the integral definit …

References to Liouville go back to his 1847 result that a doubly periodic function without poles is identically constant, which does not yet contain the generalization to either harmonic functions or …

You could just try: $A\mapsto A+\nabla \lambda$ leaves the first two equations invariant if I transform $u\mapsto e^{i\lambda}u$ and leave $V$ the same, but then the third equation no longer holds, so …

Perhaps to resolve this issue it helps to work out a simple example. Take a region $D$ consisting of the strip $|x|<1$, $0<y<1$, and a $y$-independent conductivity profile $$\sigma(x)=\begin{cases} 1 …

No, the implication in the title is not true, as a simple counterexample, take a factorized $f(t,x)=a(t)b(x)$, with $\int_0^\infty a(t)dt=0$ but $a(t)$ not identically equal to zero and $\nabla_x b(0) …

Random walk method for the two‐ and three‐dimensional Laplace, Poisson and Helmholtz's equations (paywall) Random Walk Method for Potential Problems (freely accessible) The random walk method is …

Q: Can we conclude that $Lu=\Delta u+ fu=0$ has only zero (or constant solutions) if we assume $f$ non-constant? A: No, a counter example in one dimension is the Mathieu equation, which has non-consta …

For elliptic PDE applications to options these would need be independent of time, they need to be perpetual (i.e. never expire), which is not a typical scenario. If your definition of "mathematical fi …

Back in 2012, professor Ben Chow gave some advice to a similar question; these include the Lectures on Elliptic and Parabolic Equations in Sobolev Spaces by Krylov [recommended here by Giorgio Metafu …

I made a trip to the library and scanned the relevant pages from Miranda's 1955 book: page 152-153 and page 154-155 the references are: [3] J. Leray, J.Math. pures et appl. 17, 89-104 (1938) [8] R. Ca …