Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition

Local Lipschitz continuity of solutions to a problem in the calculus of variations

In this paper we study the local Lipschitz regularity of weak solutions to certain singular elliptic equations involving the one-Laplacian. Equations treated here also contains another well-behaving elliptic operator such as the $$p$$ -Laplacian with $$1<p<\infty $$ . The problem is that the one-Laplacian is too singular on degenerate points, what is often called a facet, which makes it difficult to obtain even Lipschitz regularity of weak solutions. This difficulty is overcome by making suitable approximation schemes, and by avoiding analysis on facet for approximated solutions. The key estimate is a local a priori uniform Lipschitz estimate for classical solutions to regularized equations, which is proved by Moser’s iteration. Another local a priori uniform Lipschitz bounds can also be obtained by De Giorgi’s truncation. Proofs of local Lipschitz estimates in this paper are rather classical and elementary in the sense that nonlinear potential estimates are not used at all.

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the L2 sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the L2 sense.

Value function, relaxation, and transversality conditions in infinite horizon optimal control

We prove in this article that functions satisfying a dynamic programming principle have a local interior Lipschitz type regularity. This DPP is partly motivated by the connection to the normalized parabolic $ p $-Laplace operator.

On local Lipschitz regularity for quasilinear equations in the Heisenberg group

Motivated by the safety problem, several definitions of reachability maps, for hybrid dynamical systems, are introduced. It is well established that, under certain conditions, the solutions to continuous-time systems depend continuously with respect to initial conditions. In such setting, the reachability maps considered in this paper are locally Lipschitz (in the Lipschitz sense for set-valued maps) when the right-hand side of the continuous-time system is locally Lipschitz. However, guaranteeing similar properties for reachability maps for hybrid systems is much more challenging. Examples of hybrid systems for which the reachability maps do not depend nicely with respect to their arguments, in the Lipschitz sense, are introduced. With such pathological cases properly identified, sufficient conditions involving the data defining a hybrid system assuring Lipschitzness of the reachability maps are formulated. As an application, the proposed conditions are shown to be useful to significantly improve an existing converse theorem for safety given in terms of barrier functions. Namely, for a class of safe hybrid systems, we show that safety is equivalent to the existence of a locally Lipschitz barrier function. Examples throughout the paper illustrate the results.

SynopsisWe demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.

If \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$u : {\mathbb R}^{n}\supset \Omega \rightarrow {\mathbb R}^{M} $\end{document} locally minimizes the functional ∫Ωh(|∇u|) dx with h such that ${{h^{\prime }(t)}\over{t}} \le h^{\prime \prime }(t) \le c\, (1 + t^2)^\omega {{h^{\prime }(t)}\over{t}} $ for all t ⩾ 0, then u is locally Lipschitz independent of the value of ω ⩾ 0. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

Fuzzy delay differential equation driven by Liu's process is a type of functional differential equations. In this paper, we are going to provide and prove a novel existence and uniqueness theorem for the solutions of fuzzy delay differential equation under Local Lipschitz and linear growth conditions. Also the stability of the solutions for fuzzy delay differential is investigated. Finally, to illustrate the main results we give some examples.

Fuzzy fractional differential equations (FFDEs) driven by Liu’s process are a type of fractional differential equations. In this paper, we intend to provide and prove a novel existence and uniqueness theorem for the solutions of FFDEs under local Lipschitz and linear growth conditions. We also investigate the stability of solutions to FFDEs by a theorem. Finally, some examples are provided.

On some Lipschitz-type functions

Local and global Lipschitz constants

In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ∂tu−∑i=12nXiAi(Xu)=K(x,t,u,Xu), where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For 2≤p≤4, we establish the local Lipschitz regularity (u∈Cloc0,1), with the horizontal gradient satisfying Xu∈Lloc∞; (ii) For 2≤p&lt;3, we establish the local second-order horizontal Sobolev regularity (u∈HWloc2,2), with the second-order horizontal derivative satisfying XXu∈Lloc2. These results solve an open problem proposed by Capogna et al.

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source