Search results for "unique"
showing 10 items of 268 documents
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 …
Radial growth of solutions to the poisson equation
2001
We establish a radial growth estimate of the type of the iterated law of the logarithm for solutions to the Poisson equation in the unit ball.
On semi-fredholm properties of a boundary value problem inR + n
1988
The paper considers a boundary value problem with the help of the smallest closed extensionL ∼ :H k →H k 0×B h 1×...×B h N of a linear operatorL :C (0) ∞ (R + n ) →L(R + n )×L(R n−1)×...×L(R n−1). Here the spacesH k (the spaces ℬ h ) are appropriate subspaces ofD′(R + n ) (ofD′(R n−1), resp.),L(R + n ) andC (0) ∞ (R + n )) denotes the linear space of smooth functionsR n →C, which are restrictions onR + n of a function from the Schwartz classL (fromC 0 ∞ , resp.),L(R n−1) is the Schwartz class of functionsR n−1 →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ∼) and for the uniqueness of solutionsL ∼ U=F are expressed. In addition, ana prio…
UNIQUENESS OF PERIODIC SOLUTIONS OF THE LIENARD EQUATION
1981
This chapter analyzes the uniqueness of periodic solutions of the Lienard equation. It considers the Lienard equation = y − F ( x ) and y = − x where F (0) = 0 , F ( x ) ∈ Lip( R ). The chapter discusses the existence of periodic solutions. It highlights that the origin is the only stationary point of the system = y − F ( x ) and y = − x , and therefore all nontrivial periodic solutions must circle around the origin. The existence of at least one periodic solution is proved by constructing a Poincare–Bendixson domain. The chapter also emphasizes that to prove the uniqueness of periodic solutions, additional assumptions are also needed. In the literature there are numerous uniqueness results…
Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem
2020
Abstract We consider the Laplace equation in a domain of R n , n ≥ 3 , with a small inclusion of size ϵ . On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For ϵ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.
A differential equation approach to implicit sweeping processes
2019
International audience; In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits.
Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension
2017
A fundamental problem in the dimension theory of self-affine sets is the construction of high- dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high-dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. Whilst the existence of these equilibrium states has been well-known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular …
L’intégration Monétaire de la Roumanie entre Couts et Bénéfices
2008
Il n’y a pas dans notre pays de stratégie qui concerne bien techniquement l’adoption de l’euro. Il s’imposerait d’essayer, quoiqu’il soit trèstéméraire, d’identifier le moment opportun pour l’introduction de l’euro. Ce n’est pas facile mais il faut s’y mettre puisqu’il s’agit de faire suivrel’évolution et l’identification des tendances d’intégration de la Roumanie dans la zone euro, de faire également attention aux modèles d’intégrationmonétaires développés en Europe et bien entendu de mettre en place des scénarios alternatives pour la Roumanie. Et tout cela tout en évaluant deprès les coûts et les bénéfices de l’adoption de l’euro car l’intégration monétaire de la Roumanie est bien importa…
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…
A fixed-point problem with mixed-type contractive condition
2020
We consider a fixed-point problem for mappings involving a mixed-type contractive condition in the setting of metric spaces. Precisely, we establish the existence and uniqueness of fixed point using the recent notions of $F$-contraction and $(H,\varphi)$-contraction.