On the constrained error bound condition and the projected Levenberg–Marquardt method

In this paper, we first derive a characterization of the solution set of a continuously differentiable system of equations subject to a closed feasible set. Assuming that a constrained local error bound condition is satisfied, we prove that the solution set can locally be written as the intersection of a differentiable manifold with the feasible set. Based on the derivation of this result, we then show that the projected Levenberg–Marquardt method converges locally R-linearly to a possibly nonisolated solution under significantly weaker conditions than previously done.     

A Linearization Method for Nondegenerate Variational Conditions

We employ recent results about constraint nondegeneracy in variational conditions to design and justify a linearization algorithm for solving such problems. The algorithm solves a sequence of affine variational inequalities, but the variational condition itself need not be a variational inequality: that is, its underlying set need not be convex. However, that set must be given by systems of differentiable nonlinear equations with additional polyhedral constraints. We show that if the variational condition has a solution satisfying nondegeneracy and a standard regularity condition, and if the linearization algorithm is started sufficiently close to that solution, the algorithm will produce a well defined sequence that converges Q-superlinearly to the solution.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs

In this paper, we establish the equivalence between the half-space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions. As consequences, we prove that the corresponding marginal function is differentiable and the solution map admits a differentiable selection.     

Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations

We first prove the existence and regularity of the trajectory attractor for a three-dimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.     

Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

We study the partial regularity of weak solutions to the 2-dimensional LandauLifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C ∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.     

Some regularity results on the Ventcel-Freidlin quasi-potential function

We consider regularity properties of the quasipotential function V defined by A. D. Ventcel and M. I. Freidlin in their work on asymptotically small random perturbations of stable dynamical systems. The regularity properties of V are important for the success of various asymptotic calculations carried out in the literature. Employing classical techniques from the calculus of variations and differential equations, we prove various results about the smoothness of V and its level sets. Among other things, there exists a dense connected open set, containing the stable point for the underlying dynamical system, in which V is continuously differentiable to the same degree as the Lagrangian involved in the defining variational problem.     

Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems

We derive conditions for the local uniqueness of solutions of nonlinear complementarity problems (NCP). We then prove the existence, continuity, and directional differentiability of a locally unique parametric solution of the parametric NCP under stronger assumptions. In the absence of degeneracy this parametric solution is also shown to be continuously differentiable.     

On C1, C2, and weak type-(1,1) estimates for linear elliptic operators

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L¹-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haïm Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.     

On the entropy conditions for some flux limited diffusion equations

In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.     

Regularity Properties of <Emphasis Type="Italic">H</Emphasis>-Convex Sets

We study the first- and second-order regularity properties of the boundary of H-convex sets in the setting of a real vector space endowed with a suitable group structure: our starting point is indeed a step two Carnot group. We prove that, locally, the noncharacteristic part of the boundary has the intrinsic cone property and that it is foliated by intrinsic Lipschitz continuous curves that are twice differentiable almost everywhere.     

Second-Order Optimality Conditions in Minimax Optimization Problems

The paper primarily is concerned with the second-order optimality conditions for minimax problems, where the constraints are described by a set inclusion and a finite number of equalities, and where all the functions involved are Fréchet differentiable with locally Lipschitz derivatives. We make use of the Mangasarian Fromovitz regularity conditions and of the second-order Abadie regularity conditions.     

Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary regularity for Maxwell's equations with applications to shape optimization

The dynamic Maxwell equations with a strictly dissipative boundary condition is considered. Sharp trace regularity for the electric and the magnetic field are established for both: weak and differentiable solutions. As an application a shape optimization problem for Maxwell's equations is considered. In order to characterize the shape derivative as a solution to a boundary value problem, the aforementioned sharp regularity of the boundary traces is critical.     

Oscillation Properties of Scalar Conservation Laws

We obtain several new regularity results for solutions of scalar conservation laws satisfying the genuine nonlinearity condition. We prove that the solutions are continuous outside of the jump set, which is codimension one rectifiable. We show that the entropy dissipation vanishes away from the closure of the jump set. We prove that the solution decays algebraically in L^∞ as t → ∞, and we compute the presumably optimal decay rate. All these results are based on a local oscillation estimate that is obtained properly adapting some ideas of De Giorgi from the context of elliptic equations. © 2018 Wiley Periodicals, Inc.     

Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application

Stability properties of the solution set of generalized inequality systems with locally Lipschitz functions are obtained under a regularity condition on the generalized Jacobian and the Clarke tangent cone. From these results, we derive sufficient conditions for the optimal value function in a nonsmooth optimization problem to be continuous or locally Lipschitz at a given parameter.     

Local regularity for nonlocal equations with variable exponents

In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler–Lagrange equations. We show that weak solutions are locally bounded when the variable exponent p is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on p. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder–type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on p outside the domain.     

A Shooting Algorithm for Optimal Control Problems with Singular Arcs

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.     

Evolution Systems for Differential Equations with Variable Time Lags

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions.     

Two‐Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front

We continue to develop the regularity theory of general two‐phase free boundary problems for parabolic operators. In a 2010 paper, we establish the optimal (Lipschitz) regularity of a viscosity solution under the assumptions that the free boundary is locally a flat Lipschitz graph and a nondegeneracy condition holds. Here, on one side we improve this result by removing the nondegeneracy assumption; on the other side we prove the smoothness of the front. The proof relies in a crucial way on a local stability result stating that, for a certain class of operators, under small perturbations of the coefficients flat free boundaries remain close and flat. © 2013 Wiley Periodicals, Inc.