Local Lipschitz Continuity Research Articles

Weak–strong uniqueness of solutions to entropy-dissipating reaction–diffusion equations

We establish a weak–strong uniqueness principle for solutions to entropy-dissipating reaction–diffusion equations: As long as a strong solution to the reaction–diffusion equation exists, any weak solution and even any renormalized solution must coincide with this strong solution. Our assumptions on the reaction rates are just the entropy condition and local Lipschitz continuity; in particular, we do not impose any growth restrictions on the reaction rates. Therefore, our result applies to any single reversible reaction with mass-action kinetics as well as to systems of reversible reactions with mass-action kinetics satisfying the detailed balance condition. Renormalized solutions are known to exist globally in time for reaction–diffusion equations with entropy-dissipating reaction rates; in contrast, the global-in-time existence of weak solutions is in general still an open problem–even for smooth data–, thereby motivating the study of renormalized solutions. The key ingredient of our result is a careful adjustment of the usual relative entropy functional, whose evolution cannot be controlled properly for weak solutions or renormalized solutions.

Certain properties related to well posedness of switching diffusions

This work is devoted to switching diffusions that have two components (a continuous component and a discrete component). Different from the so-called Markovian switching diffusions, in the setup, the discrete component (the switching) depends on the continuous component (the diffusion process). The objective of this paper is to provide a number of properties related to the well posedness. First, the differentiability with respect to initial data of the continuous component is established. Then, further properties including uniform continuity with respect to initial data, and smoothness of certain functionals are obtained. Moreover, Feller property is obtained under only local Lipschitz continuity. Finally, an example of Lotka–Volterra model under regime switching is provided as an illustration.

Extensions of vector-valued functions with preservation of derivatives

Let X and Y be Banach or normed linear spaces and F⊂X a closed set. We apply our recent extension theorem for vector-valued Baire one functions to obtain an extension theorem for vector-valued functions f:F→Y with pre-assigned derivatives, with preservation of differentiability (at every point where the pre-assigned derivative is actually a derivative), preservation of continuity, preservation of (point-wise) Lipschitz property and (for finite dimensional domain X) preservation of strict differentiability and global (eventually local) Lipschitz continuity. This work depends on the paper Extensions of vector-valued Baire one functions with preservation of points of continuity (Koc and Kolář (2016) [20]).

Dependence of solutions of nonsmooth differential-algebraic equations on parameters

The well-posedness of nonsmooth differential-algebraic equations (DAEs) is investigated. More specifically, semi-explicit DAEs with Carathéodory-style assumptions on the differential right-hand side functions and local Lipschitz continuity assumptions on the algebraic equations. The DAEs are classified as having differential index one in a generalized sense; solution regularity is formulated in terms of projections of generalized (Clarke) Jacobians. Existence of solutions is derived under consistency and regularity of the initial data. Uniqueness of a solution is guaranteed under analogous Carathéodory ordinary-differential equation uniqueness assumptions. The continuation of solutions is established and sufficient conditions for continuous and Lipschitzian parametric dependence of solutions are also provided. To accomplish these results, a theoretical tool for analyzing nonsmooth DAEs is provided in the form of an extended nonsmooth implicit function theorem. The findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth DAEs.

Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F: X → 2Y; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F(x). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X,Y.

Regularity results for nonlinear parabolic obstacle problems with subquadratic growth

In [26] it was shown that the spatial gradient of the solution u to the parabolic obstacle problem with superquadratic growth is local Hölder continuous provided the obstacle is regular enough. In this paper, we extend this regularity result to the subquadratic case. This means we establish the local Hölder continuity of the spatial gradient of the solution u to the parabolic obstacle problem with subquadratic growth. More precisely, we prove thatDu∈Cloc0;α,α2 for some α∈(0,1), provided the coefficients and the obstacle are regular enough. Moreover, we use the local Hölder continuity to prove the local Lipschitz continuity of the solution u, i.e.u∈Cloc0;1,12.

Generalized Derivatives of Differential–Algebraic Equations

Nonsmooth equation-solving and optimization algorithms which require local sensitivity information are extended to systems with nonsmooth parametric differential---algebraic equations embedded. Nonsmooth differential---algebraic equations refers here to semi-explicit differential---algebraic equations with algebraic equations satisfying local Lipschitz continuity and differential right-hand side functions satisfying Caratheodory-like conditions. Using lexicographic differentiation, an auxiliary nonsmooth differential---algebraic equation system is obtained whose unique solution furnishes the desired parametric sensitivities. More specifically, lexicographic derivatives of solutions of nonsmooth parametric differential---algebraic equations are obtained. Lexicographic derivatives have been shown to be elements of the plenary hull of the Clarke (generalized) Jacobian and thus computationally relevant in the aforementioned algorithms. To accomplish this goal, the lexicographic smoothness of an extended implicit function is proved. Moreover, these generalized derivative elements can be calculated in tractable ways thanks to recent advancements in nonsmooth analysis. Forward sensitivity functions for nonsmooth parametric differential---algebraic equations are therefore characterized, extending the classical sensitivity results for smooth parametric differential---algebraic equations.

POF-Darts: Geometric adaptive sampling for probability of failure

We introduce a novel technique, POF-Darts, to estimate the Probability Of Failure based on random disk-packing in the uncertain parameter space. POF-Darts uses hyperplane sampling to explore the unexplored part of the uncertain space. We use the function evaluation at a sample point to determine whether it belongs to failure or non-failure regions, and surround it with a protection sphere region to avoid clustering. We decompose the domain into Voronoi cells around the function evaluations as seeds and choose the radius of the protection sphere depending on the local Lipschitz continuity. As sampling proceeds, regions uncovered with spheres will shrink, improving the estimation accuracy. After exhausting the function evaluation budget, we build a surrogate model using the function evaluations associated with the sample points and estimate the probability of failure by exhaustive sampling of that surrogate. In comparison to other similar methods, our algorithm has the advantages of decoupling the sampling step from the surrogate construction one, the ability to reach target POF values with fewer samples, and the capability of estimating the number and locations of disconnected failure regions, not just the POF value. We present various examples to demonstrate the efficiency of our novel approach.

Local regularity for time-dependent tug-of-war games with varying probabilities

We study local regularity properties of value functions of time-dependent tug-of-war games. For games with constant probabilities we get local Lipschitz continuity. For more general games with probabilities depending on space and time we obtain Hölder and Harnack estimates. The games have a connection to the normalized p(x,t)-parabolic equation ut=Δu+(p(x,t)−2)Δ∞Nu.

Derivative-Free Optimization via Classification

Many randomized heuristic derivative-free optimization methods share a framework that iteratively learns a model for promising search areas and samples solutions from the model. This paper studies a particular setting of such framework, where the model is implemented by a classification model discriminating good solutions from bad ones. This setting allows a general theoretical characterization, where critical factors to the optimization are discovered. We also prove that optimization problems with Local Lipschitz continuity can be solved in polynomial time by proper configurations of this framework. Following the critical factors, we propose the randomized coordinate shrinking classification algorithm to learn the model, forming the RACOS algorithm, for optimization in continuous and discrete domains. Experiments on the testing functions as well as on the machine learning tasks including spectral clustering and classification with Ramp loss demonstrate the effectiveness of RACOS.

Local Lipschitz continuity of the inverse of the fractional [formula omitted]-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains

Let p∈(1,∞), s∈(0,1) and Ω⊂RN an arbitrary bounded open set. In the first part we consider the inverse Φs,p:=[(−Δ)p,Ωs]−1 of the fractional p-Laplace operator (−Δ)p,Ωs with the Dirichlet boundary condition. We show that in the singular case p∈(1,2), the operator Φs,p is locally Lipschitz continuous on L∞(Ω) and that global Lipschitz continuity cannot be achieved. We use this result to show that in the case N>sp, if 2NN+2s<p<2, 0≤q≤NpN−sp−2 and α,β are small constants, then the nonlinear problem (−Δ)psu=α∣u∣qu+β∣u∣NpN−sp−2u+hin Ω,u=0 on RN∖Ω, has at least one weak solution. In the second part of the paper, we prove that the operator −(−Δ)p,Ωs generates a (nonlinear) submarkovian semigroup (Ss,p(t))t≥0 on L2(Ω). If p∈[2,∞) and sp<N, we obtain that Ss,p(t) satisfies the following (Lq−L∞)-Hölder type estimate: there exists a constant C>0 such that for every t>0 and u,v∈Lq(Ω) (q≥2), we have ‖Ss,p(t)u−Ss,p(t)v‖L∞(Ω)≤C∣Ω∣β(s)t−δ(s)‖u−v‖Lq(Ω)γ(s) where β(s),δ(s) and γ(s) are explicit constants depending only on N,s,p,q.

On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations

We extend the taming techniques for explicit Euler approximations of stochastic differential equations driven by Levy noise with superlinearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global assumptions and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.

Quadratic-Exponential Growth BSDEs with Jumps and Their Malliavin's Differentiability

We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bound on the possible solutions. With a bounded terminal condition and local Lipschitz continuity, we give a simple and streamlined proof for the existence as well as the uniqueness of the solution without using the comparison principle. The properties of locally Lipschitz BSDEs with coefficients in BMO space enable us to show the strong convergence of a sequence of globally Lipschitz BSDEs to the interested one, which is then used to give sufficient conditions for the Malliavin’s differentiability.

Une inégalité de Korn non linéaire dans [formula omitted], [formula omitted

Let Ω be a bounded and connected open subset of Rn that satisfies the uniform interior cone property and let p>n. We establish a nonlinear Korn inequality in W2,p, which provides an upper bound of the ‖⋅‖W2,p(Ω)-norm of the difference between two immersions φ∈W2,p(Ω) and φ˜∈W2,p(Ω) in terms of the ‖⋅‖W1,p(Ω)-norm of the difference between their associated metric tensors ∇φT∇φ∈W1,p(Ω) and ∇φ˜T∇φ˜∈W1,p(Ω).Second, let Ω be a bounded, simply-connected, open subset of Rn with a Lipschitz boundary, the set Ω being locally on the same side of its boundary. Using the above nonlinear Korn inequality in W2,p, we establish the local Lipschitz-continuity of the mapping C∈W1,p(Ω)→φ∈W2,p(Ω), which is well-defined when the components of the Riemann curvature tensor associated with C vanish in D′(Ω), the immersion φ∈W2,p(Ω) being the solution, unique up to an isometry of En, of the equation ∇φT∇φ=C in Ω.

NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

Controllability of semilinear systems with fixed delay in control

In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.

Local regularity results for value functions of tug-of-war with noise and running payoff

Abstract We prove local Lipschitz continuity and Harnack's inequality for value functions of the stochastic game tug-of-war with noise and running payoff. As a consequence, we obtain game-theoretic proofs for the same regularity properties for viscosity solutions of the inhomogeneous p-Laplace equation when p &gt; 2.

Lipschitz continuity of the optimal value function in parametric optimization

We study generalized parametric optimization problems in Banach spaces, given by continuously Frechet differentiable mappings and some abstract constraints, in terms of local Lipschitz continuity of the optimal value function. Therefore, we make use of the well-known regularity condition by Kurcyusz, Robinson and Zowe, an inner semicontinuity property of the solution set mapping and some earlier results by Mordukhovich, Nam and Yen. The main theorem presents a handy formula which can be used in order to approximate the Clarke subdifferential of the optimal value function, provided that the conditions mentioned above are satisfied and hence the optimal value function is locally Lipschitz continuous. Throughout the paper we avoid any compactness assumptions.

Asymptotically Periodic Solutions to Nonlocal Cauchy Problems Governed by Compact Evolution Families

We study a class of abstract Cauchy problems for semilinear nonautonomous evolution equations involving nonlocal initial conditions. Combining the theory of evolution families with the Krasnosel’skii fixed-point theorem and a decomposition technique, we prove the existence of asymptotically periodic mild solutions to these problems. Our results generalize and improve some previous results, since the condition of (local) Lipschitz continuity of the nonlinearity is not required. A partial differential equation is also presented as an application.

Control problems for semilinear neutral differential equations in Hilbert spaces.

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.