Search results for "Regularity"
showing 10 items of 98 documents
On Noncoercive (p, q)-Equations
2021
We consider a nonlinear Dirichlet problem driven by a (p, q)-Laplace differential operator (1 < q < p). The reaction is (p - 1)-linear near +/-infinity and the problem is noncoercive. Using variational tools and truncation and comparison techniques together with critical groups, we produce five nontrivial smooth solutions all with sign information and ordered. In the particular case when q = 2, we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.
Weak regularity of functions and sets in Asplund spaces
2006
Abstract In this paper, we study a new concept of weak regularity of functions and sets in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak regularity is equivalent to Mordukhovich regularity in finite dimension. Finally, we give characterizations of the weak regularity of epi-Lipschitzian sets in terms of their local representations.
Maximal regularity for Kolmogorov operators in L2 spaces with respect to invariant measures
2006
Abstract We prove an optimal embedding result for the domains of Kolmogorov (or degenerate hypoelliptic Ornstein–Uhlenbeck) operators in L 2 spaces with respect to invariant measures. We use an interpolation method together with optimal L 2 estimates for the space derivatives of T ( t ) f near t = 0 , where T ( t ) is the Ornstein–Uhlenbeck semigroup and f is any function in L 2 .
Singular quasilinear elliptic systems involving gradient terms
2019
Abstract In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.
Anti-powers in infinite words
2018
In combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. As a consequence, we show that in every aperiodic uniformly recurrent word, anti-powers of ev…
Modeling the Dynamics of a Financial Index after a Crash
2004
Supply and demand are perhaps the most fundamental concepts in economics. In a financial market they reflects the orders of the agents to buy or sell a given asset. In turn the fluctuations of supply and demand influence the dynamics of the price of an asset, as, for example, a stock or a financial index. Therefore the dynamics of the price of an asset is affected by the actions and of the beliefs of the agents. It is known that the dynamics of the price of an asset is far from simple, Several stylized facts has been empirically discovered such as, for example, the fat tails in the return distribution and the clustered volatility. These stylized facts has been detected by considering long t…
Conformality and $Q$-harmonicity in sub-Riemannian manifolds
2016
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
Regularity and h-polynomials of toric ideals of graphs
2020
For all integers 4 ≤ r ≤ d 4 \leq r \leq d , we show that there exists a finite simple graph G = G r , d G= G_{r,d} with toric ideal I G ⊂ R I_G \subset R such that R / I G R/I_G has (Castelnuovo–Mumford) regularity r r and h h -polynomial of degree d d . To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O’Keefe that compares the depth and dimension of toric ideals of graphs.
Improved Hölder regularity for strongly elliptic PDEs
2019
We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of H\"older regularity, higher than what is given by the classical exponent $1/K$.
Gradient regularity for elliptic equations in the Heisenberg group
2009
Abstract We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485–544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C 1 , α -regularity for p-harmonic functions in the Heisenberg group for …