Nonlinear Free Boundary Research Articles

Full and partial regularity for a class of nonlinear free boundary problems

In this paper we classify the nonnegative global minimizers of the functionalJF(u)=∫ΩF(|∇u|2)+λ2χ{u>0}, where F satisfies some structural conditions and χD is the characteristic function of a set D⊂Rn. We compute the second variation of the energy and study the properties of the stability operator. The free boundary ∂{u>0} can be seen as a rectifiable n−1 varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if n=3 and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.

Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications

We establish sharp $$C_{\text {loc}}^{1, \beta }$$ geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy, whose model equation is given by $$\begin{aligned} \left[ |Du|^p+\mathfrak {a}(x)|Du|^q\right] {\mathcal {M}}_{\lambda , \Lambda }^{+}(D^2 u)= f(x, u) \quad \text {in} \quad \Omega , \end{aligned}$$ for a bounded and open set $$\Omega \subset {\mathbb {R}}^N$$ , and appropriate data $$p, q \in (0, \infty )$$ , $$\mathfrak {a}$$ and f. Such regularity estimates simplify and generalize, to some extent, earlier ones via totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear models in the theory of elliptic PDEs, which may have their own interest. We also deliver explicit examples where our results are sharp.

Real Options with Endogenous Payoff

This paper studies a model of irreversible investment decisions in which the exercise payoff is endogenously determined by the firm's risk management choice. By obtaining the explicit solution of a non-linear free boundary problem with a stochastic control, we present the implications for the optimal investment timing and the associated optimal risk management strategy. The firm's optimal risk management strategy exhibits risk-seeking: it increases the insurance asset holdings that provide rewards (damages) when there is good (bad) news for the underlying price. As a result of the risk-seeking insurance strategy, in our model, in contrast to the standard models, investment can be hastened as the underlying uncertainty increases, depending on the economic conditions. The main force driving these results is that the firm's risk management is designed to optimize the risk-return tradeoff of the endogenous payoff.

Effects of rotation on planetary inertial gravity waves initiated by the stratospheric polar vortex

We examine stability of the polar vortex that initiates planetary-scale westerly (west to east) atmospheric flow that encircles the pole in middle or high latitude of a planet. In our studies, the polar vortex is modeled as an annular flow (zonal flow) extending radially from a rigid inner boundary to a free outer boundary on a Ek− plane, which is parallel to the equatorial plane but generally displaced in latitude. The vortex is associated with a particular class of exact solution of a nonlinear two-dimensional free boundary problem on the nonstationary motion of a perfect fluid propagating around a solid circle with a sufficiently large radius with included Coriolis effect. The intensity of the polar vortex is parametrized by means of the “average” (in vertical) zonal flow that can be associated with atmospheric flow patterns that are parallel or nearly parallel to the lines of latitude. Perturbations are allowed in the zonal and nominally vertical (inward–outward) direction. A linear stability analysis purportedly suggests that increasing latitude decreases the maximum strength of the vortex that is permitted for stability. Three essential parameters that control stability of the vortex are identified.

A fully nonlinear free boundary problem for minimizing the ruin probability

In this paper, we consider the problem of how to run an insurance company to minimize the ruin probability in a finite time horizon. The value function v with the corresponding optimal reinsure strategy a∗ is the solution of the following fully nonlinear equation vt−12σ02vxx+γvx−inf0≤a≤1(12σ2a2vxx+μavx)=0under initial–boundary condition v(0,t)=1, v(x,0)=0. The main effort is to analyze the properties of the solution and the free boundary to show the optimal decision for the company.

Free boundary theory for singular/degenerate nonlinear equations with right hand side: a non-variational approach

We use non-variational type arguments to prove optimal regularity and smoothness of the free boundary for one-phase solutions to inhomogeneous nonlinear free boundary problems (FBP) governed by singular/degenerate elliptic PDEs with a nonzero right hand side (RHS). In a precise way, we show that viscosity solutions to FBP, as previously mentioned, are locally Lipschitz continuous and under certain conditions, flat or Lipschitz free boundaries, are $$C^{1, \alpha }.$$

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

In this paper we study a nonlinear free boundary problem for the growth of radially symmetric tumor with a necrotic core. The proliferation of tumor cells depends on the concentration of nutrient which satisfies a diffusion equation within tumor and is periodically supplied by external tissues. The tumor outer surface and the inner interface of the necrotic core are both free boundaries. We give a sufficient and necessary condition for the existence and uniqueness of positive periodic solution, and show it is globally asymptotically stable under radial perturbations. Our analysis implies that tumor growth may finally synchronize the periodic external nutrient supply.

Fujita type critical exponent for a free boundary problem with spatial–temporal source

In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.

A dam break driven by a moving source: a simple model for a powder snow avalanche

We study the two-dimensional, irrotational flow of an inviscid, incompressible fluid injected from a line source moving at constant speed along a horizontal boundary, into a second, immiscible, inviscid fluid of lower density. A semi-infinite, horizontal layer sustained by the moving source has previously been studied as a simple model for a powder snow avalanche, an example of an eruption current, Carroll et al. (Phys. Fluids, vol. 24, 2012, 066603). We show that with fluids of unequal densities, in a frame of reference moving with the source, no steady solution exists, and formulate an initial/boundary value problem that allows us to study the evolution of the flow. After considering the limit of small density difference, we study the fully nonlinear initial/boundary value problem and find that the flow at the head of the layer is effectively a dam break for the initial conditions that we have used. We study the dynamics of this in detail for small times using the method of matched asymptotic expansions. Finally, we solve the fully nonlinear free boundary problem numerically using an adaptive vortex blob method, after regularising the flow by modifying the initial interface to include a thin layer of the denser fluid that extends to infinity ahead of the source. We find that the disturbance of the interface in the linear theory develops into a dispersive shock in the fully nonlinear initial/boundary value problem, which then overturns. For sufficiently large Richardson number, overturning can also occur at the head of the layer.

Radially symmetric growth of necrotic tumors and connection with nonnecrotic tumors

In this paper we study a nonlinear free boundary problem for radially symmetric growth of tumors with necrosis. We show this problem is globally well-posed and find a threshold value σ∗>0, such that if and only if external nutrient supply σ̄≥σ∗, there exists a unique necrotic stationary solution. We prove it is globally asymptotically stable. For σ̄<σ∗, asymptotic behavior of transient solution has been studied. The connection and mutual transition between necrotic and nonnecrotic tumors are also given.

The numerical analysis of controllability of EAST plasma vertical position by TSC

Vertical position controllability with EAST vertical instability fast control system in current mode is analyzed using Tokamak Simulation Code (TSC). TSC is a non-linear two-dimensional axisymmetric free boundary simulation code. Maxwell’s and MHD equations are used to describe the evolution of magnetic field in a rectangular domain for the plasma. In 2016, the model for VDE free drift and control has been benchmarked against experiments. In this paper, the power supply model for fast VDE control is extended to one with limited maximum control current. VDE controllability with the present system capability is studied. The maximum controllable displacement dZmax is numerically analyzed, for both ordinary experimental discharge and equilibria with typical βp and li in EAST. It is found that dZmax inversely grows with βp and li. Rigorous conditions of li=1, βp<0.3 or βp=0.5, li<0.8 are required for dZmax reaching the safe control level. For operation safety, system capability enhancement is needed. Effects of system delay time, the maximum control current and system current ramp time on VDE controllability are discussed.

A fully nonlinear free boundary problem arising from optimal dividend and risk control model

Focusing on the problem arising from a stochastic model of risk control and dividend optimization techniques for a financial corporation, this work considers a parabolic variational inequality with gradient constraint \begin{document}$\min\Big\{v_t-\max\limits_{0\leq a\leq1}\Big(\frac{1}{2}\sigma^2a^2v_{xx}+\mu av_x\Big)+cv,\;v_x-1\Big\} = 0.$ \end{document} Suppose the company's performance index is the total discounted expected dividends, our objective is to choose a pair of control variables so as to maximize the company's performance index, which is the solution to the above variational inequality under certain initial-boundary conditions. The main effort is to analyse the properties of the solution and two free boundaries arising from the above variational inequality, which we call dividend boundary and reinsurance boundary.

Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

This paper is concerned with long-time dynamics of a full von Karman system subject to nonlinear thermal coupling and free boundary conditions. In contrast with scalar von Karman system, vectorial full von Karman system accounts for both vertical and in plane displacements. The corresponding PDE is of critical interest in flow structure interactions where nonlinear plate/shell dynamics interacts with perturbed flows [vicid or invicid] [8,9,15]. In this paper it is shown that the system admits a global attractor which is also smooth and of finite fractal dimension. The above result is shown to hold for plates without regularizing effects of rotational inertia and without any mechanical dissipation imposed on vertical displacements. This is in contrast with the literature on this topic [15] and references therein. In order to handle highly supercritical nature of the von Karman nonlinearities, new results on 'hidden' trace regularity generated by thermal effects are exploited. These lead to asymptotic compensated compactness of trajectories which then allows to use newly developed tools pertaining to quasi stable dynamical systems [8].

Numerical Studies for Solving a Free Convection Boundary–Layer Flow Over a Vertical Plate

Abstract In this paper, The aim of this study is to present a reliable combination of the shifted Legendre collocation method to approximate of the problem of free convection boundary-layer flow over a vertical plate as produced by a body force about a flat plate in the direction of the generating body force. The proposed method is based on replacement of the unknown function by truncated series of well known shifted Legendre expansion of functions. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shift Legendre coefficients. Thus, by solving this system of equations, the shifted Legendre coefficients are obtained. Boundary conditions in an unbounded domain, i.e. boundary condition at infinity, pose a problem in general for the numerical solution methods. The obtained results are in good agreement with those provided previously by the iterative numerical method. As a result, without taking or estimating missing boundary conditions, the shifted Legendre collocation method provides a simple, non-iterative and effective way for determining the solutions of nonlinear free convection boundary layer problems possessing the boundary conditions at infinity.

The generalized Riemann problem and instability of delta shock to the chromatography equations

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

Bifurcation of traveling waves in a Keller–Segel type free boundary model of cell motility

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

Error identities for variational problems with obstacles

AbstractThe paper is devoted to analysis of a class of nonlinear free boundary problems that are usually solved by variational methods based on primal, dual or primal‐dual variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, whose minimization is the keystone of all variational numerical methods. Therefore, the identity actually sets the measure that contains maximal quantitative information on the quality of a numerical solution available through these methods. The measure has quadratic terms generated by the linear part of the differential operator and nonlinear terms associated with the free boundary. We obtain fully computable two sided bounds of this measure and show that they provide efficient estimates of the distance between the minimizer and any function (approximation) from the corresponding energy space. Several computational examples show that for different minimization sequences the balance between the quadratic and the nonlinear terms of the overall error measure may be different and essential contribution of nonlinear terms may serve as an indicator that the free boundaries are approximated very roughly.

Three-Dimensional Steady Supersonic Euler Flow Past a Concave Cornered Wedge with Lower Pressure at the Downstream

In this paper, we study the stability of the three-dimensional jet created by a supersonic flow past a concave cornered wedge with the lower pressure at the downstream. The gas beyond the jet boundary is assumed to be static. It can be formulated as a nonlinear hyperbolic free boundary problem in a cornered domain with two characteristic free boundaries of different types: one is the rarefaction wave, while the other one is the contact discontinuity, which can be either a vortex sheet or an entropy wave. A more delicate argument is developed to establish the existence and stability of the square jet structure under the perturbation of the supersonic incoming flow and the pressure at the downstream. The methods and techniques developed here are also helpful for other problems involving similar difficulties.

Infiltration of MHD liquid into a deformable porous material

Abstract We analyze the capillary rise dynamics for magnetohydrodynamics (MHD) fluid flow through deformable porous material in the presence of gravity effects. The modeling is performed using mixture theory approach and mathematical manipulation yields a nonlinear free boundary problem. Due to the capillary rise action, the pressure gradient in the liquid generates a stress gradient that results in the deformation of porous substrate. The capillary rise process for MHD fluid slows down as compared to Newtonian fluid case. Numerical solutions are obtained using a method of lines approach. The graphical results are presented for important physical parameters, and comparison is presented with Newtonian fluid case.

A nonlinear free boundary problem with a self-driven Bernoulli condition

We study a Bernoulli type free boundary problem with two phasesJ[u]=∫Ω|∇u(x)|2dx+Φ(M−(u),M+(u)),u−u¯∈W01,2(Ω), where u¯∈W1,2(Ω) is a given boundary datum. Here, M1 and M2 are weighted volumes of {u⩽0}∩Ω and {u>0}∩Ω, respectively, and Φ is a nonnegative function of two real variables.We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases.In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.As a special case, we can deal with the energy levels generated by the volume term Φ(0,r2)≃r2n−1n, which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality.In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.