The theory of energy for sub-Laplacians with an application to quasi-continuity

In this paper, we provide a suitable theory for the energy where μ is a Radon measure and Γ is the fundamental solution of a sub-Laplacian on a stratified group As a significant application, we prove the quasi-continuity of superharmonic functions related to . The proofs are elementary and mostly rely on the use of appropriate mean-value formulas and mean-integral operators relevant to the Potential Theory for .     

Liouville-type theorems for real sub-Laplacians

An inequality generalizing the classical Liouville and Harnack Theorems for real sub-Laplacians ℒ is proved. A representation formula for functions $u$ for which ℒu is a polynomial is also showed. As a consequence, some conditions are given ensuring that u is a polynomial whenever ℒu is a polynomial. Finally, an application of this last result is given: if ψ is a C ² map commuting with ℒ, then any of its component is a polynomial function. Received: 3 November 2000     

Nonlinear Liouville theorems for some critical problems on H-type groups

In this paper, we provide a non-existence result for a semilinear sub-elliptic Dirichlet problem with critical growth on the half-spaces of any group of Heisenberg-type. Our result improves a recent theorem in (Math. Ann. 315 (3) (2000) 453).     

Sobolev inequalities with remainder terms for sublaplacians and other subelliptic operators

We prove that on bounded domains Ω, the usual Sobolev inequality for sublaplacians on Carnot groups can be improved by adding a remainder term, in striking analogy with the euclidean case. We also show analogous results for subelliptic operators like $$ {\user1{\mathcal{L}}} = \Delta _{x} + |x|^{{2\alpha }} \Delta _{y} ,\,\alpha \gt 0. $$     

Semilinear subelliptic problems with critical growth on Carnot groups

We prove existence and multiplicity of solutions for the semilinear subelliptic problem with critical growth in Ω, u = 0 on ∂Ω, where is a sublaplacian on a Carnot group , 2* = 2Q/(Q − 2) is the critical Sobolev exponent for and Ω is a bounded domain of .     

On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)^α/2, 0<α<2, in a bounded C^1,1 domain D in Rⁿ. In particular, we introduce a new Kato class K_α(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.     

Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient

In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray-Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarke's gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis.     

Strichartz estimates on the quaternion Heisenberg group

The aim of this article is to prove dispersive estimates and the Strichartz estimates on the quaternion Heisenberg group. In order to obtain these results, we first study the properties of Fourier transform for radial functions and the Besov spaces, Sobolev spaces on the quaternion Heisenberg group, then we give the proofs for the main results.     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

An analytic series method for Laplacian problems with mixed boundary conditions

Mixed boundary value problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary. Historically, only a very small subset of these problems could be solved using analytic series methods (“analytic” is taken here to mean a series whose terms are analytic in the complex plane). In the past, series solutions were obtained by using an appropriate choice of axes, or a co-ordinate transformation to suitable axes where the boundaries are parallel to the abscissa and the boundary conditions are separated into pure Dirichlet or Neumann form. In this paper, I will consider the more general problem where the mixed boundary conditions cannot be resolved by a co-ordinate transformation. That is, a Dirichlet condition applies on part of the boundary and a Neumann condition applies along the remaining section. I will present a general method for obtaining analytic series solutions for the classic problem where the boundary is parallel to the abscissa. In addition, I will extend this technique to the general mixed boundary value problem, defined on an arbitrary boundary, where the boundary is not parallel to the abscissa. I will demonstrate the efficacy of the method on a well known seepage problem.     

A multiplicity result for generalized Laplacian systems with multiparameters

In this paper, we consider the existence, nonexistence and multiplicity of a positive solution for a Gelfand type generalized Laplacian system with a singular indefinite weight and a vector parameter. By using the upper and lower solution method and fixed point index theory, we obtain a global multiplicity result with respect to the parameter.     

An Invariant Harnack Inequality for a Class of Hypoelliptic Ultraparabolic Equations

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related stationary operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators.     

Harmonic measures for a point may form a square

Let X be a Green domain in Rd, d?2, x∈X, and let Mx(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, x∈U, which is contained in the set of extreme points of Mx(P(X)), is dense in Mx(P(X)). In this paper, it is shown that Mx(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U₀, U₁, U₂, U₃ of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural L²-structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations).     

Convexity properties of harmonic measures

It is shown that any convex combination of harmonic measures , where U₁,…,Uk are relatively compact open neighborhoods of a given point x∈Rd, d?2, can be approximated by a sequence of harmonic measures such that each Wn is an open neighborhood of x in U₁∪?∪Uk.This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures.This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner.These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hörmander's condition, for example, sub-Laplacians on stratified Lie algebras).     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups

We prove global sharp estimates for the heat kernel related to certain sub-Laplacians on a real semisimple Lie group, from which we deduce an estimate for the corresponding Green function.     

Measurable Riemannian geometry on the Sierpinski gasket: the Kusuoka measure and the Gaussian heat kernel estimate

We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype of “measurable Riemannian geometry”. The shortest pathmetric on the harmonic Sierpinski gasket is shown to be thegeodesic distance associated with the “measurable Riemannianstructure”. The Kusuoka measure is shown to have the volumedoubling property with respect to the Euclidean distance and alsoto the geodesic distance. Li–Yau type Gaussian off-diagonal heatkernel estimate is established for the heat kernel associated withthe Kusuoka measure.