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Understanding Singularities in Free Boundary Problems

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Understanding Singularities in Free Boundary Problems UNDERSTANDING SINGULARITIES IN FREE BOUNDARY PROBLEMS XAVIER ROS-OTON AND JOAQUIM SERRA Abstract. Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. The most classical example is the melting of ice to water (the Stefan problem). In this case, the free boundary is the liquid-solid interface between ice and water. A central mathematical challenge in this context is to understand the regularity and singularities of free boundaries. In this paper we provide a gentle introduction to this topic by presenting some classical results of Luis Caffarelli, as well as some important recent works due to Alessio Figalli and collaborators. Abstract. I problemi di frontiera libera sono quelli descritti da EDP che mostrano inter- facce o frontiere a priori sconosciuti (liberi). L'esempio pi`uclassico `elo scioglimento del ghiaccio in acqua (problema di Stefan). In questo caso, la frontiera libera `el'interfaccia solido-liquido tra acqua e ghiaccio. Una sfida matematica centrale in questo contesto `ecomprendere la regolarit`ae le singo- larit`adelle frontiere libere. In questo articolo introduciamo questo argomento presentando alcuni risultati classici di Luis Caffarelli, oltre ad alcuni importanti lavori recenti dovuti ad Alessio Figalli e collabo- ratori. 2010 Mathematics Subject Classification. 35R35. Key words and phrases. Free boundaries; obstacle problem. This paper will appear in a special issue of the journal `Matematica, Cultura e Societ`a' (published by the Unione Matematica Italiana) dedicated to the work of Alessio Figalli. 1 2 XAVIER ROS-OTON AND JOAQUIM SERRA 1. Introduction while in the complement θ is just zero. Determining where is the interphase or 1.1. The Stefan problem. free boundary that separates the two re- The Stefan problem, dating back to the gions (i.e., the surface @fθ > 0g) is part XIXth century, is the most classical and im- of the problem. For it, an extra equation portant free boundary problem. First consid- (or boundary condition) on the interface is ered by Lam´eand Clapeyron in 1831, aims needed. This is the so-called Stefan condi- to describe the temperature distribution in tion: a homogeneous medium undergoing a phase 2 change, typically a body of ice at zero de- @tθ = jrxθj on @fθ > 0g: (1.1) grees centigrade submerged in water. It is This extra relation comes from two consid- named after Josef Stefan, a Slovenian physi- erations. First, the normal velocity of the cist who introduced the general class of such interphase, V , is proportional to the amount problems around 1890; see [52, 53, 37]. of heat absorbed by it (which must be used The most classical formulation of the Ste- to melt the ice). In turn, this heat which 3 fan problem is as follows: Let Ω ⊂ R be \enters" the interphase is, by Fourier law, some some bounded domain. For concrete- proportional to the gradient of temperature. ness, we let us think that Ω is cylindric water Thus, we have jV j = Cjrθj . Second, since tank as depicted in Figure 1. We denote θ = 0 on the moving interphase we ob- θ = θ(x; t) tain that, on it, V and rθ are parallel and (@ + V · r)θ = 0. Combining the two pre- the temperature of the water at the point t vious formations and choosing the physical x 2 Ω at time t 2 + := [0; +1). We as- R units to make C = 1, we obtain the Stefan sume that θ ≥ 0 in Ω × +. Given are the R condition (1.1). (nonnegative) initial temperature and tem- One can also see that, by the \maximum perature at the boundary of the tank. principle", the ice fθ = 0g shrinks with time. In other words, if at some point of the tank there is liquid water for some given time then Ice that point will remain liquid at all future times. Water It can be shown that, after the transfor- mation Z t u(x; t) := θ(x; τ)dτ; 0 (see [2, 18]) the new function The Stefan problem. + + Figure 1. u :Ω × R ! R satisfies The set f(x; t) 2 Ω× + : θ(x; t) > 0g, de- R @ u − ∆u = −χ noted for brevity fθ > 0g, represents the wa- t fu>0g ter while its complement, denoted fθ = 0g, u ≥ 0 (1.2) represents the ice. The temperature θ satis- @tu ≥ 0; fies the heat equation where χA denotes the characteristic function @tθ − ∆θ = 0 in the region fθ > 0g; of the set A. UNDERSTANDING SINGULARITIES IN FREE BOUNDARY PROBLEMS 3 Since we can easily recover θ form u by obstacle problem (1.3), and the free bound- computing its time derivative, we see that ary corresponds to the interface between the (1.2) is an equivalent formulation of the Ste- wet and dry parts of the dam. fan problem. The new formulation is use- ful because it enjoys better mathematical properties (it has the structure of \varia- tional inequality") than the original formu- lation with θ. For instance, while from the original formulation with θ it is unclear how to show existence and uniqueness of solution (there was no rigurous proof for more than a Dam century!), it is much easier to do it with the Water equivalent formulation (1.2). The stationary version of (1.2) is the well- known obstacle problem: ∆u = χfu>0g (1.3) Figure 2. The Dam problem. u ≥ 0: It is among the most famous problems in el- liptic PDE, as it arises in a variety of situa- tions. 1.2. Motivations and applications. Both the Stefan problem and the obstacle • Hele-Shaw flow. This model, dating problem appear in many different models in back to 1898, describes a fluid flow between physics, industry, biology, or finance. We two flat parallel plates separated by a very next briefly comment on some of them, and thin gap. Various problems in fluid mechan- refer to the books [19, 34, 43, 27, 42] for more ics can be approximated to Hele-Shaw flows, details and further applications of obstacle- and that is why understanding these flows is type problems. important. • Phase transitions. In the classical Stefan A Hele-Shaw cell (see Figure 3) is an ex- problem, as explained in the previous sub- perimental device in which a viscous fluid section, the solution u of (1.2) is the integral is sandwiched in a narrow gap between two of the temperature of a solid undergoing a parallel plates. In certain regions, the gap phase transition, such as ice melting to wa- is filled with fluid while in others the gap is ter. filled with air. When liquid is injected inside • Fluid filtration. The so-called Dam the device though some sinks (e.g. though a problem aims to describe the filtration of small hole on the top plate) the region filled water inside a porous dam. One considers with liquid grows. In this context, an in- a porous dam separating two reservoirs of tegral of the pressure solves, for each fixed water at two different heights; see Figure 2. time t, the obstacle problem (1.3). Simi- Then, the interior of the dam has a wet part, larly as in the Dam problem, the free bound- where water flows, and a dry part. In this ary corresponds to the interface between the context, an integral of the pressure solves the fluid and the air regions. 4 XAVIER ROS-OTON AND JOAQUIM SERRA In some some models the particles attract Air each other when they are far, but experience a repulsive force when they are close [14]. In Injection other related models in statistical mechan- Fluid point ics, the particles (e.g. electrons) repel with a Coulomb force and one wants to understand their behaviour in presence of some external field that confines them [48]. Figure 3. A Helle-Shaw cell. In the previous models, a natural and in- teresting question is to determine the \equi- librium configurations”. For instance, in Coulomb systems the charges accumulate in • Optimal stopping, finance. In proba- some region with a well definite boundary; bility and finance, both the Stefan problem see Figure 4. Interestingly, this problems (1.2) and the obstacle problem (1.3) appear are equivalent to obstacle problems of the when considering optimal stopping problems type (1.3) | for instance the electric poten- for stochastic processes. tial u = u(x) generated by the charges solves A typical example is the Black-Scholes a problem like (1.3) and free boundary cor- model for pricing of American options. An responds to the boundary of the region in American option is a contract that entitles which the particles concentrate. its owner to buy some financial asset (typi- cally a share of some company) at some spec- ified price (the\strike price") at any time before some specified date (the \maturity date"). This option has some value, since External field in case that the always fluctuating market price of the asset goes higher than the strike price then the option can be \exercised" to Charges buy the asset at the lower price. The Black-Sholes model aims to calculate the rational price u = u(x; t) of an option at any time t prior to the maturity date and de- pending on the price x 2 R+ of the financial Figure 4. The equilibrium asset. Since the option can be exercised at configuration for a Coulomb any time before maturity, determining the system. \exercise region", i.e. the pairs (x; t) for which it is better to exercise the option, is a part of the problem. Interestingly, this prob- lem leads to an Stefan problem of the type (1.2), and the free boundary corresponds to • Elasticity.
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