Mathematics Subject Classiﬁcation. 35R35, 49Q10, 35Q30, 76D07.
Received November 10, 2014. Revised August 26, 2015. Accepted September 17, 2015.
1. Introduction
This paper aims at studying a **free** **boundary** **problem** for ﬂuid ﬂows governed by the Stokes equations. This **problem** originates from numerous applications such as magnetic shaping processes, where the shape of the ﬂuid is determined by the Lorentz force. For such conﬁgurations, the model is described by the ﬂuid ﬂow equations, here the Stokes equations, and a pressure balance equation on the unknown **boundary** in the case where surface tension eﬀects are neglected. Depending on the application, two types of models can be considered: an interior **problem** where the ﬂuid is conﬁned in a mould and has an internal unknown **boundary** and an exterior **problem** where a part of the ﬂuid **boundary** adheres to a solid and the remaining (unknown) part is **free** and is in contact with the ambient air. We shall focus our study on this last case for two-dimensional geometries.

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In the next Section 2, we present the main results of the paper, in terms of biological and mathe- matical insights. We then present in Section 3 the main arguments that lead to well-posedness. For the sake of simplicty, we focus on the 2D case. Note that the difficulty lies in the fact that two domains are involved (the cell cytoplasm and the extracellular medium) and the velocity is the gradient of the inner chemical signal. As usual for such **free**-**boundary** problems [13,33], the proof for well-posedness is based on appropriate quasilinearization. Interestingly, this leads to a parabolic type equation that prevents the a priori loss of regularity. More precisely, thanks to Lagrangian formalism and complex analysis tools we provide the Dirichlet-to-Neumann operators in both inner and outer domains that make it possible to rewrite equivalently the **free** **boundary** **problem** on the torus T as provided in Lemma 10. The quasi- linearization is presented in Subsection 3.2, leading to the well-posedness result under a sign condition of the datum g. The sign condition under which holds the well-posedness is a kind of Taylor criterion 3 adapted to our model. Section 4 is devoted to present our first order finite difference method on Cartesian grid, which ensures the accuracy of the simulations. The numerical method relies on immersed **boundary** and ghost fluid methods. We introduce a new continuous stencil for the gradients at the interface –the cell membrane– which stabilizes the standard schemes such as this of Cisternino and Weynans [4]. We then illustrate numerically that instabilities appear if the positivity of g is not satisfied and we show numerically the protrusions formation, similarly to biological images. Interestingly, numerical simulations for invadopodium and pseudopodial protrusions, are qualitatively similar to the biological observations. In Appendix A we show how the well-posedness of the Hele-Shaw model (also called Muskat model) can be tackled thanks to our approach.

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The divergence of E 1 (z) at z → 0 is logarithmic and thus integrable. Usually
this means that the integration can be performed by the Gauss method. However, since our **problem** is singular due to the contact line effects, many Gauss points are needed to attain the required accuracy in the contact line region and a more sophisticated algorithm is necessary to get both accuracy and speed. The analytical integration [10] is used when y = 0 i.e. when the singularity occurs. Although there is no singularity when y 6= 0, the integrand varies sharply near the point u = −x when y ≪ l. For the case |x| < l/2, the integration interval can be split by the point u = −x and changes of variables can be done in both integrals to produce

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A very interesting - and open - question is whether for general convex bounded sets Ω the structure of the solutions of Bernoulli **problem** enjoys similar features. In this paper we try to provide some positive evidence towards this conjecture. Let us first recall that, for any bounded and convex domain Ω, and for any fixed volume σ > 0, there is at least one convex solution to Bernoulli **free** **boundary** **problem** of volume σ. This has already been proved in [2], Theorem 3 and in [10], Theorem 5.1. In the first part of this paper, we give a new - and we hope enlighting - proof of this result. In [19], Henrot and Shahgholian proved the existence, for any λ ≥ λ Ω , of a maximal convex solution f D λ of level

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2010 Mathematics Subject Classification. Primary 35R35.
Key words and phrases. almost-minimizer, **free** **boundary** **problem**, uniform rectifiability.
G. David was partially supported by the Institut Universitaire de France and the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082 and the University of Chicago RTG grant DMS 1246999. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823 and by the Robert R. & Elaine F. Phelps Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

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1 ).
5. Shallow water model with a floating body on the water surface
We turn to analyze other examples of wave-structure interaction in which the fluid occupies an infinite canal and a floating rigid body is placed on the water surface. We follow the approach proposed in [Lan17] where the **free** surface Euler equations are reformulated in terms of the **free** surface elevation and of the horizontal water flux. Under this approach, the pressure exerted by the fluid on the floating body can be viewed as the Lagrange multiplier associated to the constraint that, under the body, the surface of the fluid coincides with the bottom of the body. As shown in [Lan17], this approach can be used also in the shallow water approximation, replacing the **free** surface Euler equations by the much simpler nonlinear shallow water equations. This is the framework that we shall consider here, addressing three cases; the floating body is fixed, the motion of the body is prescribed, and the body moves freely according to Newton’s laws under the action of the gravitational force and the pressure from the air and from the water. The case of a floating body moving only vertically and with vertical lateral walls has been considered in [Lan17] in 1D, in [Boc18] for a 2D configuration with radial symmetry, and numerical computations have been proposed in [BEKER]. For such configurations, the horizontal projection of the portion of the solid in contact with the water is independent of time. We consider here the more complex situation of nonvertical lateral walls: even in the case of a fixed object, determining the portion of the solid in contact with the water is then a **free** **boundary** **problem** that is difficult to handle; in the numerical study [GPSMW] for instance, the authors use a compressible approximation of the equations in order to remove this issue. The configuration under study here is described in Figure 2.

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More precisely, we are interested in this paper in a shape optimization **problem** for solving numerically the so-called Bernoulli’s **free** **boundary** **problem** or Bernoulli’s **problem**. Let us note that many applications to fluid dynamic and industrial application lead to such a **free** **boundary** **problem**. As an example, let us quote the **problem** where the design of an annular capacitor is required in which one of the plates is prescribed while the other must be determined, so that the intensity of the electrostatic field remains constant thereon. Depending on whether we describe the internal or external plate, we have an exterior or an interior Bernoulli **problem**. This class of problems has been extensively studied theoretically by many authors, see for example [1, 11, 16, 22] and references therein. A practical way to study this type of **problem** is to transform it into a shape optimization **problem** where one of the Dirichlet or Neumann **boundary** condition on the **free** **boundary** is included into a cost functional while the other **boundary** condition is considered as part of an appropriate state **problem**. The question of existence for such shape optimization formulations is studied for example in [17] or [9], using the C 2 or C 1 regularity of the

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[2] ——-, Interior **free** **boundary** problems for the Laplace equation, Arch. Rational Mech. Anal. 75 (1980/81), n. 2, 157-168.
[3] H. W. Alt, L. A. Caffarelli, Existence and regularity for a minimum **problem** with **free** **boundary**, J. Reine Angew. Math. 325 (1981), 105-144.
[4] ——-, A **free** **boundary** **problem** for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 1-44. [5] A. Acker, R. Meyer, A **free** **boundary** **problem** for the p-Laplacian: uniqueness, convexity, and successive approximation of solutions, Electron.

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We consider in this paper the numerical solution method of the so-called Bernoulli **problem**. This **problem** can be considered as a typical example of a stationary **free** **boundary** **problem**. Its applications appear in fluid dynamics as in Friedman[2], optimal design as in Flusher and Rumpf[3], or in electromagnet- ics as in Crouzeix [4] and Descloux[5] and various other engineering fields. The presented numerical method is based on a level set formulation of the Bernoulli **problem**. More specifically, since this **problem** is time independent we derive a simpler formulation of the method, the basic tool being an adequate extension of the propagation velocity of the unknown **boundary**.

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How to use three-in-a-tree is discussed in [3] and further evidences of its generality are given in [6]. Because of the power and deepness of three-in-a-tree, it would be interesting to generalise it. Here we study four-in-a-tree: the **problem** whose instance is a graph G together with four of its vertices, and whose question is “Does G contain an induced tree covering the four vertices ?”. Since this **problem** seems complicated to us, we restrict ourselves to triangle- **free** graphs. Our approach is similar to that of Chudnovsky and Seymour for three-in-a-tree. We give a structural answer to the following question: how does look like a triangle-**free** graph such that no induced tree covers four given vertices x 1 , x 2 , x 3 , x 4 ? On Fig. 1 and 2,

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To avoid these discontinuities, a new approach is proposed based on the propagation of a moving layer inside which the damage is a continuous function of the position. The evolution of damage is then associated to the motion of a layer of finite length [5]. The initial material and the damaged material are separated by a surface Γ. This **boundary** is a moving interface. A surface is an isopotential or a level-set. Through the interface the material changes its mechanical properties. In the proposed description, this transition is continuous.

4 Discussion
The processes contributing to the “topographic venting” under fair weather daytime conditions are summarized in Fig. 13, expanding a similar figure proposed by Seibert (1996) in the context of ALPTRAC. Pollutants are emitted mainly at the valley floor or advected horizontally by the up- valley wind from the forelands. Within the valley, a well- mixed **boundary** layer (NO x mixing ratios of about 10 ppbv) is capped by a rather stable layer indicated by an increase of potential temperature (dashed-dotted line). Up-slope winds are able to penetrate this layer and lift polluted air from lower levels. Additional shallow cumulus cloud formation above the crests further maintains vertical motion. Above the sta- ble layer horizontal airflow is mainly synoptically driven. In contrast to the ABL, the upper layer is only partly mixed and is indirectly connected to the surface, therefore the term “in- jection layer” is used (NO x ∼ 1 ppbv). The injection layer is capped by a strong inversion that marks the transition to the FT (NO x ∼ 0.1 ppbv). The mass balance of the slope flow sys- tem is not closed in the two-dimensional valley cross section but in a three-dimensional way by the up-valley wind.

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Given a graph G = (V, E), a matching in G of maximum cardinality is called a maximum matching. Its size corresponds to the matching number of G which is denoted by ν(G). One of the most attractive and studied problems in combinatorial optimization is the maximum matching **problem**, which consists, given a graph G, in finding a maximum matching in G [3, 8]. This **problem** can be solved in polynomial time using the algorithm developed by Edmonds [3]. If the graph is bipartite, the **problem** is much simpler. It reduces to a maximum flow **problem**. A vertex cover of a graph G is a set T of nodes such that every edge of G has at least one end in T . A well known min-max relation in graph theory and combinatorics is the following. A stable set of a graph is a subset of nodes S such that no two nodes in S are adjacent. Given a graph G = (V, E), the stable set **problem** in G consists in finding a stable set of maximum cardinality. Theorem 1. (K¨ onig [8]) For a bipartite graph, the maximum cardinality of a matching is equal to the minimum cardinality of a vertex cover. As the complementary of a vertex cover in a graph is a stable set, a conse- quence of Theorem 1 is the following.

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2 Proof of Theorem 1
We will start with perturbations of the level of the **free** **boundary** and con- siderations on its regularity. The method essentially goes as in [2], so we shall simply give a sketch of the proofs. Then we will prove that Theo- rem 1 is a consequence of Theorem 3. Before, simply notice that Equations (1.1)-(1.2)-(1.3) are such that Assumptions (A0)-(A3) are satisﬁed.

well-constrained. The structural analysis **problem** for a DAS consists in verifying
if the system is well-constrained.
In many practical situations, physical systems yield differential-algebraic sys- tems with conditional equations. A conditional equation is an equation whose from depends on the value (true or false) of a condition. A conditional equation can generate several equations. A conditional differential-algebraic system may then have different forms depending on the set of conditions that hold. Here we consider conditional DAS’s such that any conditional equation may take two possible values, depending on whether the associated condition is true or false and may generate only one equation. Consider for example the following DAS : eq1 : if a > 0

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Abstract
This paper is concerned with the existence of solutions for a non local fractional **boundary** value **problem** with integral conditions. New existence and uniqueness results are established using Banach fixed point theorem. Other existence results are obtained using Schauder and Krasnoselskii theorems. As an application, we give an example to illustrate our results.

2 + s =
r > 0, they are often called Strichartz estimates with loss of r derivatives. The derivation of such estimates (and the associated well-posedness results) for NLS on a domain with the Dirichlet (or Neuman) Laplacian has been intensively studied over the last decade in various geometric settings. We will only cite results in the case where Ω is the exterior of a non trapping obstacle since it is the one studied here. Roughly speaking, a non-trapping obstacle is an obstacle such that any ray propagating according to the laws of geometric optic leaves any compact set in finite time (for a a mathematical definition of the rays, see [21]). In their seminal work [9], Burq, G´ erard and Tzvetkov proved a local smoothing property similar to the one on R d (see [12]) and deduced Strichartz estimates with loss of 1/p derivative. Since then numerous improvements were obtained [2][3][8] and eventually led to scale invariant Strichartz estimates : see Blair-Smith-Sogge [7] in the general non-trapping case (s > 0 and limited range of exponents), Ivanovici [15] for the exterior of a convex obstacle (s = 0, all exponents except endpoints). The methods used relied heavily on spectral localization and construction of parametrices. As such they are not very convenient for the study of non homogeneous **boundary** value problems when the **boundary** data are not smooth enough to reduce the **problem** to a homogeneous one.

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[6] P. Gérard. The Cauchy **problem** for the Gross-Pitaevskii equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(5):765779, 2006.
[7] P. Gérard. The Gross-Pitaevskii equation in the energy space. In Stationary and time dependent Gross- Pitaevskii equations, volume 473 of Contemp. Math., pages 129148. Amer. Math. Soc., Providence, RI, 2008. [8] M. Hayashi and T. Ozawa. Well-posedness for a generalized derivative nonlinear Schrödinger equation. J.

Another interesting feature of the proposed formulation comes from its ability to properly handle the pre- scription of inlet/outlet **boundary** conditions at artificial boundaries. This situation may occur for instance when modeling flow through a network of pipes truncated to a region of interest. The challenging issue of prescribing suitable **boundary** conditions at the artificial sections is reviewed in [ 16 ] and different strategies to cope with this difficulty are proposed for instance in [ 35 ] by means of the Nitsche method or in [ 30 ] in the context of blood flow modeling. Moreover, as noted in [ 25 , 22 ], using the symmetric gradient 1 2 (∇u + ∇u T ) and prescribing the normal stress at the outlet lead to a non-physical representation of the flow: the velocity vectors “spread” like at the end of a pipe, instead of mimicking the fact that the network continues after this artificial section. Alternatively, the non symmetric tensor ∇u can be used to recover the Poiseuille exact solution in a cylinder, but the physical meaning of such a **boundary** condition is not clear. In the present work, the mathematical and computational models share both advantages: the fluid model is described in terms of the mechanical stress tensor, which is more appropriate from the modeling viewpoint, and it is able to properly take into account the fact that the flow continues beyond the boundaries, thanks to the specific form of the **boundary** conditions.

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