Lipschitz Continuous Solutions Research Articles

Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

Lipschitz Continuous Solutions of the Vlasov–Maxwell Systems with a Conductor Boundary Condition

Lipschitz Continuous Solutions of the Vlasov–Maxwell Systems with a Conductor Boundary Condition

Axi-symmetric solutions for active vector models generalizing 3D Euler and electron–MHD equations

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂tB+V⋅∇B=B⋅∇V,V=−∇×(−Δ)−aB,∇⋅B=0,where B is a time-dependent vector field in R3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1/2<a<1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.

Error estimates for total-variation regularized minimization problems with singular dual solutions

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix–Raviart element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.

Singular solutions, graded meshes,and adaptivity for total-variation regularized minimization problems

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems require the existence of a Lipschitz continuous dual solution. We discuss the validity of this condition and devise numerical methods using locally refined meshes that lead to improved convergence rates despite the occurrence of discontinuities. It turns out that linear convergence is possible on suitably constructed meshes.

A dynamical approach for the quantitative stability of parametric bilevel equilibrium problems and applications

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results.

The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $\mu$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\mu)$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\Delta)^{\alpha/2}$, $\alpha\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

Statistical Inference of Second-Order Cone Programming

In this paper, we study the stability of stochastic second-order programming when the probability measure is perturbed. Under the Lipschitz continuity of the objective function and metric regularity of the feasible set-valued mapping, the outer semicontinuity of the optimal solution set and Lipschitz continuity of optimal values are demonstrated. Moreover, we prove that, if the constraint non-degeneracy condition and strong second-order sufficient condition hold at a local minimum point of the original problem, there exists a Lipschitz continuous solution path satisfying the Karush–Kuhn–Tucker conditions.

Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem.

NONLINEAR MEAN-VALUE FORMULAS ON FRACTAL SETS

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem [Formula: see text] in the Sierpiński gasket with prescribed values [Formula: see text], [Formula: see text] and [Formula: see text] at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle.

Lipschitz solutions of initial boundary value problems for balance laws

The flow of gas through networks of pipes can be modeled by the isothermal Euler equations and algebraic node conditions that model the flow through the vertices of the network graph. This motivates our analysis of the well-posedness of a coupled system of [Formula: see text] conservation laws on a network. We consider initial data and control functions that are Lipschitz continuous and compatible with the node and boundary conditions. We show the existence of semi-global Lipschitz continuous solutions of the initial boundary value problem on a network. The construction of the solution is based upon a fixed point iteration along the characteristic curves. The solutions of the initial boundary value problem on arbitrary networks satisfy a maximum principle in terms of the Riemann invariants that states that the maximum of the absolute values is attained for the initial or the boundary data.

Global solution for a non-local eikonal equation modelling dislocation dynamics

In this paper, we study a non-local eikonal equation that arises from the theory of dislocation dynamics. For this equation, the global existence and uniqueness are available only for Lipschitz continuous viscosity solutions in some particular cases. Based on a new gradient entropy estimate, we prove the global existence of a continuous viscosity solution of this equation, considering non-signed velocity.

Existence of Lipschitz continuous solutions to the Cauchy–Dirichlet problem for anisotropic parabolic equations

The Cauchy–Dirichlet and the Cauchy problem for the degenerate and singular quasilinear anisotropic parabolic equations are considered. We show that the time derivative ut of a solution u belongs to L∞ under a suitable assumption on the smoothness of the initial data. Moreover, if the domain satisfies some additional geometric restrictions, then the spatial derivatives uxi belong to L∞ as well. In the singular case we show that the second derivatives uxixj of a solution of the Cauchy problem belong to L2.

Parabolic equations and the bounded slope condition

In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations{∂tu−divDf(Du)=0in ΩT,u=uoon ∂PΩT. The only assumptions needed are the convexity of the generating function f:Rn→R, and the classical bounded slope condition on the initial and the lateral boundary datum uo∈W1,∞(Ω). We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – uo could be any Lipschitz initial and boundary datum, vanishing at the boundary ∂Ω, and the boundary may contain flat parts, for instance Ω could be a rectangle in Rn.

Global existence results for eikonal equation with BV initial data

In this paper, we study a local and a non-local eikonal equations in one dimensional space describing the evolution of interfaces moving with non-signed velocity. For these equations, the global existence and uniqueness are available only of Lipschitz continuous viscosity solutions in some particular cases. In the present paper, we are interested in the study of the global in time existence of these equations, considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed propagation property of these equations, we show, in a particular setting, global existence results of discontinuous viscosity solutions of this problem. An interesting application of these results is shown in the case of dislocation dynamics.

Wigner measures supported on weak KAM tori

In the setting of the Weyl quantization on the flat torus \(\mathbb{T}^n \), we exhibit a class of wave functions with uniquely associated Wigner probability measure, invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in phase space. These sets are the graphs of Lipschitz-continuous weak KAM solutions of negative type of the stationary Hamilton-Jacobi equation. Such Wigner measures are, in fact, given by the Legendre transform of Mather’s minimal probability measures.

Asymptotic behavior of solutions of a system of q-functional-difference equations

The asymptotic behavior and the existence of a bounded Lipschitz continuous solution of a system of nonlinear q-functional-difference equations on a half-interval of the real line are studied here. The main results compare the behavior of such a solution of the system with bounded solutions of a linear q-difference equation.

On the De Giorgi Type Conjecture for an Elliptic System Modeling Phase Separation

In this paper we study the one dimensional symmetry problem of entire solutions to the problem for all n ≥ 2. We prove that, if a solution (u, v) is a local minimizer and has a linear growth at infinity, then it is one dimensional, i.e. depending only on one variable. In the proof we also obtain the global Lipschitz continuity of solutions only under the linear growth assumption.

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sem Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $$C(t)$$ C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $$C(t)$$ C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset $$C(t)$$ C ( t ) has a continuous variation for every $$t\in [0,T]$$ t ? [ 0 , T ] with $$C(0)$$ C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Kreja?i in Eur J Appl Math 2:281---292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179---186, 1973 and Cornet in J. Math. Anal. Appl. 96:130---147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau's seminal work.

Near Field Photometric Stereo with Point Light Sources

Shape recovery of an object based on shading variations resulting from different light sources has recently been reconsidered. Improvements have been made that allow for the photometric stereo approach to serve as a competitive alternative to other shape reconstruction methods. However, most photometric stereo methods tend to ignore factors that are relevant in practical applications. The setup considered in this paper tackles photometric stereo reconstruction in the case of a specific near-field imaging. This means that both the camera and the light sources are close to the imaged object, where close can be loosely considered as a setup having similar distances between lights, camera, and object. The known challenges that characterize the problem involve perspective viewing geometry, point light sources, and images that may include shadowed regions. Here, we pay special attention to the question of how to faithfully model these aspects and at the same time design an efficient and robust numerical solver. We present a mathematical formulation that integrates the above assumptions into a single coherent model based on quasi-linear PDEs. The well-posedness is proved showing uniqueness of a weak (i.e., Lipschitz continuous) solution. The surface reconstruction in our near-field scenario can then be executed efficiently in linear time. The merging strategy of the irradiance equations provided for each light source allows us to consider a characteristic expansion model which enables the direct computation of the surface. We evaluate several types of light attenuation models with a nonuniform albedo and noise on synthetic data. We also demonstrate the proposed method on surface reconstruction of real data using three images, each one taken with a different light source by a working prototype. We demonstrate the accuracy of the proposed method compared to other methods that ignore the near-field setup and assume distant, parallel beam light sources.