31,439 research outputs found
Regularity for quasilinear PDEs in Carnot groups via Riemannian approximation
 We study the interior regularity of weak solutions to subelliptic quasilinear PDEs in Carnot groups of the formΣi=1m1Xi (Φ(|∇Hu|2)Xiu) = 0.Here ∇Hu = (X1u,...,Xmiu) is the horizontal gradient, δ > 0 and the exponent p ∈ [2, p*), where p* depends on the step ν and the homogeneous dimension Q of the group, and it is given byp* = min {2ν ∕ ν-1 , 2Q+8 ∕ Q-2}.Sunto. Studiamo la regolarità interna delle soluzioni deboli di EDP, quasilineari subellittiche in gruppi di Carnot, della formaΣi=1m1Xi (Φ(|∇Hu|2)Xiu) = 0.Qui ∇Hu = (X1u,...,Xmiu) è il gradiente orizzontale, δ > 0 e l'esponente p ∈ [2, p*), dove p* dipende dal passo ν e dalla dimensione omogenea Q del gruppo ed è dato dap* = min {2ν ∕ ν-1 , 2Q+8 ∕ Q-2}
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are
spatially coupled across a finite window along the chain direction. We
investigate their phase diagram at zero temperature using the survey
propagation formalism and the interpolation method. We prove that the SAT-UNSAT
phase transition threshold of an infinite chain is identical to the one of the
individual standard model, and is therefore not affected by spatial coupling.
We compute the survey propagation complexity using population dynamics as well
as large degree approximations, and determine the survey propagation threshold.
We find that a clustering phase survives coupling. However, as one increases
the range of the coupling window, the survey propagation threshold increases
and saturates towards the phase transition threshold. We also briefly discuss
other aspects of the problem. Namely, the condensation threshold is not
affected by coupling, but the dynamic threshold displays saturation towards the
condensation one. All these features may provide a new avenue for obtaining
better provable algorithmic lower bounds on phase transition thresholds of the
individual standard model
Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations
We investigate the asymptotic behavior of solutions of anisotropic equations
of the form
in , where for all and is a
Caratheodory function with critical Sobolev growth. This problem arises in
particular from the study of extremal functions for a class of anisotropic
Sobolev inequalities. We establish decay estimates for the solutions and their
derivatives, and we bring to light a vanishing phenomenon which occurs when the
maximum value of the exponents exceeds a critical value.Comment: Final version to appear in Advances in Mathematic
-Regularity of Quasilinear equations on the Heisenberg Group
In this article, we reproduce results of classical regularity theory of
quasilinear elliptic equations in the divergence form, in the setting of
Heisenberg Group. The conditions encompass a very wide class of equations with
isotropic growth conditions, which are a generalization of the -Laplace type
equations in this respect; these also include all equations with polynomial or
exponential type growth. In addition, some even more general conditions have
also been explored.Comment: long versio
Maximum Principle for Quasi-linear Backward Stochastic Partial Differential Equations
In this paper we are concerned with the maximum principle for quasi-linear
backward stochastic partial differential equations (BSPDEs for short) of
parabolic type. We first prove the existence and uniqueness of the weak
solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral
boundary. Then using the De Giorgi iteration scheme, we establish the maximum
estimates and the global maximum principle for quasi-linear BSPDEs. To study
the local regularity of weak solutions, we also prove a local maximum principle
for the backward stochastic parabolic De Giorgi class
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