In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.

In Chen and Liang previous work, a model, together with its well-posedness, was established for credit rating migrations with different upgrade and downgrade thresholds (i.e. a buffer zone, also called dead band in engineering). When positive dividends are introduced, the model in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) may not be well-posed. Here, in this paper, a new model is proposed to include the realistic nonzero dividend scenarios. The key feature of the new model is that partial differential equations in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) are replaced by variational inequalities, thereby creating a new free boundary, besides the original upgrading and downgrading free boundaries. Well-posedness of the new model, together with a few financially meaningful properties, is established. In particular, it is shown that when time to debt paying deadline is long enough, the underlying dividend paying company is always in high grade rating, that is, only when time to debt paying deadline is within a certain range, there can be seen the phenomenon of credit rating migration.

This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined by spectral radius of the next infection operator. By introducing the generalized principal eigenvalue defined in general domain, we obtain an upper bound of the limit value of basic reproduction number. We discuss the spreading and vanishing phenomena in terms of the basic production number. By employing the perturbed approximation method and monotone iteration method, we establish the existence, uniqueness and monotonicity of solution to semi-wave problem. When spreading occurs, we determine the asymptotic spreading speeds of free boundaries by constructing suitable upper and lower solutions from the semi-wave solutions. Moreover, spreading speeds for partially degenerate diffusion case are provided in a similar way.

A delayed reaction-diffusion system with free boundaries is investigated in this paper to understand how the bacteria spread spatially to larger area from the initial infected habitat. Under the assumptions that the nonlinearities are of monostable type and the initial values satisfy some compatible condition, we show that the free boundary problem is well-posed and discuss the long-time behaviour of solution (including spreading and vanishing) in terms of the spatial-temporal risk index. Furthermore, to determine the spreading speed of free boundaries when spreading occurs, we first study the distribution of roots of a transcendental equation containing a polynomial of degree four and then establish the existence and uniqueness of monotone solution to a delay-induced nonlocal semi-wave problem by employing the approximation method, lower-upper solutions technique and Schauder fixed point theorem. It is shown that time delays slow down the spreading of bacteria.

Two mathematical models under so-called intensity and structure frameworks to pricing a double defaultable interest rate swap are established. The default could happen or jump to a high probability in both fixed and floating parties on the predetermined boundaries. The models lead to a new and interesting mathematical problem. As the intensity approaches infinity in designated regions, the solutions of the intensity models converge to a solution of a structure-type model which is an initial value problem of a partial differential equation coupled with two obstacles problem in their restricted regions. According to the value of the fixed rate, three cases are discussed. The free boundary that determines the swap rate and the free boundaries that determine the earlier termination of the contract (due to counterparty’s default) are analysed.

This paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.

In this paper we study a finite-fuel two-dimensional degenerate singular stochastic control problem under regime switching motivated by the optimal irreversible extraction problem of an exhaustible commodity. A company extracts a natural resource from a reserve with finite capacity and sells it in the market at a spot price that evolves according to a Brownian motion with volatility modulated by a two-state Markov chain. In this setting, the company aims at finding the extraction rule that maximizes its expected discounted cash flow, net of the costs of extraction and maintenance of the reserve. We provide expressions for both the value function and the optimal control. On the one hand, if the running cost for the maintenance of the reserve is a convex function of the reserve level, the optimal extraction rule prescribes a Skorokhod reflection of the (optimally) controlled state process at a certain state and price-dependent threshold. On the other hand, in the presence of a concave running cost function, it is optimal to instantaneously deplete the reserve at the time at which the commodity's price exceeds an endogenously determined critical level. In both cases, the threshold triggering the optimal control is given in terms of the optimal stopping boundary of an auxiliary family of perpetual optimal selling problems with regime switching.

We investigate a degenerate parabolic variational inequality arising from optimal continuous exercise perpetual executive stock options. It is also shown in Qin et al. (Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN: http://dx.doi.org/10.2139/ssrn.2757541) that to make this problem non-trivial the stock's growth rate must be no smaller than the discount rate. Well-posedness of the problem is established in Lai et al. (2015, Mathematical analysis of a variational inequality modeling perpetual executive stock options, Euro. J. Appl. Math., 26 (2015), 193–213), Qin et al. (2015, Regularity free boundary arising from optimal continuous exercise perpetual executive stock options, Interfaces and Free Boundaries, 17 (2015), 69–92), Song & Yu (2011, A parabolic variational inequality related to the perpetual American executive stock options, Nonlinear Analysis, 74 (2011), 6583-6600) for the case when the underlying stock's expected return rate is smaller than the discount rate. In this paper, we consider the remaining case: the discount rate is bigger than the growth rate but no bigger than the return rate. The existence of a unique classical solution as well as a continuous and strictly decreasing free boundary is proved.

We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike the previously known results, the surface diffusion coefficient is not assumed constant and depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting-time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations.

The isothermal, non-sliding shallow-ice approximation, combined with mass conservation, is a fundamental model for ice-sheet and glacier flow. It determines the ice extent, geometry and velocity by the solution of a free-boundary problem. In this paper, the steady-state form of this problem is solved directly, without time-stepping, thereby demonstrating a fully implicit scheme with no stability restrictions. The classical Mahaffy (1976) finite difference method is first reinterpreted as a ‘finite volume element’ scheme that has both an everywhere-defined approximate thickness function and an approximation of the conservation equation in flux integral form. From this reinterpretation an improved scheme is built by using better quadrature in the integral and upwinding on that part of the flux which is proportional to the bed gradient. The discrete equations are then solved by a parallel Newton scheme which respects the constraint that ice thickness is non-negative. The results show good accuracy on both flat-bed and bedrock-step exact solutions. The method is then applied at high resolution to model the steady-state geometry of the Greenland ice sheet, using only bedrock elevation and present-day surface mass balance as input data.

In this paper, we establish the existence and uniqueness of a classical solution of a degenerate parabolic variational inequality of which a strong solution was shown to exist by Song and Yu [21]. The problem arises from optimal stochastic control of exercising continuously perpetual executive stock options (ESOs). We also characterize the basic graph, continuity, and monotonicity properties of the free boundary from which the optimal control strategy can be described precisely.

An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz function g(z, t) of the free boundary. This generalises the well-known equation ∂g/∂t = 2∂w/∂z, where w is the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

We consider the following nonlinear elliptic equations

\begin{gather*} \begin{cases} \Delta u+u_{+}^{N/(N-2)}=0\amp\quad\text{in }\sOm, \\ u=\mu\amp\quad\text{on }\partial\sOm\quad(\mu\text{ is an unknown constant}), \\ \dsty\int_{\partial\sOm}\biggl(-\dsty\frac{\partial u}{\partial n}\biggr)=M, \end{cases} \end{gather*}

where $u_{+}=\max(u,0)$, $M$ is a prescribed constant, and $\sOm$ is a bounded and smooth domain in $R^N$, $N\geq3$. It is known that for $M=M_{*}^{(N)}$, $\sOm=B_R(0)$, the above problem has a continuum of solutions. The case when $M>M_{*}^{(N)}$ is referred to as supercritical in the literature. We show that for $M$ near $KM_{*}^{(N)}$, $K>1$, there exist solutions with multiple condensations in $\sOm$. These concentration points are non-degenerate critical points of a function related to the Green's function.

AMS 2000 Mathematics subject classification: Primary 35B40; 35B45. Secondary 35J40

In this paper, we prove someregularity results for the boundary of an open subset of $\xR^d$ whichminimizes the Dirichlet's energy among all open subsets withprescribed volume. In particular we show that, whenthe volume constraint is “saturated”,the reduced boundary of the optimal shape (and even the wholeboundary in dimension 2)is regular if the state function is nonnegative.

In this paper we study a free boundary problem appearing inelectromagnetism and its numerical approximation by means ofboundary integral methods. Once the problem is written in aequivalent integro-differential form, with the arcparametrization of the boundary as unknown, we analyse it inthis new setting. Then we consider Galerkin and collocationmethods with trigonometric polynomial and spline curves asapproximate solutions.

We solve the following three optimal stopping problems for different kinds of options, based on the Black-Scholes model of stock fluctuations. (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more difficult than the closely related one for the Russian option, and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary differential equation. (ii) A new type of stock option, for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a fixed constant l as a ‘hedge’ on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at any exercise time. We show that the optimal policy depends on the ratio of x/l, where x is the current stock price.