Dario D. Monticelli

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation. A good example of such an operator is the Grushin operator on R dCk , to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the result of Gidas-Ni-Nirenberg (12), (13), and a nonexistence result for classical solutions of semilinear equa- tions with subcritical growth defined on the whole space, which is a generalization of the result of Gidas-Spruck (14) and Chen-Li (6). We use the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

Conformal Ricci solitons and related integrability conditions

Abstract We introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of (0, 3)-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor D recently introduced by Cao and Chen. We derive commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.

Spectral Theory for Linear Operators of Mixed Type and Applications to Nonlinear Dirichlet Problems

For a class of linear partial differential operators L of mixed elliptic-hyperbolic type in divergence form with homogeneous Dirichlet data on the entire boundary of suitable planar domains, we exploit the recent weak well-posedness result of [8] and minimax methods to establish a complete spectral theory in the context of weighted Lebesgue and Sobolev spaces. The results represent the first robust spectral theory for mixed type equations. In particular, we find a basis for a weighted version of the space comprised of weak eigenfunctions which are orthogonal with respect to a natural bilinear form associated to L. The associated eigenvalues {λ k } k∈ℕ are all non-zero, have finite multiplicity and yield a doubly infinite sequence tending to ± ∞. The solvability and spectral theory are then combined with topological methods of nonlinear analysis to establish the first results on existence, existence with uniqueness and bifurcation from (λ k , 0) for associated semilinear Dirichlet problems.

Classification of expanding and steady Ricci solitons with integral curvature decay

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar curvature decay. As a corollary, we obtain that the only three dimensional steady solitons with less than quadratic volume growth are quotients of

A variational characterization of flat spaces in dimension three

\mathbb{R}\times\Sigma^{2}

On the relation between conformally invariant operators and some geometric tensors

, where

An extension problem for the CR fractional Laplacian

\Sigma^{2}

On the geometry of gradient Einstein-type manifolds

is Hamiltons cigar.

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the

BOUNDEDNESS OF WEAK SOLUTIONS OF DEGENERATE QUASILINEAR EQUATIONS WITH ROUGH COEFFICIENTS

\sigma_{2}