Alcuni progressi sull'Ipoellitticità analitica

We present a brief survey on the theory of the real analytic regularity for the solutions to sums of squares of vector fields satisfying the Hörmander condition.Presentiamo una breva rassegna sulla regolarità reale analitica delle soluzioni di operatori somme di quadrati di campi vettoriali che soddisfano la condizione di Hörmander

A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

We consider the boundary value problem \cases{ -\Delta_\gamma u = \lambda u + \left\vert u \right\vert^{2^*_\gamma-2}u &in $\Omega$\cr u = 0 &on $\partial\Omega$,\cr } where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, while $\Delta_\gamma$ is the Grushin operator $$\Delta_ \gamma u(z) = \Delta_x u(z) + \vert x \vert^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0).$$ We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe and of Fiscella, Molica Bisci and Servadei

On solutions to a class of degenerate equations with the Grushin operator

The Grushin Laplacian −Δα is a degenerate elliptic operator in Rh+k that degenerates on {0}×Rk. We consider weak solutions of −Δαu=Vu in an open bounded connected domain Ω with V∈W1,σ(Ω) and σ>Q/2, where Q=h+(1+α)k is the so-called homogeneous dimension of Rh+k. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of Ω. As an application we derive strong unique continuation properties for solutions

Asymptotic analysis of an elastic rod with rounded ends

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m is an element of(0,1). Error estimates are obtained in the case m is an element of(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m is an element of(0,1/4) but play a crucial role in the formulation of the limit problem for m >= 1/4.Peer reviewe

A p-specific spectral multiplier theorem with sharp regularity bound for Grushin operators

Our approach avoids the usage of weighted restriction type estimates which played a key role in the work of Chen and Ouhabaz, and is rather based on a careful analysis of the underlying sub-Riemannian geometry and restriction type estimates where the multiplier is truncated along the spectrum of the Laplacian on R d2

Spectral multipliers on two-step structures

In harmonic analysis and partial differential equations, the topics of Fourier restriction estimates, wave estimates for linear wave equations, and spectral multiplier problems associated with elliptic and sub-elliptic linear differential operators are closely related. The aim of this thesis is to further explore these connections and to contribute to a deeper understanding of these phenomena. We focus on certain classes of differential operators which are sub-Laplacians defined as divergence form operators associated with a two-step sub-Riemannian structure on a smooth manifold. The analysis of these operators is closely related to their underlying sub-Riemannian geometry, which is a major challenge in understanding analytic properties of these differential operators. The specific question addressed in this thesis is as follows. The functional calculus for the sub-Laplacian L provided by the spectral theorem allows to define the operator F(L) for every spectral multiplier F : R → C. The L^p-spectral multiplier problem asks to identify spectral multipliers F for which F(L) extends to a bounded operator on the Lebesgue space L^p. Usually this question is answered by so-called Mikhlin–Hörmander type theorems, which require a smoothness condition on the multiplier F. In this thesis we prove spectral multiplier theorems where this smoothness condition is even p-specific. This is done for two specific classes of sub-Laplacians, namely Grushin operators and left-invariant sub-Laplacians on certain subclasses of two-step stratified Lie groups. The proof of these spectral multiplier theorems relies on a careful analysis of the underlying sub-Riemannian geometry and exploiting appropriate restriction type estimates, an idea that goes back to C. Fefferman. A novelty in the restriction type estimates proved in this thesis is an additional truncation along the spectrum of a Laplacian on the second layer of the associated two-step structure

Analysis of degenerate elliptic operators of Grušin type

34 page(s