Related papers: Eigenvalue Inequalities for Fully Nonlinear Ellipt…
We present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan…
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations…
In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the…
We prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack…
We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying…
We introduce the notion of $ P -$functions for fully nonlinear equations and establish a general criterion for obtaining such quantities for this class of equations. Some applications are gradient bounds, De Giorgi-type properties of entire…
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 consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $ L = \tfrac{1}{ \rho } ( -\Delta + V ) $ and divergence form operators $ L = -\operatorname{div} A \nabla $, on bounded…
We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…
In this article we review a new method for proving the nonexistence of positive solutions of elliptic inequalities in unbounded domains in $\rn$, which was recently introduced by the authors. We expose our method and new results on the two…
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-\'Emery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate…
Let $\Omega$ be a bounded, smooth domain of $\mathbb R^N$, $N\ge 2$. In this paper, we prove some inequalities involving the first Robin eigenvalue of the $p$-laplacian operator. In particular, we prove an upper bound for the first Robin…
In this paper, we compute universal estimates of eigenvalues for a class of coupled systems of elliptic differential equations in divergence form on a bounded domain in Euclidean space, which includes the well-known Lam\'e and the Laplacian…
In this work we provide an Aleksandrov-Bakelman-Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a…
In this paper we deal with existence and uniqueness of solution to super-linear problems for the Pucci operator: $$ -\M^+(D^2u)+|u|^{s-1}u=f(x) \quad {in} \RR^n, $$ where $s>1$ and $f$ satisfies only local integrability conditions. This…
We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based…
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, \alpha}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem…
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with…
In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabr\'e's proof of the classical isoperimetric inequality in…
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general…