Search results for "Lipschitz continuity"
showing 10 items of 93 documents
Generalized countable iterated function systems
2011
One of the most common and most general way to generate fractals is by using iterated function systems which consists of a finite or infinitely many maps. Generalized countable iterated function systems (GCIFS) are a generalization of countable iterated function systems by considering contractions from X ? X into X instead of contractions on the metric space X to itself, where (X, d) is a compact metric space. If all contractions of a GCIFS are Lipschitz with respect to a parameter and the supremum of the Lipschitz constants is finite, then the associated attractor depends continuously on the respective parameter.
Quantitative uniqueness estimates for pp-Laplace type equations in the plane
2016
Abstract In this article our main concern is to prove the quantitative unique estimates for the p -Laplace equation, 1 p ∞ , with a locally Lipschitz drift in the plane. To be more precise, let u ∈ W l o c 1 , p ( R 2 ) be a nontrivial weak solution to div ( | ∇ u | p − 2 ∇ u ) + W ⋅ ( | ∇ u | p − 2 ∇ u ) = 0 in R 2 , where W is a locally Lipschitz real vector satisfying ‖ W ‖ L q ( R 2 ) ≤ M for q ≥ max { p , 2 } . Assume that u satisfies certain a priori assumption at 0. For q > max { p , 2 } or q = p > 2 , if ‖ u ‖ L ∞ ( R 2 ) ≤ C 0 , then u satisfies the following asymptotic estimates at R ≫ 1 inf | z 0 | = R sup | z − z 0 | 1 | u ( z ) | ≥ e − C R 1 − 2 q log R , where C > 0 depends …
Bounded Palais–Smale sequences for non-differentiable functions
2011
The existence of bounded Palais-Smale sequences (briefly BPS) for functionals depending on a parameter belonging to a real interval and which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, is obtained when the parameter runs in a full measure subset of the given interval. Specifically, for this class of non-smooth functions, we obtain BPS related to mountain pass and to global infima levels. This is done by developing a unifying approach, which applies to both cases and relies on a suitable deformation lemma. © 2011 Elsevier Ltd. All rights reserved.
Image compression based on a multi-directional map-dependent algorithm
2007
Abstract This work is devoted to the construction of a new multi-directional edge-adapted compression algorithm for images. It is based on a multi-scale transform that is performed in two steps: a detection step producing a map of edges and a prediction/multi-resolution step which takes into account the information given by the map. A short analysis of the multi-scale transform is performed and an estimate of the error associated to the largest coefficients for a piecewise regular function with Lipschitz edges is provided. Comparisons between this map-dependent algorithm and different classical algorithms are given.
Empirical measures and Vlasov hierarchies
2013
The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the i…
Signal Restoration via a Splitting Approach
2012
International audience; In the present study, a novel signal restoration method from noisy data samples is presented and is termed as "signal split (SSplit)" approach. The new method utilizes Stein unbiased risk estimate estimator to split the signal, the Lipschitz exponents to identify noise elements and a heuristic approach for the signal reconstruction. However, unlike many noise removal techniques, the present method works only in the non-orthogonal domain. Signal restoration was performed on each individual part by finding the best compromise between the data samples and the smoothing criteria. Statistical results are quite promising and suggest better performance than the conventional…
Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian
2014
Abstract We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype ∂ t u = − ∑ i = 1 m X i ⁎ ( | X u | p − 2 X i u ) where p ⩾ 2 , X = ( X 1 , … , X m ) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and X i ⁎ denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M ; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincare inequality involving X and μ. Our results extend the recent work in [16] , [36] , to a more general setting including the model cases of (1)…
Regularity properties of spheres in homogeneous groups
2015
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an alg…
Regularity of sets with constant horizontal normal in the Engel group
2012
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic horizontal upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the intrinsic horizontal upper-graph of a function t…
A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term
2017
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary H\"older continuity and a boundary Harnack type inequality.