Related papers: Elliptic complexes and generalized Poincar\'e ineq…

This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{exactly solvable};…

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between…

Even if the Fourier symbols of two constant rank differential operators have the same nullspace for each non-trivial phase space variable, the nullspaces of those differential operators might differ by an infinite dimensional space. Under…

We prove the following operator-coefficients free Poincar\'{e} inequality: $$|f(X)-E[f(X)]|_{2}\leq2|X|_{2}\|\widehat{\partial}_{X:B}[f(X)]\|_{\pi}, \quad f(X) \in \mathrm{dom}(\widehat{\partial}_{X:B}),$$ where $\|\cdot\|_{\pi}$ is the…

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, $\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator…

We establish several sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor in this paper. The first presented sufficient condition is an extension of positive definite matrices, which states that the strong…

The general theory of boundary value problems for linear elliptic wedge operators (on smooth manifolds with boundary) leads naturally, even in the scalar case, to the need to consider vector bundles over the boundary together with general…

Tools of the intrinsic analysis on manifolds, helpful in solving the invariant inverse problem of the calculus of variations are being presented comprising a combined approach which consists in the simultaneous imposition of symmetry…

We develop an elliptic theory based in $L^2$ of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum. In particular, we do not require the indicial roots to be…

On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphism satisfying an arbitrary polynomial equation with real constant coefficients. We study the compatibility of these structures with the de Rham…

In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed…

Poincar\'e gave a criterion which determines the shape of equilibrium for planar differential equations. In his statement, he excluded the case of repeated eigenvalues. In fact, in such a case, we can give a $C^1$ counter-example to his…

We consider general symmetric systems of first order linear partial differential operators on domains $\Omega \subset \mathbb{R}^d$, and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The…

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…

We study existence and Lorentz regularity of distributional solutions to elliptic equations with either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise…

We study the generalized eigenvalue problem in $\mathbb{R}^N$ for a general convex nonlinear elliptic operator which is locally elliptic and positively $1$-homogeneous. Generalizing article of Berestycki and Rossi in [Comm. Pure Appl. Math.…

We deal with regular Lagrangian constrained systems which are invariant under the action of a symmetry group. Fixing a connection on the higher-order principal bundle where the Lagrangian and the (independent) constraints are defined, the…

We give sufficient conditions, on data including the monodromy representation, the Stokes matrices and the Poincare ranks at prescribed singularities, to solve the generalized Riemann-Hilbert problem with irregular singularities. We recover…

This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant $p \geq 1$. We consider ordinary and elliptic problems. The results obtained in the…

Given a complex, elliptic coefficient function we investigate for which values of $p$ the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly…