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As your title indicates, you would do this using the implicit function theorem for Banach space applied to an appropriately defined functional. This in turn requires proving that there is a bounded ri …

You might find it helpful to study previous work on specific functionals like this. I suggest searching for journal articles that talk about extremal Kahler metrics (which usually means studying the i …

With a linear elliptic PDE, there's no way to bootstrap. What you see is what you get. The regularity of $A^{ij}\partial_j u$ cannot be made any better than the regularity of $g^i$. So if all you assu …

I just stumbled onto this: On fully nonlinear elliptic equations of second order by Louis Nirenberg which seems to answer your question in the affirmative. Also, are you really studying a general f …

In general nothing is known. Only local solvability is known for some special cases.

I've seen only the first. It is indeed used mostly for identifying whether a nonlinear PDE is elliptic, hyperbolic, or parabolic. If so, one can use the respective linear theory, along with the approp …

If you assume that the coefficients $a^{ij}$ are smooth functions and let $$b^{ij} = \frac{1}{2}(a^{ij} + a^{ji}),$$ then the PDE can be written as $$ b^{ij}\partial^2_{ij}u + \partial_ia^{ij}\partial …

Previous answer replaced by new one: Fix a finite relatively open cover $U_1, \dots, U_m$ of $M$, where each $U_i$ is diffeomorphic to either the open ball or the half-ball obtained by intersecting …

First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$, $$ \|u\|_{\frac{2n}{n-2 …

Start with the linear problem: Let $K \subset \Gamma(E)$ denote the kernel of $P$ and $\hat{K} \subset \Gamma(F)$ the cokernel. Assume that there are inner products on the vector bundles $E$ and $F$ a …

Any Lie bracket structure that is a first order differential operator is, by bilinearity and skew-symmetry, of the form $$ \mathrm{ad}_X Y = A^{kp}_{ij}(X^i\partial_pY^j- Y^i\partial_pX^j)e_k, $$ wher …

Assuming that $P^{-1}$ is a right inverse and $\Omega$ an open subset of $\mathbb{R}^n$ or an open manifold, then you can proceed as follows: 1) An operator $Q: L^2(\Omega) \rightarrow L^2\Omega$ is …

Let me turn my comment into a general discussion for which a special case is an answer to your question. First, on an open domain $D \subset \mathbb{R}^n$, there is a standard elliptic regularity res …

This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on ellipti …

Use a partition of unity to reduce the statement to a local one for a function compactly supported on a coordinate chart. At that point, any elliptic regularity theorem on an open domain in Rn can be …