Related papers: The Landis Conjecture For Nonlocal Elliptic Operat…

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary.…

We establish a global boundedness result for Lane-Emden systems involving general second-order elliptic operators in divergence form and arbitrary positive exponents whose product equals one. Furthermore, we observe that, for this class of…

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…

A generalization of Callias' index theorem for self adjoint Dirac operators with skew adjoint potentials on asymptotically conic manifolds is presented in which the potential term may have constant rank nullspace at infinity. The index…

We study exponential decay rates of eigenfunctions of self-adjoint higher order elliptic operators on R^n. We are interested in decay rates as a function of direction. We show that the possible decay rates are to a large extent determined…

This paper concerns the non-degeneracy and uniqueness of ground states to the following nonlinear elliptic equation with mixed local and nonlocal operators, $$ -\Delta u +(-\Delta)^s u + \lambda u=|u|^{p-2}u \quad \mbox{in} \,\,\, B, \quad…

We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for…

Motivated by applications to stochastic differential equations, an extension of H\"{o}rmander's hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established…

In 1974, Landis and Oleinik conjectured that if a bounded solution of a parabolic equation decays fast at a time, then the solution must vanish identically before that time, provided the coefficients of the equation satisfy appropriate…

We prove that every weak solution to a certain class of infinitely degenerate quasilinear equations is continuous. An essential feature of the operators we consider is that their Fefferman-Phong associated metric may be non doubling with…

We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles…

The asymptotic behavior of solutions to the second order elliptic equations in exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space $L^{p,q}$ or the weak Lebesgue space…

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse…

Motivated by the study of layer potentials on manifolds with straight conical or cylindrical ends, we introduce and study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of…

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound…

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have…

We investigate the behaviour of the regularized determinant of the Laplace-Beltrami operator on compact hyperbolic surfaces when the genus goes to infinity. We show that for all popular models of random surfaces, with high probability as…

We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for…

We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the…