Frequency functions, monotonicity formulas, and the thin obstacle problem

Frequency functions, monotonicity formulas, and the thin obstacle problem IMA - University of Minnesota March 4, 2013

Thank you for the invitation!

In this talk we will present an overview of the parabolic Signorini (or thin obstacle) problem. In particular, we will focus on a new approach to this classical problem, based on monotonicity properties for a family of so-called frequency functions, first introduced by Almgren.

The Classical Obstacle Problem The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems.

The Classical Obstacle Problem The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.

Figure : Example of a one-dimensional obstacle problem

Mathematically, the obstacle problem consists of studying the properties of minimizers of the Dirichlet integral J(u) = u 2 dx in a domain D R n, among all configurations u(x) (representing the vertical displacement of the membrane) with prescribed boundary values u D = f (x), and constrained to remain above the obstacle ϕ(x). D The solution breaks down into a region where the solution is equal to the obstacle function, known as the coincidence set, and a region where the solution is above the obstacle. The interface between the two regions is the so-called free boundary.

The obstacle problem can be reformulated as a standard problem in variational inequalities on Hilbert spaces. In fact, solving the obstacle problem is equivalent to seeking a function such that u K = {v W 1,2 (D) v D = f (x), v ϕ} D u (v u) dx 0 for all v K. Variational arguments show that the solution to the obstacle problem is harmonic away from the contact set, and that it is superharmonic on the contact set. Hence, the solution is a superharmonic function.

The study of the classical obstacle problem, initiated in the 60 s with the pioneering works of G. Stampacchia, H. Lewy, J. L. Lions, has led to beautiful and deep developments in calculus of variations and geometric partial differential equations. The crowning achievement has been the development, due to L. Caffarelli, of the theory of free boundaries.

Optimal regularity of the solution The solution to the obstacle problem is C 1,1 (i.e. it has bounded second derivatives) when the obstacle itself has such regularity. In general, the second derivatives of the solutions are discontinuous across the free boundary. The free boundary The free boundary is characterized as a Hölder continuous surface except at certain singular points, which are either isolated or contained on a C 1 manifold.

Background: The time-independent thin obstacle problem Let Ω be a domain in R n and M a smooth (n 1)-dimensional manifold in R n that divides Ω into two parts: Ω + and Ω. For given functions ϕ : M R and g : Ω R satisfying g > ϕ on M Ω, consider the problem of minimizing the Dirichlet integral D Ω (u) = u 2 dx on the closed convex set K = {u W 1,2 (Ω) : u = g on Ω, u ϕ on M Ω}. This problem is known as the lower dimensional, or thin obstacle problem. The thin obstacle is the function ϕ. Ω

Figure : Graphs of Re(x 1 + i x 2 ) 3/2 and Re(x 1 + i x 2 ) 6

When u is constrained to stay above an obstacle ϕ which is assigned in the whole domain Ω, i.e. when M = Ω, then we obtain the classical obstacle problem. Whereas the latter is by now well-understood, the thin obstacle problem still presents considerable challenges and only recently there has been some significant progress on it.

When M and ϕ are smooth, it has been proved by Caffarelli (1979) that the minimizer u in the thin obstacle problem is of class C 1,α loc (Ω ± M). The function u satisfies u = 0 in Ω \ M = Ω + Ω, but in general u does not need to be harmonic across M. Instead, on M, one has the following complementary conditions u ϕ 0, ν +u + ν u 0, (u ϕ)( ν +u + ν u) = 0, where ν ± are the outer unit normals to Ω ± on M.

One of the main goals in the study of this problem is to understand the properties of the coincidence set Λ(u) := {x M : u = ϕ} and its boundary (in the relative topology of M) i.e., the free boundary. Γ(u) := M Λ(u), Simplifying assumptions: 1. Vanishing thin obstacle ϕ. 2. The manifold M is a flat portion of the boundary of the relevant domain: M=R n 1 {0}. In this case the thin obstacle problem is known as the Signorini problem.

This problem, which arises in linear elasticity, was first proposed by Signorini in 1959. Signorini called it the problem with ambiguous boundary conditions. The existence and uniqueness of solutions was proved by Fichera in 1963. It was Fichera who renamed it as Signorini problem. In the original formulation, it consists of finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to its mass forces.

Figure : What will be the equilibrium configuration of an elastic body resting on a rigid frictionless plane?

Other applications include optimal control of temperature across a surface, in the modeling of semipermeable membranes where some saline concentration can flow through the membrane only in one direction, and financial math (when the random variation of underlying asset changes in a discontinuous fashion, as a Levi process).

Normalization Since we are interested in properties of minimizers near free boundary points, after translation, rotation and scaling arguments we may consider a function u defined in the upper half-ball B 1 + := B 1 R n + satisfying u = 0 in B + 1 (0.1) u 0, xn u 0, u xn u = 0 on B 1 (0.2) 0 Γ(u) = Λ(u) := {(x, 0) B 1 u(x, 0) = 0}, (0.3) where Λ(u) is the coincidence set and the boundary is in the relative topology of B 1. Here B 1 := B 1 (R n 1 {0}). We denote by S the class of solutions of the normalized Signorini problem (0.1) (0.3).

What is the meaning of the boundary condition u 0, xn u 0, u xn u = 0 on B 1? In the original formulation of the problem, it was requested that the solution satisfied one of the two sets of conditions or u = 0, xn u 0 (0.4) u > 0, xn u = 0 (0.5) The set of conditions (0.4) applies to points of the boundary of the body which do not leave the contact set in the equilibrium configuration. The set of conditions (0.5) applies instead to points of the boundary of the body which leave that set in the equilibrium configuration.

Recent Developments Athanasopoulos-Caffarelli (2006): Optimal C 1,1/2 interior regularity with flat M and ϕ = 0. Athanasopoulos-Caffarelli-Salsa (2008): Fine regularity properties of the free boundary. Namely, the set of regular free boundary points is locally a C 1 -manifold of dimension n 2. Crucial tool: Monotonicity of Almgren s Frequency Function N(r, u) = r B r u 2 B r u 2 The name comes from fact that if u is a harmonic function in B 1, homogeneous of degree κ, then N(r, u) = κ. When ϕ = 0 and M is flat, then a free boundary point is called regular if at such point the frequency attains its least possible value N(0+, u) = 3/2.

In the particular case Ω = R n 1 (0, ) and M = R n 1 {0}, the Signorini problem can be interpreted as an obstacle problem for the fractional Laplacian on R n 1 : u ϕ 0, ( x ) s u 0, (u ϕ)( x ) s u = 0, with s = 1/2. Silvestre (2007): Almost optimal regularity of solutions, namely u C 1,α (R n 1 ) for any α < s, 0 < s < 1. Caffarelli-Salsa-Silvestre (2008): Optimal regularity C 1,s (R n 1 ), free boundary regularity. Interesting aspect: In the above results, the thin obstacle ϕ is allowed to be nonzero, thanks to a suitable generalization of Almgren s monotonicity of the frequency. Garofalo-Petrosyan (2009): Structure of the singular set of solutions to the thin obstacle problem by construction of two one-parameter families of monotonicity formulas (of Weiss and Monneau type).

Similarly to the classical obstacle problem, in the lower dimensional obstacle problem the analysis of the free boundary revolves around the behavior of the so-called blowups. In the Signorini problem Athanasopoulos, Caffarelli and Salsa considered the rescalings u(rx) u r (x) := ( 1 ) 1/2, r n 1 B r u 2 and studied the limits as r 0+, known as the blowups.

Monotonicity of the Frequency Theorem 1 Let u S, then the function N(r, u) := r B r u 2 B r u 2 is monotone increasing in r for 0 < r < 1. Moreover, N(r, u) κ for 0 < r < 1 iff u is homogeneous of order κ in B 1, i.e. x u κu = 0 in B 1. When u is a harmonic function this is a classical result of Almgren (1979).

The Blowups It follows easily from the monotonicity formula that, for r 1 B 1 u r 2 = N(1, u r ) = N(r, u) N(1, u), where in the last inequality we have used the monotonicity of the frequency N(r, u) claimed in the previous theorem. The above inequality, and the C 1,α loc estimates of Caffarelli, imply that there exists a nonzero function u 0 W 1,2 (B 1 ), called a blowup of u at the origin, such that for a subsequence r = r j 0+ u rj u 0 in W 1,2 (B 1 ) u rj u 0 in L 2 ( B 1 ) u rj u 0 in C 1 loc (B 1 B ± 1 )

The monotonicity of the frequency easily implies the following Proposition 2 (Homogeneity of blowups) Let u S and denote by u 0 any blowup of u as described above. Then, u 0 S and it is a homogeneous function of degree κ = N(0+, u). The following result was proved in part by Luis Silvestre in his Ph.D. Dissertation, and in part by Caffarelli, Salsa and Silvestre Lemma 1 (Minimal homogeneity) Let u S. Then N(0+, u) 2 1 2 = 3 2. Moreover, one has either N(0+, u) = 2 1 2 or N(0+, u) 2.

Statement of the Parabolic Signorini Problem Given a domain Ω in Rx n with a sufficiently regular boundary Ω, a relatively open subset M Ω, S = Ω \ M, consider the problem v t v = 0 in Ω T := Ω (0, T ] (0.6) v ϕ, ν v 0, (v ϕ) ν v = 0 on M T := M (0, T ], (0.7) v = g on S T := S (0, T ] (0.8) v(, 0) = ϕ 0 on Ω 0 := Ω {0} (0.9) where ν is the outer normal derivative on Ω and ϕ : M T R, ϕ 0 : Ω 0 R, g : S T R are prescribed functions satisfying the compatibility conditions ϕ 0 ϕ on M {0}, g ϕ on S (0, T ], g = ϕ on S {0}. The condition (0.7) is known as the Signorini boundary condition and the problem (0.6) (0.9) as the Signorini problem for the heat equation.

As a weak (or generalized) solution of the Signorini problem we understand the solution v K of the variational inequality Ω T v (w v) + t v(w v) 0 for every w K, v(, 0) = ϕ 0, where K = {w W 1,0 2 (Ω T ) w ϕ on M T, w = g on S T }.

Known Results Regularity of the solution v H α,α/2, 0 < α < 1, on compact subsets of Ω T M T Athanasopoulos (1982) Uraltseva (1985) Arkhipova-Uraltseva (1996)

Known Results Regularity of the solution v H α,α/2, 0 < α < 1, on compact subsets of Ω T M T Athanasopoulos (1982) Uraltseva (1985) Arkhipova-Uraltseva (1996) Poon s Monotonicity Formula Poon (1996): If u is a solution of the heat equation in a unit strip, the parabolic frequency N u (r) = r 2 R u 2 (x, r 2 )ρ(x, r 2 )dx n R u(x, r 2 ) 2 ρ(x, r 2 )dx n is monotone in r (0, 1). Here ρ denotes the backward heat kernel on S = R n (, 0], i.e. ρ(x, t) = ( 4πt) n 2 e x 2 4t.

Notations R n = {x = (x 1, x 2,..., x n ) : x i R} (Euclidean space) R n + = R n {x n > 0} (positive halfspace) R n = R n {x n < 0} (negative halfspace) R n 1 R n 1 {0} R n x = (x 1, x 2,..., x n 1 ), x = (x 1, x 2,..., x n 2 ) x = (x, x n ), x = (x, x n 1 )

For x 0 R n 1, t 0 R we let B r (x 0 ) = {x R n x < r} (Euclidean ball) B r ± (x 0 ) = B r (x 0 ) R n ± (Euclidean halfball) B r (x 0 ) = B r (x) R n 1 ( thin ball) Q r (x 0, t 0 ) = B r (x 0 ) (t 0 r 2, t 0 ] (parabolic cylinder) Q r (x 0, t 0 ) = B r (x 0 ) (t 0 r 2, t 0 ] ( thin parabolic cylinder) Q r ± (x 0, t 0 ) = B r ± (x 0 ) (t 0 r 2, t 0 ] (parabolic halfcylinders) Q r (x 0, t 0 ) = B r (x 0 ) [t 0, t 0 + r 2 ) (upper parabolic cylinder) Q r (x 0, t 0 ) = B r (x 0 ) (t 0 r 2, t 0 + r 2 ) (full parabolic cylinder) S r = R n ( r 2, 0] (parabolic strip) S r ± = R n ± ( r 2, 0] (parabolic halfstrip) S r = R n 1 ( r 2, 0] ( thin parabolic strip)

Solutions in Half-Cylinders The following is joint work with N. Garofalo, A. Petrosyan, and T. To. Definition 3 The class S ϕ (Q 1 + 2,1 ) consists of functions v W2 (Q 1 + ), with v H α,α/2 (Q 1 + Q 1 ) for some 0 < α < 1, satisfying v t v = 0 in Q + 1 v ϕ 0, xn v 0, (v ϕ) xn v = 0 on Q 1, and (0, 0) Γ(v) = {(x, t) Q 1 v(x, 0, t) > ϕ(x, t)}.

Reduction to Vanishing Obstacle The difference v(x, t) ϕ(x, t) satisfies the Signorini conditions on Q 1 with zero obstacle, but at an expense of solving a nonhomogeneous heat equation instead of the homogeneous one. This difference may be extended to the strip S 1 + = Rn + ( 1, 0] by multiplying it by a suitable cutoff function ψ. The resulting function will satisfy with u t u = f (x, t) in S + 1, f (x, t) = ψ(x)[ ϕ t ϕ] + [v(x, t) ϕ(x, t)] ψ + 2 v ψ. For smooth enough ϕ, the function f is bounded in S + 1!

Solutions in Half-Strips A function u is in the class S f (S 1 + ), for f L (S 1 + 2,1 ), if u W2 (S 1 + ), u H α,α/2 (S 1 + S 1 ), u has a compact support and solves u t u = f in S 1 +, u 0, xn u 0, u xn u = 0 on S 1, and (0, 0) Γ(u) = {(x, t) S 1 : u(x, 0, t) > 0}.

Generalized Monotonicity Formula Problem Poon s monotonicity formula requires the function u to be caloric in an entire strip, and it is not immediately applicable to caloric functions in the unit cylinder Q 1. (Partial) Solution Extend the function u, caloric in Q 1, to the entire strip S 1 by multiplying it by a spatial cutoff function ψ, supported in B 1 : v(x, t) = u(x, t)ψ(x).

Introduce quantities h u (t)= R n + i u (t)= t u(x, t) 2 ρ(x, t)dx R n + u(x, t) 2 ρ(x, t)dx, for any function u on S 1 + for which they make sense. Poon s parabolic frequency function is given by N u (r) = i u( r 2 ) h u ( r 2 ). There are many substantial technical difficulties involved in working with this function directly. To overcome such difficulties, consider averaged versions of h u and i u : H u (r)= 1 r 2 I u (r)= 1 r 2 0 0 h u (t)dt = 1 r 2 r 2 i u (t)dt = 1 r 2 r 2 S + r S + r u(x, t) 2 ρ(x, t)dxdt t u(x, t) 2 ρ(x, t)dxdt

The New Generalized Monotonicity Formula Theorem 4 Let δ > 0. Then there exists C > 0, depending only on δ and n, such that the function Φ u (r) = 1 2 recr δ d dr log max{h u(r), r 4 2δ } + 3 2 (ecr δ 1) is nondecreasing for r (0, 1). Remark 5 H u (r) > r 4 2δ Φ u (r) 1 2 rh u(r)/h u (r) = 2N u, when f = 0. The truncation of H u (r) with r 4 2δ controls the error terms caused by the right-hand-side f.

With the previous theorem in hands, we can study the existence and the homogeneity properties of the blow-up s. Since the function Φ u (r) = 1 2 recr δ d dr log max{h u(r), r 4 2δ } + 3 2 (ecr δ 1) is nondecreasing for r (0, 1), the limit exists. κ := Φ u (0+) = lim r 0+ Φ u(r)

We have the following basic result concerning the values of κ. Lemma 2 Let u S f (S 1 + ) and satisfy the conditions of the monotonicity theorem, and let κ be as above. Then κ 2 δ. Moreover, if κ < 2 δ, then there exists r u > 0 such that H u (r) r 4 2δ for 0 < r r u. In particular, κ = 1 2 lim rh u(r) r 0+ H u (r) = 2 lim I u (r) r 0+ H u (r).

Similarly to what was done in the elliptic case by Athanasopoulos, Caffarelli and Salsa, we introduce the parabolic rescalings. Definition: For u S f (S 1 + ) and r > 0 define the rescalings u r (x, t) := u(rx, r 2 t) H u (r) 1/2, (x, t) S + 1/r = Rn + ( 1/r 2, 0]. Our main result shows that, unless we are in the borderline case κ = 2 δ, we can study the blow-up s of u at the origin.

Existence and Homogeneity of Blowups It is easy to see that u r solves the nonhomogeneous Signorini problem u r t u r = f r (x, t) in S + 1/r u r 0, xn u r 0, u r xn u r = 0 on S 1/r with f r (x, t) = r 2 f (rx, r 2 t) H u (r) 1/2.

Theorem 6 There is a subsequence r j 0+ and a function u 0 in S + = R n + (, 0] such that ( u rj u 0 2 + t (u rj u 0 ) 2 )ρ 0. S + R We call any such u 0 a blowup of u at the origin. u 0 is a nonzero global solution of the Signorini problem: u 0 t u 0 = 0 in S + u 0 0, xn u 0 0, u 0 xn u 0 = 0 on S in the sense that it solves the Signorini problem in every Q + R. u 0 is parabolically homogeneous of degree κ: u 0 (λx, λ 2 t) = λ κ u 0 (x, t), (x, t) S, + λ > 0

Homogeneous global solutions of homogeneity 1 < κ < 2: Let u be a nonzero κ-parabolically homogeneous solution of the Signorini problem in S + = R n + (, 0] with 1 < κ < 2. Then κ = 3/2 and u(x, t) = C Re(x e + ix n ) 3/2 + in S + for some tangential direction e B 1.

Optimal Regularity Theorem 7 Let v S f ϕ(q + 1 ) and f L (Q + 1 ). Then v H 1 2, 1 4 (Q + 1/2 Q 1/2 ) with v H 1 2, 1 4 (Q + 1/2 Q 1/2 ) C n ( v W 1,0 (Q + 1 ) + f L (Q + 1 ) + ϕ H 2,1 (Q 1 ) ). Regularity is optimal! (compare with elliptic case)

The Free Boundary: The Regular Set Analysis of homogeneous global solutions κ = Φ u (0+) 3 2. Definition 8 Let v S ϕ (Q 1 + ) with ϕ Hl,l/2 (Q 1 ), l 2. We say that a free boundary point (x 0, t 0 ) is regular it has a minimal homogeneity κ = 3/2. R(v) denotes the set of regular free boundary points. Theorem 9 Let v S ϕ (Q 1 + ) with ϕ Hl,l/2 (Q 1 ), l 3 and such that (0, 0) R(v). Then there exist δ = δ v > 0 such that Γ(v) Q δ = R(v) Q δ = {x n 1 = g(x, t) (x, t) Q δ }. after a possible rotation in R n 1, where g H 1,1/2 (Q δ ) (i.e., g is a parabolically Lipschitz function).

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