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Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Let $u\in C^2(\bar \Omega)$ be a solution of the Dirichlet problem for the non-linear equation \begin{eqnarray} F( …

Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier …

In his book "Some non-linear problems in Riemannian geometry" T. Aubin states the following result (Theorem 3.56): Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order differential opera …

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the …

Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions: \begin{eqnarray*} \int_B |f_\lambda(x)|^2dx\leq 1,\\ …

Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions $$X=\{u\in C^2(\bar B) \mid u|_{\partial B}=0 \text{ and } \|\Delta u\|_{L^2(B)}\leq 1\},$$ where $\Delta$ …

Let $A$ be a linear operator between two Hilbert spaces. Let $A^*$ be its adjoint. Question. Under what conditions the non-zero spectra of $A^*A$ and $AA^*$ coincide counting multiplicities? In my s …

Let $D$ be a linear differential elliptic operator of second order with infinitely smooth coefficients acting on real valued functions on a compact manifold $M$. Let us assume that $D$ has no free ter …

In the paper in quantum fields theory by Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19; in Section 3 the author makes the following claim from PDE and op …

This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian …

EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a compact comp …

Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting o …