Local Lipschitz Continuity Research Articles

General Cauchy–Lipschitz theory for Δ-Cauchy problems with Carathéodory dynamics on time scales

The aim of this paper was to complete some aspects of the classical Cauchy–Lipschitz (or Picard–Lindelöf) theory for general nonlinear systems posed on time scales. Despite a rich literature on Cauchy–Lipschitz type results on time scales, most of the existing results are concerned with rd-continuous dynamics (and -solutions) and do not cover the framework of general Carathéodory dynamics encountered for instance in control theory with measurable controls (which are not rd-continuous in general). In this paper, our main objective was to study -Cauchy problems with general Carathéodory dynamics. We introduce the notion of absolutely continuous solution (weaker regularity than ) and then the notion of maximal solution. We state and prove a Cauchy–Lipschitz theorem, providing existence and uniqueness of the maximal solution of a given -Cauchy problem under suitable assumptions such as regressivity and local Lipschitz continuity. Three new related issues are also discussed in this paper: the boundary value is not necessarily an initial or a final condition, the solutions are constrained to take their values in a non-empty open subset and the behaviour of maximal solutions at terminal points is studied.

A minimum problem with two-phase free boundary in Orlicz spaces

We consider the optimization problem of minimizing \(\mathcal{J }_{\gamma }(u)=\int _{\Omega }(G(|\nabla u|)+\lambda _{+}(u^{+})^{\gamma }+\lambda _{-}(u^{-})^{\gamma }+fu)\,\text {d}x\) in the class of functions \(W^{1,G}(\Omega )\) with \(u-\varphi \in W^{1,G}_{0}(\Omega )\) for a given function \(\varphi \), where \(W^{1,G}(\Omega )\) is the class of weakly differentiable functions with \(\int _{\Omega }G(|\nabla u|)\,\text {d}x<\infty \). The conditions on the function \(G\) allow for a different behavior at \(0\) and at \(\infty \). We give a rather complete description of regularity theory for a family of two-phase variational free boundary problems. For \(0<\gamma \le 1\), we prove that every minimizer \(u_{\gamma }\) of \(\mathcal{J }_{\gamma }(u)\) is \(C^{1,\alpha }_{loc}\) continuous. For \(\gamma =0\), we obtain local Lipschitz continuity for any minimizer \(u_{0}\) of \(\mathcal{J }_{0}(u)\). Here we also consider an asymptotic problem as \(\gamma \rightarrow 0\). At last, we establish sharp geometric estimates for the free boundary corresponding to the minimizer \(u_{0}\) of \(\mathcal J _{0}(u)\).

Blow-up analysis of stationary two-valued functions

Abstract. We consider two-valued real functions of two variables which are stationary with respect to both domain and range transformations. We prove their local Lipschitz continuity and use it to establish strong convergence in W1,2 to their unique blow-up at any point. The main theorem of this paper states that the branch set of any such function consists of finitely many real analytic curves meeting at nod points with equal angles. We also provide an example showing that stationarity with respect to domain transformations only does not imply continuity.

A note on tamed Euler approximations

Strong convergence results on tamed Euler schemes, which approximate stochastic differential equations with superlinearly growing drift coefficients that are locally one-sided Lipschitz continuous, are presented in this article. The diffusion coefficients are assumed to be locally Lipschitz continuous and have at most linear growth. Furthermore, the classical rate of convergence, i.e. one--half, for such schemes is recovered when the local Lipschitz continuity assumptions are replaced by global and, in addition, it is assumed that the drift coefficients satisfy polynomial Lipschitz continuity.

A study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we derive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this: if interior gradient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

Sensitivity Analysis for Two-Level Value Functions with Applications to Bilevel Programming

This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization called the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not---to a large extent---differ from those known for the conventional problem. It has already been well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This is related to the local Lipschitz continuity of the optimal value funct...

Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces

Let (X, d) be a pathwise connected metric space equipped with an Ahlfors Q-regular measure μ, Q ∈ [1, ∞ ). Suppose that (X, d, μ) supports a 2-Poincaré inequality and a Sobolev–Poincaré type inequality for the corresponding “Gaussian measure”. The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation Δu = f, where \(f\in L^{p}_{\rm{loc}}\). When p > Q, the local Lipschitz continuity of u is established.

Stability Analysis of Two-Stage Stochastic Mathematical Programs with Complementarity Constraints via NLP Regularization

This paper presents numerical approximation schemes for a two-stage stochastic programming problem, where the second stage problem has a general nonlinear complementarity constraint. First the complementarity constraint is approximated by a parameterized system of inequalities with a well-known regularization approach by Scholtes [SIAM J. Optim., 11 (2001), pp. 918–936] in deterministic mathematical programs with equilibrium constraints; the distribution of the random variables of the regularized two-stage stochastic program is then approximated by a sequence of probability measures. By treating the approximation problems as a perturbation of the original (true) problem, we carry out a detailed stability analysis of the approximated problems, including continuity and local Lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points. A particular focus is given to the case where the probability distribution is approximated by the empirical probability measure which is also known as sample average approximation.

Nonlinear A-Dirac Equations

This paper is a study of solutions to nonlinear Dirac equations, in domains in Euclidean space, which are generalizations of the Clifford Laplacian as well as elliptic equations in divergence form. A Caccioppoli estimate is used to prove a global integrability theorem for the image of a solution under the Euclidean Dirac operator. Oscillation spaces for Clifford valued functions are used which generalize the usual spaces of bounded mean oscillation, local Lipschitz continuity or local order of growth of real-valued functions.

Energy methods for stochastic differential equations

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.

GENERALIZED RELAXED PROXIMAL POINT ALGORITHMS INVOLVING RELATIVE MAXIMAL ACCRETIVE MODELS WITH APPLICATIONS IN BANACH SPACES

General models for the relaxed proximal point algorithm using the notion of relative maximal accretiveness (RMA) are developed, and then the convergence analysis for these models in the context of solving a general class of nonlinear inclusion problems differs significantly than that of Rockafellar (1976), where the local Lipschitz continuity at zero is adopted instead. Moreover, our approach not only generalizes convergence results to real Banach space settings, but also provides a suitable alternative to other problems arising from other related fields.

Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs

The paper concerns the study of new classes of parametric optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints, infinitely many inequality constraints. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We focus on DC infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth) value functions in DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well-posedness and sensitivity. In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.

Properties of a Class of Nonlinear Transformations Over Euclidean Jordan Algebras with Applications to Complementarity Problems

For any function φ from ℜr to ℜr, Tao and Gowda [Math. Oper. Res., 30 (2005), pp. 985–1004] introduced a corresponding nonlinear transformation Rφ over a Euclidean Jordan algebra (which is called a relaxation transformation) and established some useful relations between φ and Rφ. In this paper, we further investigate some interconnections between properties of φ and properties of Rφ, including the properties of continuity, (local) Lipschitz continuity, directional differentiability, (continuous) differentiability, semismoothness, monotonicity, the P0-property, and the uniform P-property. As an application, we investigate the symmetric cone complementarity problem with a relaxation transformation. A property of the solution set of this class of problems is given. We also investigate a smoothing algorithm for solving this class of problems and show that the algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty.

Weak convergence in the Prokhorov metric of methods for stochastic differential equations

We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen–Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions, assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding.

Optimal control of Markovian jump processes with partial information and applications to a parallel queueing model

We consider a stochastic control problem over an infinite horizon where the state process is influenced by an unobservable environment process. In particular, the Hidden-Markov-model and the Bayesian model are included. This model under partial information is transformed into an equivalent one with complete information by using the well-known filter technique. In particular, the optimal controls and the value functions of the original and the transformed problem are the same. An explicit representation of the filter process which is a piecewise-deterministic process, is also given. Then we propose two solution techniques for the transformed model. First, a generalized verification technique (with a generalized Hamilton–Jacobi–Bellman equation) is formulated where the strict differentiability of the value function is weaken to local Lipschitz continuity. Second, we present a discrete-time Markovian decision model by which we are able to compute an optimal control of our given problem. In this context we are also able to state a general existence result for optimal controls. The power of both solution techniques is finally demonstrated for a parallel queueing model with unknown service rates. In particular, the filter process is discussed in detail, the value function is explicitly computed and the optimal control is completely characterized in the symmetric case.

Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian

We prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the p -Laplacian.

Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles

We are interested in the superposition operators T f (g):=f∘g on vector valued Besov and Lizorkin-Triebel spaces of positive smoothness exponent s. As a first step towards the characterization of functions which operate, we establish that the local Lipschitz continuity of f is necessary if the space B p,q s (ℝ n ,ℝ m ) or F p,q s (ℝ n ,ℝ m ) is imbedded into L ∞ (ℝ n ,ℝ m ), and that the uniform Lipschitz continuity of f is necessary if the space is not imbedded into L ∞ (ℝ n ,ℝ m ). We prove also that the local membership to the same space is necessary for m≤n. We finally study the regularity of the superposition operator T f .

On the Lipschitz Continuity of the Solutions of a Class of Elliptic Free Boundary Problems

In this article we prove local interior and boundary Lipschitz continuity of the solutions of a general class of elliptic free boundary problems in divergence form.

Regularity results for non smooth parabolic problems

In this paper we deal with the study of some regularity properties of weak solutions to non-linear, second-order parabolic equations and systems of the type \[ u_{t}-{\operatorname{div}} A(Du)=0 \;,\;\;\; (x,t)\in \Omega \times (-T,0)=\Omega_{T}, \] where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain, $T>0$, $A:{\mathbb{R}}^{nN}\to {\mathbb{R}}^{N}$ satisfies a $p$-growth condition and $u:\Omega_{T}\to {\mathbb{R}}^{N}$. In particular, we focus our attention on local regularity of the spatial gradient of solutions of problems characterized by weak differentiability and ellipticity assumptions on the vector field $A(z)$. We prove the local Lipschitz continuity of solutions in the scalar case ($N=1$). We extend this result in some vectorial cases under an additional structure condition. Finally, we prove higher integrability and differentiability of the spatial gradient of solutions for general systems.

Local Lipschitz continuity of graphs with prescribed Levi mean curvature

We prove interior gradient estimates of viscosity solutions of the prescribed Levi mean curvature equation.