Degenerate Monge-Ampère equations and the smoothness of the eigenfunction

Transcription

1 Degenerate Monge-Ampère equations and the smoothness of the eigenfunction Ovidiu Savin Columbia University November 2nd, 2015 Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

2 1) Introduction 2) Degenerate Monge-Ampère equations at the boundary 3) Global Schauder estimates 4) Smoothness of the first eigenfunction { (det D 2 u ) 1 n = λ u in Ω, u = 0 on Ω. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

3 The Dirichlet problem for the Monge-Ampère equation: Find u : B 1 R convex, such that { det D 2 u = f in B 1, u = ϕ on B 1. Theorem (Caffarelli-Nirenberg-Spruck, Krylov) There exists a unique solution to the Dirichlet problem and u C 2,α (B 1 ) C(f, ϕ), provided that f > 0 in B 1, and f C 3, ϕ C 4. Important contributions due to: Calabi, Pogorelov, Cheng-Yau, Ivockina, Lions, Evans, Trudinger, Urbas, X.J. Wang. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

4 Idea of proof: Use the concavity of the equation (det M) 1 n concave in M 0 = u ee is a subsolution. Step 1 (C 1,1 boundary estimates): D ττ u, D τν C on B 1 = D 2 u C on B 1 Step 2 (C 1,1 interior estimates): D 2 u C on B 1 = D 2 u C in B 1 Step 3 (C 1,1 bounds by below): D 2 u C in B 1 = c I D 2 u C I in B 1 Step 4 (continuity of D 2 u): Evans-Krylov theorem + boundary Harnack Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

5 Interior estimates: { det D 2 u = 1 in Ω, u = 0 on Ω. Theorem (Pogorelov 1971) D 2 u C(δ, Ω) in Ω δ := {x Ω dist(x, Ω) > δ}. Affine invariance: solves the same equation if A linear. ũ(x) = (det A) 2 n u(ax) John s lemma: For any convex bounded set Ω R n, there exists A linear such that B 1 A 1 Ω n B 1. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

6 The sections of u: S h (x 0 ) = {u < u(x 0 ) + u(x 0 ) (x x 0 )} Pogorelov estimate says that D 2 u(x 0 ) is controlled by the eccentricity of S h (x 0 ) as long as S h (x 0 ) Ω. Degenerate Monege-Ampère equations: with f = 0 at some points in Ω. det D 2 u = f 0 If f 0 then u is the convex envelope of its boundary data. Regularity results were obtained by Trudinger-Urbas, Caffarelli-Nirenberg-Spruck. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

7 Assume f degenerates at a single point: det D 2 u = x 2 in B 1 R 2. Weyl problem (1916): Does there exist a C 2 isometric embedding from (S 2, g) (R 3, ds 2 ) if the Gauss curvature of g is positive? Nirenberg (1953) - yes. P. Guan- Y.Y.Li (1994) - C 1,1 isometric embedding if the Gauss curvature nonnegative. Two different types of behavior at 0: 1) radial u(x) = c x 3, (or x 2+ 2 n in R n ) 2) C solutions u(x 1, x 2 ) = ax bx l.o.t. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

8 P. Guan ( 97) : u C (B 1 ) if u 11 c > 0 near 0. P. Guan - Sawyer ( 09): u C (B 1 ) if u c > 0 near the origin. Daskalopoulos - S. ( 09): u C 2,1 and has only two possible behaviors near the origin: either u C or u is radial (unstable). Open problem: C 2,α estimates for det D 2 u = x 2 in dimension n 3. Rios-Sawyer-Wheeden ( 07): u C if u C 2 and uniformly convex on an n 1 dimensional subspace. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

9 Degenerate Monge-Amperé equation at the boundary det D 2 u = f [dist(x, Ω)] α, α > 0. Example: the eigenvalue problem { (det D 2 u ) 1 n = λ u in Ω, u = 0 on Ω. Theorem (Lions 86) There exist a unique λ > 0 and u (up to multiplication by a constant) and u C 1,1 (Ω) C (Ω). In fact λ = inf λ A L A where λ A is the 1st eigenvalue for linear operators L A u = tr(a(x)d 2 u), det A(x) c n. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

10 Tso gave a variational characterization of λ c n λ n = inf v Ω v det D2 v Ω v n+1. If Ω 1 Ω 2 then λ(ω 1 ) λ(ω 2 ). Question: Assume Ω is smooth and uniformly convex. Is u C (Ω)? Main difficulty - lower bounds for the curvatures of the level sets of u. Hong J.- Huang G. - Wang W. ( 11) : u C (Ω) in 2 D. Le N. - S. ( 15): u C (Ω). Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

11 Basic problem (P ) Particular solution: { det D 2 u = x α n in B + 1, u = 1 2 x 2 on {x n = 0}. U 0 (x, x n ) = 1 2 x (1 + α)(2 + α) x2+α n. Affine invariance: Scaling: x Ax := (x τ x n, x n ). ũ(x) = 1 h u(h 1 2 x, h 1 2+α xn ). Does there exist a R, A as above such that near 0 u(x) = ax n + (1 + o(1))u 0 (Ax)? Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

12 Theorem (S. 13) The solution to problem (P) satisfies u C 2 (B + 1/2 ) and near 0 u(x) = ax n + (1 + o(1))u 0 (Ax) with a, A bounded by C( u L, n, α). Theorem (global solutions) Assume u solves problem (P) in (R n ) +. If u grows less than x 3+α at then u = U 0 modulo a linear map and an affine deformation. This result is new even for α = 0 and it is sharp. { det D (P ) 2 u = 1 in (R n ) +, u = 1 2 x 2 on {x n = 0}. A non-quadratic solution in 2D is x 2 1 u(x) = x 2 2 x x3 2. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

13 Lower bound for tangential derivatives Assume that u(0) = 0 and we look at the shape of the sections S h = {u < h} as h 0. Let d x (h) denote the diameter of S h in the x -direction. Then is scale invariant. b(h) := d x (h) h b(h) the average of the smallest tangential eigenvalue of D2 u in S h. Key lemma: If b(h) > C sufficiently large then b(τh) b(h) 2 with τ, C depending only on n, α. b(τh) b(h) = τ 1 2 d x (τh) d x (h). Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

14 If b(h) and we scale S h to unit size then we obtain a solution to a Dirichlet problem with discontinuous data at the origin: det D 2 ū = x α n in S 1, ū = 1 on S 1 \ {0}, ū(0) = 0, ū(0) = 0. Graph of ū has a conical singularity at 0. We want to show that Model for a conical singularity is d x (τ) 1 2 τ 1 2 dx (1) τ 1 2 for small τ. x + c n,α x n+1+α. We need to study the smoothness of the tangent cone, which is related to the obstacle problem for the MA equation. The case α > 0 and α = 0 are similar. This problem is simpler than problem (P): the smallest eigenvalue should occur along the radial direction. S. ( 04) - Obstacle problem for MA equation (the case α = 0.) Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

15 Let ū be the Legendre transform of ū. Then det D 2 ū = χ {ū >0}. The free boundary {ū = 0} is a C 1,1 uniformly convex set. This gives the bound on b(h) lower bounds of tangential derivatives near 0. Continuity of D 2 u Assume u is a global solution to problem (P) with cu 0 u CU 0. Then u = U 0. Idea: Look for the maximum of the scale invariant quantity w = u n x 1+α n which solves an elliptic equation and show that w is constant. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

16 Schauder estimates Suppose that near the origin u satisfies det D 2 u = g(x) d α Ω, α > 0, where g is a positive function, and Ω is convex, 0 Ω. Theorem (Le - S. 15) 2 Let β (0, 2+α ), and β α. Assume that Ω C 2,β, u Ω C 2,β, g C γ where γ = β 2 + α. 2 If u separates quadratically from its tangent plane at 0 then u C 2,β (Ω B δ (0)) for some small δ > 0. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

17 The proof is by perturbation methods and relies on the Schauder estimates for the linearized equation for U 0 : L w := x α n x w + w nn = 0 in B + 1, w = 0 on {x n = 0}. Smoothness of the eigenfunction Step 1 Assume that 0 Ω and e n is the inner normal of Ω at 0. We make the rotation of coordinates y n = x n+1, y n+1 = x n, y k = x k (1 k n 1). In the new coordinates, the graph of u near the origin can be represented as y n+1 = ũ(y) in the upper half-space R n + = {y R n : y n > 0}. K = det D 2 u(ν n+1 ) n+2 = det D 2 ũ ν n+2 n { det D2ũ = y n n ũ n+2 n in B + r 0, ũ = ϕ on {y n = 0} B r0. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

18 Step 2: We perform the partial Legendre transformation x i = ũ i (y) (i n 1), x n = y n, u (x) = y y ũ ũ(y). The function u is obtained by taking the Legendre transform of u on each slice y n = constant. Note that (u ) = u. Then u is convex in x and concave in x n, and satisfies { x α n ( u n) n+2 det D 2 x u + u nn = 0 in B + δ u = ϕ on {y n = 0} B δ, where α = n. Moreover u C 2,β (B + δ ), u n > c and ϕ C. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

19 Fix k < n. Then v = u k solves the linearized equation x α n a ij v ij + v nn = x α nf(x) in B + δ where i,j n 1 a ij = ( u n) n+2 U ij x, f(x) = (n + 2)( u n) n+1 u nk det D2 x u, and U x denotes the cofactor matrix of D 2 x u. Using Schauder estimates for the linear equation above we can bound in B + δ/2 the derivatives D m x Dl x n u with l {0, 1, 2}, m 0. In order to obtain the continuity of these derivatives for all values of l we differentiate the equation for u and use that α = n is a nonnegative integer. Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20

20 Thank you! Ovidiu Savin (Columbia University) Degenerate Monge-Ampère equations November 2nd, / 20