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Main Question

Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation: $$ -\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0 $$ Here $a_i:\mathbb{R}^m \to \mathbb{R}$ and $B:\mathbb{R}^m \to \mathbb{C}^{n \times n}$ are $C^2$, bounded functions. Furthermore, $B$ is valued in Hermitian matrices. Also define: $$ \Lambda(B) = \min_{x \in \mathbb{R}^m, v \in \mathbb{C}^n, |v| = 1} \langle v,B(x)v\rangle $$ Then the following is true (proofs are below).

Proposition: $|F|^2 \le \min(-\Lambda(B),0)$.

My question is about the following slightly modified equation: $$ -\Delta F(x) + \sum_i A_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0 $$ Here the $A_i:\mathbb{R}^m \to \mathbb{C}^{n \times n}$ are $C^2$, bounded functions valued in Hermitian matrices. I have 4 main questions, and I would appreciate partial answers to any of them.

  1. Are there known examples of equations like this where a $C^0$ estimate explicitly does not hold?
  2. Are there known conditions on the matrix $A_i$ that make a $C^0$ estimate hold?
  3. Is there any literature on this problem? I'm having trouble finding anything very helpful on my own.
  4. Does anyone have approaches to recommend?

A Little Context

I'm trying to prove compactness of the space of solutions of a related system of PDE, where the PDE is formulated for sections of some vector bundle with a Riemannian connection. I'm using the above Euclidean space formulation to avoid being too theory heavy.

Proofs

Here are two proofs of the proposition. They are essentially the same, but the second one reveals a maximum principle going on in the background that's sort of interesting.

Proof 1: Let $x_0$ be a point where $|F|^2$ takes its maximum. Then since: $$-\Delta |F|^2 + 2|\nabla F|^2 = -2\text{Re}\langle F,\Delta F\rangle$$ and $-\Delta |F|^2 \ge 0$ at $x_0$, we have $-\text{Re}\langle F,\Delta F\rangle \ge 0$ at $x_0$. Also, $\nabla_i |F|^2 = \text{Re}\langle F,\nabla_i F\rangle = 0$ for all $i$ at $x_0$.

Now by taking the inner product of the above equation with $F$ and taking the real part we have: $$ \Lambda(B)|F(x_0)|^2 + |F(x_0)|^4 \le$$ $$ -\text{Re}\langle F,\Delta F\rangle + \sum_i a_i(x_0)\text{Re}\langle F(x_0),\nabla_i F(x_0)\rangle + \langle F(x_0),B(x_0)F(x_0)\rangle + |F(x_0)|^4 = 0 $$ Suppose that $|F(x_0)| \neq 0$. Then we have $\Lambda(B) + |F(x_0)|^2 \le 0$ and our result follows.

Proof 2: The idea of this proof is to apply the maximum principle for elliptic systems (see [Wang90]).

Write the equation in question as: $$\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + \Psi(x,F(x)) = 0$$ with $\Psi:\mathbb{R}^m \times \mathbb{C}^n \to \mathbb{C}^n$ defined by $v \mapsto -B(x)v - v|v|^2$. Now suppose that the proposition were false, i.e suppose that there existed an $F$ such that: $$ \max(|F|^2) > \min(-\Lambda(B),0) $$ Let $K$ be the ball of radius $\max(|F|^2)$ in $\mathbb{C}^n$ and for any $v \in \partial K$ let $\nu_K(v) = \frac{-v}{|v|}$ denote the (radial) inward normal vector. Then observe that for any $x \in \mathbb{R}^n$ and $v \in \partial K$ we have: $$ \langle \nu_K(v),\Psi(x,v)\rangle = -\frac{1}{|v|}\langle v,\Psi(x,v)\rangle \ge \frac{1}{|v|}(\Lambda(B)|v|^2 + |v|^4) = \max(|F|)(\Lambda(B) + \max(|F|^2)) > 0 $$

The maximum principle of [Wang90] then implies that since $F(x) \in \partial K$ for some $x$, $F(y) \in \partial K$ for all $y$, i.e $|F(x)|^2$ is constant and $|F(x)|^2 > \min(-\Lambda(B),0)$ everywhere. However, observe that: $$ \int \Lambda(B)|F(x)|^2 + |F(x)|^4 \le \int |\nabla F|^2 + \langle F,BF\rangle^2 + |F|^4 = $$ $$\int \langle F,-\Delta F + BF + F|F|^2\rangle = \int \text{Re}\langle F,-\Delta F + \sum_i a_i\nabla_iF + BF + F|F|^2\rangle = 0 $$ Here we use integration by parts from the 2nd to 3rd line. Then we use the facts that $\text{Re}\langle F,\nabla_iF\rangle = 0$ and that the 2nd line is real to go from the 3rd to 4th. This implies that $|F(x)|^2 > \min(-\Lambda(B),0)$ is impossible.

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