Regularity and classification of the free boundary for a Monge-Ampère obstacle problem

Soil disinfection is of great significance in reducing soil pests and weeds, overcoming the problem of crop continuous cropping obstacles, and ensuring the quality and safety of agricultural products. Soil flame disinfection technology, as a supplementary soil disinfection method that can be incorporated into an integrated plant protection system, has attracted much attention in recent years due to its characteristics of low resistance, greenness, environmental friendliness, and high efficiency. However, soil flame disinfection can also have a certain impact on soil organic matter and microbial communities, which is a core challenge that limits the promotion of flame disinfection technology. Clarifying the mechanism and temperature distribution of flame disinfection, accurately controlling flame disinfection parameters, can not only kill harmful organisms in soil, but also minimize damage to soil organic matter and microbial communities is the current research focus. This paper presents a comprehensive summary and discussion of the research progress regarding the mechanism of soil flame disinfection technology, the distribution of temperature fields, and the precise control technology for disinfection machines. It thoroughly elaborates on the efficacy of flame in eliminating typical soil-borne diseases and pests, the destructive impact of flame on soil organic matter and beneficial microbial communities, as well as the current status of research and development on soil flame disinfection devices. Additionally, it explores the pressing technical challenges that remain to be addressed. The article then discusses the future market prospects of soil flame disinfection equipment, focusing on key technological breakthroughs and opportunities, providing theoretical support for the next research, optimization and promotion of soil flame disinfection technology.

We develop a monotone, two-scale discretization for a class of integro-differential operators of order 2s , s \in (0,1) . We apply it to develop numerical schemes, and derive pointwise convergence rates for linear and obstacle problems governed by such operators. As applications of the monotonicity, we provide error estimates for free boundaries and a convergent numerical scheme for a concave fully nonlinear, nonlocal, problem.

Abstract We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure q -superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and q -superlinear convergence of the developed solution algorithm.

Weighted Lorentz estimates with a variable power for non-uniformly elliptic two-sided obstacle problems

An epiperimetric inequality for odd frequencies in the thin obstacle problem

In this paper we are concerned with the obstacle problem related to an operator with a drift-type lower order term that in the linear case represents the one related to the Fokker–Plank equation, whose (normalized) solution describes the evolution of the probability density for a stochastic process. The main novelty is the presence in the coefficient of the lower order term of a singularity in the spatial variable and minimal-in-time integrability assumption. We prove the well-posedness of a global solution to the obstacle problem and we describe the asymptotic behavior of such a solution. In particular, in the autonomous case, we prove that the global solution of our obstacle problem converges to the solution of the corresponding elliptic obstacle.

Abstract In this paper, we derive a priori error estimates for variational inequalities of the first kind in an abstract framework. This is done by combining the first Strang Lemma and the Falk Theorem. The main application consists of the derivation of a priori error estimates for Galerkin methods, in which “variational crimes” may perturb the underlying variational inequality. Different types of perturbations are incorporated into the abstract framework and are discussed in various examples. For instance, the perturbation caused by an inexact quadrature is examined in detail for the Laplacian obstacle problem. For this problem, guaranteed rates for the approximation error resulting from the use of a higher-order finite element method are derived. In numerical experiments, the influence of the number of quadrature points on the approximation error and on the quadrature-related error itself is studied for several discretization methods.

Optimality of smallness conditions in Willmore obstacle problems under Dirichlet boundary conditions

Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: An entropy formulation

ABSTRACTThis paper presents an efficient computational framework for solving convection‐diffusion obstacle problems, designed for convection‐dominated regimes while ensuring local and global mass conservation. The method relies on an operator‐splitting strategy that decouples the problem into convection and diffusion sub‐problems, treated, respectively, in Lagrangian and Eulerian settings. The convective transport is handled by a particle‐in‐cell method, while the diffusion, formulated as a parabolic variational inequality, is discretized using mixed finite elements. This leads to symmetric saddle‐point systems with complementarity conditions, solved efficiently via a primal‐dual active set algorithm. To ensure conservative coupling between particles and mesh, a PDE‐constrained projection is employed. The effectiveness and performance of the overall approach have been established by rigorous benchmarks with analytical solutions from the literature, covering both structured and unstructured meshes.

Abstract In the present paper, we study a Crouzeix–Raviart approximation of the obstacle problem, which imposes the obstacle constraint in the midpoints (i.e., barycenters) of the elements of a triangulation. We establish a priori error estimates imposing natural regularity assumptions, which are optimal, and the reliability and efficiency of a primal-dual type a posteriori error estimator for general obstacles and involving data oscillation terms stemming only from the right-hand side. Numerical experiments are carried out to support the theoretical findings.

We consider the nonconforming discrete Raviart-Thomas mixed finite element method (dRT-MFEM) for obstacle problems with $p$-Laplacian differential operator. The a posteriori and a priori error analysis were presented in a new sense of measurement. A number of experiments confirm the effective decay rates of the proposed dRT-MFEM.

The case of autologous cartilage engineering has revolutionized ear and nasal reconstruction. This event has resulted in an improved aesthetic and functional outcome in the surgical treatment of congenital, traumatic, and oncologic defects. As described in this review, the usage of tissue engineering treatments, which use autologous chondrocytes to alleviate immunogenicity and donor morbidity of traditional modalities such as rib graft cartilage or alloplastic implants, is of great importance. New technologies, such as 3D bioprinting and nanofibrous scaffolds, have made it possible to reproduce intricate auricular and nasal designs accurately, and bioinks such as nanofibrillated cellulose-alginate can be used to provide high-fidelity constructs. The microtia and nasal alar reconstruction with clinical translations are promising, and in the case of engineered cartilage, integration and minimal adverse outcomes were found in a 12-30-month follow-up. Nonetheless, issues that still exist are the long-term shape fidelity, biomechanical inferiority of the regenerating cartilage compared to that of the native cartilage and limited vascularisation in larger constructs. The problem of considerable prices and regulatory obstacles also hinders the mass use. The workarounds timely emerge in the form of emerging technologies, including dynamic, patient-specific structures with 4D bioprinting and machine learning optimized scaffold design. The directions of the future lie in prevascularized grafts, cost-effective biofabrication, and an increased variety of clinical trials to validate the long-term efficacy in various populations. Combining biomimetic scaffolds, powerful imaging technology, and a customised approach, autologous cartilage engineering exists to revolutionise reconstructive surgery, so long as the effort behind the continued research continues to solve the questions of enhanced scalability and regulatory hurdles. This review reviews the latest advances, critically assesses the limitations, and suggests ways of clinical translation to functional, long-lasting, and aesthetically better outcomes.

Abstract This paper develops a comprehensive framework for estimating discontinuous or rapidly varying coefficients in evolutionary hemi-quasi -variational inequalities involving multi-valued monotone, semi-monotone, and pseudo-monotone maps. To establish that the coefficient-to-solution map is well-defined, we present new solvability results and demonstrate the weak compactness of the solution set for the considered hemi-quasi -variational inequalities. We introduce a novel variational selection to circumvent the commonly adopted but highly restrictive assumption that the sum of a monotone map and a pseudo-monotone map is monotone. Additionally, we relax the compactness assumption on the involved embedding operators to make the results readily applicable to evolutionary problems. Subsequently, we establish the existence of solutions for the inverse problem by developing a general regularization framework to counter the ill-posedness of such problems. The feasibility and efficacy of the developed framework are tested on three applied models: nonlinear implicit obstacle problems, a variational model with nonlocal constraints, and contact problems.

Abstract Utilising a generic framework, this paper develops a new convergence analysis of various approximation methods for fourth-order variational inequalities (obstacle problems). For the convergence proof, an unrestricted approach is adopted to avoid non-physical conditions on continuous solutions. This analysis presents the first study to investigate a unified framework for both conforming and non-conforming schemes for the studied problem. It also enables us to apply a polytopal scheme for the fourth-order VIs and produce numerical results that highlight the scheme’s accuracy.

Abstract This study is concerned with the finite element approximation of the elastoplastic torsion problem. We focus on the case of a nonconstant source term, which cannot be easily recast into an obstacle problem as can be done in the case of a constant source term. We present a simple formulation that penalizes the constraint directly on the gradient norm of the solution. We study its well-posedness, derive error estimates and present numerical results to illustrate the theory.

Existence and regularity results for the penalized thin obstacle problem with variable coefficients

On a class of superlinear obstacle problems

Regularity of weak solutions to generalized quasilinear elliptic obstacle problems