There are classifications theorems that single out the flat equatorial disk and the critical catenoid among **free** **boundary** minimal surfaces in B 3 . Some of them will be reviewed in Section 2. In this work, we sought for a new characterisation of the flat equatorial disk and the critical catenoid in terms of the length of their second fundamental forms. It turns out that a pinching condition on this quantity, that takes into

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Calculations of the properties of equilibria consisting of nested toroidal magnetic sur faces are simplified by the use of a coordinate system {xp,6,r)) made up of a surface label, xp, and poloidal and toroidal angles 9 and rj respectively. The angle coordinates define a grid within each constant-*/» surface. The converged solution to the equilibrium equations is usually m apped to such a flux coordinate system before performing a stability analysis (described shortly), b ut some codes have been w ritten which solve the equilibrium equa tions in flux coordinates directly. The so-called “inverse equilibrium ” **free** **boundary** code of Delucia et al. (1980) puts r) = (ptor and iterates the m apping from {xp,6y(ptor) to (r,<ptor, z ) w ithin the plasm a region until a converged (axisymmetric) equilibrium is obtained. At each iteration level xpnew is found by solving Eqs. (1.9) and (1.10) on a curvilinear mesh defined by xpold and 9oid at the previous iteratio n level. The outer flux contour is taken as defining the plasm a **boundary** which is then reestim ated (once the “fixed **boundary**” iterations have converged) by contouring the new values of the “**boundary**” fluxes given by G reen’s theorem. The formulation is similar to th a t described above except th a t in this case the **boundary** integral is taken over th e plasm a itself.

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In summary, we considered an attraction–repulsion chemotaxis system with a **free** bound- ary in one space dimension. In model (1.1), the so-called **free** **boundary** x = h(t) character- izes the change of the expanding front for the mobile species. Our conclusions not only provide suﬃcient conditions for species spreading success and spreading failure, but also the long time behavior of the mobile species, chemo-attraction and chemo-repulsion. Pre- cisely, we prove a spreading–vanishing dichotomy for this model, that is, either the species fails to establish and vanishes eventually, or the species successfully spreads to inﬁnity as t → ∞ and stabilizes at a constant equilibrium state under some suﬃcient conditions. Not only that, we also discuss the criteria for spreading and vanishing. These analytical ﬁndings disclose that the change of invasion region to species can determine whether the invasion is successful or not.

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Proof of Theorem 1.1 We will use the contraction mapping principle on some functional spaces arranged after rewriting the problem in a domain without a **free** **boundary**. We follow the same steps, leading to local existence, as in [10]. But we have to pay attention to the facts that our model is diﬀerent from the one in [10] (especially the nonlinearities) and the spatial domain in our work is bounded.

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In the present paper, we present a new variant of the complex velocity-complex potential pair method for studying the seepage from asymmetric soil channels. The symmetric case, which herein is considered as a particular case, was already investigated in 14, 15. We consider the conformal mapping fζ of the unit half-disk onto the half-strip from the complex potential plane. We shall use Levi-Civit´a’s function ωζ in order to construct the conformal mapping zζ of the unit half-disk onto the flow domain. The radii −1, 0 and 0, 1 of the unit half-disk correspond through the conformal mapping zζ to the **free** phreatic lines of the flow domain. On these radii, the imaginary part of ωζ vanishes by virtue of the conditions imposed on the **free** lines. According to Schwarz’s principle of symmetry, we may extend the domain of definition of ωζ to the whole unit disk. The analytic function ω is afterwards expanded into a Taylor series. In comparison to the above mentioned inverse method, our method is more general; it is not restricted to special classes of contours of the channel. We have to give only the expression of the function ωζ in fact we shall give the coeﬃcients of the Taylor series of ωζ and some additional terms for the case of profiles with angular points in order to construct the channel profile and solve the corresponding **free** **boundary** seepage problem. In fact, an inverse method has the maximum eﬃciency if it can be employed to solve the direct problem. Our method satisfies this requirement; by successive attempts, for every a priori given contour, we may endeavor to guess the corresponding coeﬃcients of the Taylor series and so, to use the inverse method in order to solve the direct problem.

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giving rise to a **free** **boundary**. Our purpose is to analyze the **free** **boundary** for a large class of obstacle problems associated with degenerate ( < p < ∞) and non-degenerate ( < p ≤ ) parabolic equations. Therefore, let us start with the formulation of the problem in the weak sense. Let be an open bounded domain of R N (N ≥ ), T = × (, T ).

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• Higher order **boundary** element methods can be used [4]. Second order con- vergence is observed in Section 4 (partially as a result of the solution being constant on the outer **free** **boundary**). To the authors’ knowledge, the present work is one of very few published results regarding the accuracy of a combined level-set **boundary** element method, see for instance [11].

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5. Conclusion. Solutions of the Bernoulli **free** **boundary** problem can be eﬃ- ciently computed by the method presented here. Providing a Green’s function is available, the method can be used to solve other **free** **boundary** problems. For in- stance, it can be applied with only minor modiﬁcations to the Prandtl–Batchelor problem (see [1] and the references therein), which consists in looking for a domain A which is now interior to the ﬁxed domain Ω such that for a given function σ,

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In this section we will state the main results, for convenience we will divide this section into two parts. In the first part, we will give three propositions and one theorem. Where, Proposition 1 and Proposition 2 are useful in proving Proposi- tion 3, and Proposition 3 proves Theorem 1. Theorem 1 represents the solution of σ in **free** **boundary** problem (27). In the second part, we will derive the so- lution of **boundary** velocity v ± ( ) t and **boundary** position x ± ( ) t by using con- dition (12) and Theorem 1.

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The class of elliptic partial differential equations includes many important systems encountered in mechanics and geometry. Indeed an industrial set- ting time is spent setting up the allowable set of shapes (domains of equa- tions) in order to get a feasible solution. The structural optimization of such systems has more commonly been applied in the automobile, marine and aerospace industries designing and even in a simple mechano-chemical model of a biomolecular processes (see [9], [1] and [2]). A large part of these problems deals with the **free** **boundary** problems when a part of domain , s

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Another is that the spreading speed approaches to a positive constant if the spreading occurs. On the other hand, they derived the criteria for spreading and vanishing. Later on, Du and Guo [2, 3], studied a **free** **boundary** problem similar to (1) in higher dimension space, Kaneko and Yamada [4] discussed (1) and the case of bistable nonlinearity with u x (t, 0) = 0 replaced by u(t, 0) = 0.

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This paper is concerned with the one-dimensional **free** **boundary** problem for quasilinear reaction-diﬀusion systems arising in the ecological models with N-species, where some of the species are made up of two separated groups and the mankind?s inﬂuence is taken into account. In the problem under consideration, there are n **free** boundaries, the coeﬃcients of the equations are allowed to be discontinuous on the **free** boundaries and the reaction functions are mixed quasimonotone. The aim is to show the local existence of the solutions for the **free** **boundary** problem by the ﬁxed point method, and the global existence and uniqueness of the solutions for the corresponding diﬀraction problem by the approximation and estimate methods. MSC: 35R35; 35R05; 35K57; 35K20

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Several classical results from functional analysis have been extended to this non-Euclidean setting in [7, 25, 10]. Here we state some fundamental embedding and a priori estimates for the variable coefficients operator L in (3.11). These results provide basic tools for the study of the **free** **boundary** problem in Section 4. In the following inequalities, Q denotes the homogeneous dimension in (5.7), D 0 is a domain contained, with its closure, in D and c is a constant only

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The **free** **boundary** condition (4) is firstly established by Lin [11] from an ecological point by using the Fick’s first law. This condition coincides with the well-known one-phase Stefan condition arising from the investigation of the melting of ice in contact with water, and has been applied in many other application field, for example, the modeling of wound healing [12], tumor growth [13], spreading of disease [14-16] and spreading of species [7, 11].

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Abstract. We consider the optimal stopping and optimal control problems related to stochastic vari- ational inequalities modeling elasto-plastic oscillators subject to random forcing. We formally derive the corresponding **free** **boundary** value problems and Hamilton-Jacobi-Bellman equations which belong to a class of nonlinear partial of differential equations with nonlocal Dirichlet **boundary** conditions. Then, we focus on solving numerically these equations by employing a combination of Howard’s al- gorithm and the numerical approach [A backward Kolmogorov equation approach to compute means, moments and correlations of non-smooth stochastic dynamical systems; Mertz, Stadler, Wylie; 2017] for this type of **boundary** conditions. Numerical experiments are given.

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Many results on the existence and uniqueness of weak periodic solutions are already available in the literature see the biographical comments collected in Section 1. Nevertheless those interesting questions are not our main aim here but only the study of the **free** **boundary** generated by the solution under suitable additional conditions on the data.

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Remark 20. The above estimate follows from the curvature estimates of Schoen-Simon [30] which imply that the only complete, properly embedded, **free** **boundary** and stable minimal hypersurface in the upper half space R n+1 + which has Euclidean volume growth is a half plane. This theorem was proved under slightly different hypotheses in [21], in particular they consider immersed and stable M , at which point the conclusion is only known to hold for n ≤ 5 using the estimates of Schoen-Simon-Yau [31]. However, under the assumption of embeddedness of M , and a lower bound on the first eigenvalue, the conclusion holds also for 2 ≤ n ≤ 6 by the curvature estimates of Schoen-Simon (see page 13 of [21]). Of course if p ∈ M \∂M and ρ < dist(p, ∂N ) then this is just the (usual) interior curvature estimate of Schoen-Simon [30, Corollary 1].

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paper have gone on to be embraced as the dogma of option pricing [1]. Regardless of the type of options, the available measures are in a way either directly or indirectly re- lated to Black & Scholes equation (sometimes known as Black-Scholes-Merton equa- tion). Nevertheless, the pricing of options put forward by Black & Scholes does not perfectly suit the American type of options. This has triggered scholars to search for appropriate pricing measures for the American type of options. American call options have relatively been handled and pricing strategies developed as compared to the put options. American calls on stock with proportional dividends were treated as a **free** **boundary** problem by [2] [3] after [4] had also addressed a related problem. If the asset does not pay dividends, early exercise on American calls is useless [2] since they would then be yielding the same as their European equals. Moreover even with discrete divi- dends, American calls can still be valued by using analytic expressions [5]. In [6], they provided an analytic characterization for a non-dividend paying put option. The prob- lem that remains is the valuation of put dividend paying options. Analytic pricing re- presentations have been independently supplied by [7]. The papers [7] [8] are so far the most popular among the analytic characterizations but unfortunately none of them does solve the analytic valuation developed therein. [8] put forward results on optimal stopping and the put option and decomposed the price of the option as the price of the European equal and the premium [7] [9], the latter in a way bettering the work of [10]. The lower and upper bounds for the American option prices are provided in [11]. A version of Monte Carlo methods for pricing American options that is claimed to per- form better than the already existing one has been put forward in [12]. The study of the **free** **boundary** for American butterfly options is done in [13]. The pricing of American options with the help of front fixing finite differences which is quite a better scheme has been handled in [14]. However, it still underestimates the prices at certain instances. Quite a variety of the recent literature on American option valuation have been about trying to study the properties of the **free** **boundary**, at least to the authors, and those that attempt to price the options tend to provide quite interesting and yet somewhat hard algorithms. This necessitates the need for an article that does not concentrate on the properties of the **free** **boundary** but attempts to look into the **free** **boundary** itself with an aim of acquiring approximates not only at specific times (as most papers have demonstrated) but in a general case (at all times) and with a straight forward approxi- mation algorithm. This work is intended to develop an easier way of approximating op- tion values, which can be applied easily in the market. This article is divided into four sections.

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namely hypersurfaces that are critical points of the area functional when the **boundary** ∂M is not fixed (like in Plateau’s problem) but subject to the sole constraint ∂M ⊂ ∂Ω. Due to their self-evident geometric interest (which can be traced back at least to Courant [3]), these variational objects have been widely studied and a number of existence results have been obtained via surprisingly diverse methods (see, among others, [4,9–11,17–20,28,30] and references therein). **Free** **boundary** minimal hypersurfaces also naturally arise in partitioning problems for convex bodies, in capillarity problems for fluids and, as has significantly emerged in recent years, in connection to extremal metrics for Steklov eigenvalues for manifolds with **boundary** (see primarily the works by Fraser-Schoen [7–9] and references therein). From an analytic perspective, it should also be mentioned that their **boundary** regularity has been the object of extensive investigations (let us mention, for instance, [12, 14–16]).

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