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Free Boundary Regularity in Obstacle Problems

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Free Boundary Regularity in Obstacle Problems Free boundary regularity in obstacle problems Alessio Figalli Abstract. These notes record and expand the lectures for the \Journ´ees Equations´ aux D´eriv´eesPartielles 2018" held by the author during the week of June 11-15, 2018. The aim is to give a overview of the classical theory for the obstacle problem, and then present some recent developments on the regularity of the free boundary. Contents 1. Introduction: the obstacle problem 1 2. The functional setting: existence and uniqueness 2 3. Euler-Lagrange equation and consequences 3 4. Optimal regularity 5 5. A new formulation of the obstacle problem 8 6. Non-degeneracy of solutions 9 7. Blow-up analysis and Caffarelli’s dichotomy 10 7.1. Regular free boundary points 11 7.2. Singular free boundary points 11 7.3. Caffarelli’s dichotomy theorem 13 8. Uniqueness of blow-up at singular points 16 9. Stratification and C1 regularity of the singular set 19 10. Recent developments 20 11. Future directions 24 Bibliography 25 1. Introduction: the obstacle problem The classical obstacle problem aims to describe the shape of an elastic membrane lying above an obstacle. n Mathematically, this problem can be described as follows: given a domain Ω ⊂ R , a function ' :Ω ! R (the obstacle), and a function f : @Ω ! R satisfying f ≥ 'j@Ω (the boundary condition), one wants to minimize the Dirichlet integral among all functions that coincide with f on @Ω and lie above the obstacle ' (see Figure 1). Hence, one is lead to minimize Z jrvj2 min : vj@Ω = f; v ≥ ' ; (1.1) v Ω 2 R jrvj2 where the Dirichlet integral Ω 2 represents the elastic energy of the membrane corre- sponding to the graph of v. The main goal consists in understanding the regularity properties of the minimizer, as well as the structure of the contact set between the minimizer and the obstacle. In the next section we shall describe all these questions in detail. A.F. is supported by ERC Grant \Regularity and Stability in Partial Differential Equations (RSPDE)". A.F. is thankful to Yash Jhaveri for useful comments on a preliminary version of this manuscript. 1 A. Figalli Figure 1. An elastic membrane lying above an obstacle. 2. The functional setting: existence and uniqueness The proof of existence and uniqueness of minimizers is very similar to the one for the clas- sical Dirichlet problem without obstacle. Since our focus will be in understanding how the obstacle influence the solution, instead of working under minimal regularity assumptions, we shall assume that all the data are smooth in order to emphasize the main ideas. Hence, 1 1 we assume that Ω is a bounded domain of class C ;' : Ω ! R is of class C , and that 1 f : @Ω ! R is C as well. Note that, because of these hypotheses, one can extend f to a 1 C function F : Ω ! R. Recall that, by assumption, 'j@Ω ≤ f. Under these hypotheses, we can show existence and uniqueness of minimizers in W 1;2(Ω). Proposition 2.1. There exists a unique minimizer u 2 W 1;2(Ω) for the minimization problem (1.1). Proof. The existence follows by observing that (1.1) corresponds to minimizing the Dirich- let functional among all functions v that belong to the convex set 1;2 K' := fv 2 W (Ω) : vj@Ω = f; v ≥ 'g; where the relation vj@Ω = f must be intended in the sense of traces of Sobolev functions. 1;2 Note that K' is closed in the strong W topology, hence it is also closed for the weak W 1;2 topology (as a consequence of Hanh-Banach separation theorem). Thus, to find a minimizer of (1.1) it suffices to argue as in the case of the classical Dirichlet problem. More precisely, consider a minimizing sequence fvkgk≥1, namely a sequence of functions fvkgk≥1 ⊂ K' such that Z jrv j2 Z jrvj2 k ! α := inf : (2.1) Ω 2 v2K' Ω 2 Also, fix a Lipschitz function V 2 K' (for instance, one can define V := maxf'; F g, where F is a C1 extension of f as explained above). R jrV j2 Note that α is finite since α ≤ Ω 2 < 1. Thus, thanks to (2.1), there exists k0 > 0 such that Z 2 jrvkj ≤ α + 1 8 k ≥ k0: (2.2) Ω 2 1;2 Furthermore, by Poincar´einequality applied to the function vk − V 2 W0 (Ω), kvk − V kL2(Ω) ≤ CΩkrvk − rV kL2(Ω) (2.3) 2 FREE BOUNDARY REGULARITY IN OBSTACLE PROBLEMS Hence, combining (2.2) and (2.3), for all k ≥ k0 we get kvkkL2(Ω) + krvkkL2(Ω) ≤ kvk − V kL2(Ω) + kV kL2(Ω) + krvkkL2(Ω) ≤ CΩkrvk − rV kL2(Ω) + kV kL2(Ω) + krvkkL2(Ω) ≤ (CΩ + 1)krvkkL2(Ω) + kV kL2(Ω) + CΩkrV kL2(Ω) p ≤ (CΩ + 1) 2(α + 1) + kV kL2(Ω) + CΩkrV kL2(Ω): 1;2 This proves that the functions vk are uniformly bounded in W (Ω), hence there exists a 1;2 subsequence vkj that converges weakly to a function u 2 W (Ω). Since vkj 2 K' and the set K' is weakly closed (by the discussion above), it follows that u 2 K'. Finally, since the L2 norm is lower semicontinuous under weak convergence, 2 2 Z jruj Z jrvk j ≤ lim inf j = α; Ω 2 j!1 Ω 2 which proves that u is a minimizer. For the uniqueness, it suffices to observe that if u1; u2 2 K' then Z 2 Z Z ru1 + ru2 1 2 2 ≤ jru1j + jru2j ; Ω 2 2 Ω Ω with equality if and only if ru1 ≡ ru2. In particular, if u1 and u2 are two minimizers then u1+u2 equality must hold (otherwise 2 would have strictly less Dirichlet energy), therefore r(u1 − u2) = 0 in Ω; u1 − u2 = 0 on @Ω: By Poincar´einequality this implies that u1 − u2 = 0, as desired. 3. Euler-Lagrange equation and consequences It is a well-known fact that minimizers of the Dirichlet energy are harmonic. However, in our case, the presence of the obstacle ' plays an important role. Proposition 3.1. Let u :Ω ! R be the minimizer of (1.1). Then ∆u ≤ 0 inside Ω: 1 Proof. Let 2 Cc (Ω) be nonnegative, and for > 0 consider the function u := u + ψ. Since ≥ 0 it follows that u ≥ u ≥ '. Also, because is compactly supported in Ω, uj@Ω = uj@Ω = f: This shows that u is admissible in the minimization problem (1.1), thus (by the minimality of u) Z jruj2 Z jru j2 Z jru + r j2 ≤ = Ω 2 Ω 2 Ω 2 Z jruj2 Z Z jr j2 = + ru · r + 2 : Ω 2 Ω Ω 2 R jruj2 Simplifying the term Ω 2 from the first and last expression, and then dividing by , we obtain Z Z jr j2 0 ≤ ru · r + : Ω Ω 2 Thus, letting ! 0+ we get Z Z 0 ≤ ru · r = − ∆u ; Ω Ω where the last equality must be intended in the sense of distribution. Since is an arbitrary nonnegative smooth function, the inequality above implies that −∆u ≥ 0, as desired. As a consequence of the previous result, we can find a \nice" representative for u. 3 A. Figalli Corollary 3.2. Let u :Ω ! R be the minimizer of (1.1). Then, up to changing u in a set of measure zero, u is lower semicontinuous. Proof. By the mean value formula for superharmonic functions Z ∆u ≤ 0 in Ω ) r 7! − u is decreasing on (0;Rx); (3.1) Br(x) where Rx := dist(x; @Ω). Define the function Z u^(x) := lim − u 8 x 2 Ω r!0 Br(x) (note that the limit exists thanks to (3.1)). Thenu ^(x) = u(x) whenever x is a Lebesgue point for u, thereforeu ^ = u a.e. Also, if xk ! x1 then, using (3.1) again, we get Z Z − u = lim − u ≤ lim inf u^(xk) 8 r 2 (0;Rx1 =2); k!1 k!1 Br(x1) Br(xk) where the first equality follows by the Lebesgue dominated convergence theorem noticing that Z 1 Z − u = uχBr(xk) and uχBr(xk) ! uχBr(x1) a.e. Br(xk) jBrj Ω Letting r ! 0 we obtain u^(x1) ≤ lim inf u^(xk); k!1 as desired. From now on we will implicitly assume that u coincides with its lower semicontinuous representativeu ^. In particular, u is pointwise defined at every point. An important consequence of the previous result is the following: Corollary 3.3. Let u :Ω ! R be the minimizer of (1.1). Then the set fu > 'g \ Ω is open: Proof. This is a direct consequence of the fact that, since u is lower semicontinuous and ' is C1 (hence continuous), then also u − ' is lower semicontinuous inside Ω. Hence, if xk ! x1 2 Ω and xk 2 fu − ' ≤ 0g \ Ω, then (u − ')(x1) ≤ lim inf(u − ')(xk) ≤ 0: k!1 This proves that the set fu ≤ 'g is closed inside Ω, thus the set fu > 'g \ Ω is open. Thanks to the previous result, we can now show that u is harmonic away from the contact set fu = 'g. Corollary 3.4. Let u :Ω ! R be the minimizer of (1.1). Then ∆u = 0 inside fu > 'g \ Ω: 1 Proof. Fix a ball Br(x0) ⊂⊂ fu > 'g\Ω, and consider 2 Cc (Br(x0)). For > 0, define the function u := u + ψ. Since Br(x0) ⊂⊂ fu > 'g \ Ω and u − ' is lower semicontinuous, it must attain a positive minimum inside Br(x0), namely min (u − ') =: c0 > 0 Br(x0) 1 Hence, if > 0 is small enough (the smallness depending on k kL (Br(x0))), it follows that 1 1 u(x) ≥ u(x) − k kL (Br(x0)) ≥ '(x) + c0 − k kL (Br(x0)) ≥ '(x) 8 x 2 Br(x0): 4 FREE BOUNDARY REGULARITY IN OBSTACLE PROBLEMS Since u = u outside Br(x0), this shows that u is admissible in the minimization problem (1.1).
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