A class of domains with fractal boundaries: Functions spaces and numerical methods

A class of domains with fractal boundaries: Functions spaces and numerical methods Yves Achdou joint work with T. Deheuvels and N. Tchou Laboratoire J-L Lions, Université Paris Diderot École Centrale - 14 juin 2012

Contents Geometrical construction 1 Geometrical construction A class of self-similar sets A class of ramified domains 2 Function spaces on Γ, available results on trace and extension Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case 3 A self-similar construction of a trace operator W 1,p (Ω) L p µ(γ) Haar wavelets on Γ New trace results Extensions 4

A class of self-similar sets A class of ramified domains A class of self-similar sets Γ Define two similitudes f 1 and f 2 with ratio a < 1 and opposite angles ±θ (0 θ < π 2 ).

A class of self-similar sets A class of ramified domains A class of self-similar sets Γ Define two similitudes f 1 and f 2 with ratio a < 1 and opposite angles ±θ (0 θ < π 2 ). The set Γ Γ is said to be the self-similar set associated to the similitudes f 1 and f 2, i.e. the unique compact set in R 2 such that Γ = f 1 (Γ) f 2 (Γ).

A class of self-similar sets A class of ramified domains The self-similar measure on Γ Theorem (J. E. Hutchinson, 1981) There exists a unique Borel invariant probability measure µ on the self-similar set Γ, in the sense that for every Borel set B in Γ, µ(b) = µ(f1 1 1 (B)) + µ(f2 (B)).

A class of self-similar sets A class of ramified domains The critical value a There exists a critical value a depending on θ such that: if a < a, then Γ is totally disconnected, a < a

A class of self-similar sets A class of ramified domains The critical value a There exists a critical value a depending on θ such that: if a < a, then Γ is totally disconnected, if a = a, then Γ is connected and has multiple points. a < a a = a

Γ 0 Geometrical construction A class of self-similar sets A class of ramified domains A class of ramified domains (1/2)

A class of self-similar sets A class of ramified domains A class of ramified domains (1/2) f 1 (Γ 0 ) Γ 0

A class of self-similar sets A class of ramified domains A class of ramified domains (1/2) f 1 (Γ 0 ) f 2 (Γ 0 ) Γ 0

A class of self-similar sets A class of ramified domains A class of ramified domains (1/2) f 1 (Γ 0 ) f 2 (Γ 0 ) Y 0 Γ 0

A class of self-similar sets A class of ramified domains A class of ramified domains (1/2) f 1 (Γ 0 ) f 2 (Γ 0 ) Y 0 Y 0 Γ 0

A class of self-similar sets A class of ramified domains A class of ramified domains (1/2) f 1 (Γ 0 ) f 2 (Γ 0 ) f 1 (Y 0 ) f 2 (Y 0 ) Y 0 Y 0 Γ 0

A class of self-similar sets A class of ramified domains A class of ramified domains (2/2) Γ If σ = (σ(1),..., σ(n)) A n, A n := {1, 2} n, write Define f σ = f σ(1)... f σ(n). Ω = Interior ( n N σ A n f σ (Y 0 ) ). Ω

Function spaces on closed sets Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Definition (d-set) If F is a closed set in R n and 0 < d n, a Borel measure m on F is said to be a d-measure if there exist constants c 1, c 2 > 0 s.t. x F, r (0, 1), c 1 r d m(b(x, r)) c 2 r d, where B(x, r) is the euclidean metric ball. A closed set having a d-measure is referred to as a d-set.

A trace theorem on d-sets Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Theorem (A. Jonsson, H. Wallin, 1984) If F R n is a d-set, 0 < d < n, 1 < p <, and 1 n d p > 0, then W 1,p (R n n d 1 ) F = W p,p (F ). Definition of the trace: if u L 1 loc (Rn ) then u is strictly defined at x R n if 1 ū(x) := lim u(y) dy r 0 B(x, r) B(x,r) exists. We define the trace of u on F to be ū F, defined only on those points where u is strictly defined.

A trace theorem on d-sets Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Theorem (A. Jonsson, H. Wallin, 1984) If F R n is a d-set, 0 < d < n, 1 < p <, and 1 n d p > 0, then W 1,p (R n n d 1 ) F = W p,p (F ). Examples d = n 1, p = 2: W 1,2 (R n ) F = W 1/2,2 (F ) = H 1/2 (F ). n = 2, F = Cantor set in a line segment, d = log 2/ log 3, W 1,2 (R 2 ) F = W d/2,2 (F ).

Consequence: trace results on Γ Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Proposition The set Γ endowed with the self-similar measure µ is a d-set, where d = log 2 log a. Corollary If p > 1, then W 1,p (R 2 ) Γ = W 2 d 1 p,p (Γ). Question: can we characterize the trace on Γ of W 1,p (Ω)?

Extension domains Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Definition (W k,p -extension domain) A domain D R n is called a W k,p -extension domain (k N, 1 p ) if there exists a continuous linear extension operator Λ : W k,p (D) W k,p (R n ). Definition (Sobolev extension domain) If D is a W k,p -extension domain for every k N and 1 p, then D is said to be a Sobolev extension domain.

(ε, δ)-domains Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Definition ((ε, δ)-domain) A domain D in R n is said to be an (ε, δ)-domain if there exist ε, δ > 0 such that for every x, y D satisfying x y < δ, there exists a rectifiable arc γ D joining x and y such that : l(γ) 1 ε x y, d(z, D c ) ε x z y z, for all z γ. x y

An Example Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case The Koch flake is an (ε, δ)-domain.

Extension theorems Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case Theorem (P. W. Jones, 1974) Every (ε, δ)-domain in R n is a Sobolev extension domain. The result is almost sharp in R 2 : Theorem (P. W. Jones, 1974) In R 2, if a domain D is finitely connected, then D is a Sobolev extension domain if and only if D is an (ε, δ)-domain.

The subcritical case Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case In the subcritical case where a < a, the set Γ is totally disconnected, and the ramified domain Ω is an (ε, δ)-domain. Hence, Ω is a Sobolev extension domain, and the trace result holds for all p (1, ): W 1,p (Ω) Γ = W n d 1 p,p (Γ).

The critical case Function spaces on d-sets Available trace and extension theorems Available trace and extension theorems The subcritical case The critical case In the critical case where a = a, the set Γ is connected. In this case, Ω is not an (ε, δ)-domain. x y γ Ω In this case, Ω is not a W 1,p -extension domain for p > 2. The trace space W 1,p (Ω) Γ cannot be easily characterized.

A new trace operator Haar wavelets on Γ New trace results Extensions Questions raised by the previous analysis In the critical case, we shall see that the traces of functions of W 1,p (Ω) on Γ cannot always be characterized as Sobolev spaces. The critical case: questions characterize the trace space of W 1,p (Ω) on Γ: new function spaces?

A new trace operator Haar wavelets on Γ New trace results Extensions Construction of a trace operator Ω Γ: a sequence of operators l n : W 1,p (Ω) L p µ(γ) u Γ 0 l 0 (u) Γ 0

A new trace operator Haar wavelets on Γ New trace results Extensions Construction of a trace operator Ω Γ: a sequence of operators l n : W 1,p (Ω) L p µ(γ) u Γ 0 u Γ (1) u Γ (2) l 0 (u) Γ (1) Γ (2) l 1 (u) Γ 0

A new trace operator Haar wavelets on Γ New trace results Extensions Construction of a trace operator Ω Γ: a sequence of operators l n : W 1,p (Ω) L p µ(γ) u Γ 0 u Γ (1) u Γ (2) l 0 (u) u Γ (11) u Γ (12) u Γ (21) u Γ (22) Γ (1) Γ (2) l 1 (u) Γ 0 Γ (12) Γ (21) Γ (11) Γ (22) l 2 (u)

A new trace operator Haar wavelets on Γ New trace results Extensions The trace operator on Γ Consider the sequence of operators l n : W 1,p (Ω) L p µ(γ) defined by l n (u) = u Γ σ f σ(γ), σ A n where u Γ σ = 1 Γ σ u and Γ σ = f σ (Γ). Γ σ

A new trace operator Haar wavelets on Γ New trace results Extensions Haar wavelets on Γ We define the Haar wavelets on Γ by { g 0 = f1 (Γ) f2 (Γ) g σ fσ(γ) = 2 n 2 g 0 f σ 1 and g σ Γ\fσ(Γ) = 0 for σ A n Mother wavelet g 0 g σ for σ = (1) g σ for σ = (12) Every function f L p µ(γ), 1 p < can be expanded in the Haar wavelet basis {g σ, σ A}: f = f Γ + β σ g σ. n 0 σ A n

A new trace operator Haar wavelets on Γ New trace results Extensions We distinguish two different cases. Call Ξ the self-intersection set of Γ. θ { π 2k, k N }, Ξ is countable,

A new trace operator Haar wavelets on Γ New trace results Extensions We distinguish two different cases. Call Ξ the self-intersection set of Γ. θ { π 2k, k N }, Ξ is countable, θ = π 2k, k N, dim H Ξ = d 2.

A new trace operator Haar wavelets on Γ New trace results Extensions Regularity of the Haar wavelets Proposition (dim H (Ξ) = 0) In the case when θ π 2k, k N, g 0 W s,p (Γ) if s < d p, g 0 W s,p (Γ) if s > d p. Proposition (dim H (Ξ) = d/2) In the case when θ = π 2k, k N, g 0 W s,p (Γ) if s < d 2p, g 0 W s,p (Γ) if s > d 2p.

A new trace operator Haar wavelets on Γ New trace results Extensions New function spaces Definition (JLip spaces, A. Jonsson 2004) The space JLip s,p (Γ) consists of all the functions f L p µ(γ) s.t. 2 ns p d 2 n( p 1) 2 n 0 σ A n β σ p <, where β σ are the Haar wavelet coefficients of f. Equivalent definition f p L p µ + n=0 2 n sp d σ A n Γ σ f f fσ(γ) p dµ <.

A new trace operator Haar wavelets on Γ New trace results Extensions A characterization of the trace space on Γ Theorem (Y.A., N. Tchou, 2010) If 1 p <, then l (W 1,p (Ω)) = JLip 2 d 1 p,p (Γ).

A new trace operator Haar wavelets on Γ New trace results Extensions Idea of the proof JLip 2 d 1 p,p (Γ) l (W 1,p (Ω))? Explicit construction of a lifting E : JLip using the expansion of a funct. f JLip l (W 1,p (Ω)) JLip 2 d 1 p,p (Γ)? 2 d 1 p 2 d 1 p,p (Γ) W 1,p (Ω), by,p (Γ) on Haar wavelets. Prove the strenghtened trace inequality: ρ (2 (p 1)(1 2 d ), 1), C s.t. v W 1,p (Ω), l (v) l (v) Γ p L p µ ( C v p L p (Y 0 ) + ρ n 2 ) n(p 1)(1 2 d ) v p L p (f σ(y 0 )). n=1 σ A n Use self-similarity to conclude.

A new trace operator Haar wavelets on Γ New trace results Extensions For the strenghtened trace inequality, map the domain Ω to a fractured domain Ω with vertical boundaries (Weierstrass curve): 6 5 4 3 2 1 0 1.0 0.8 0.6 0.4 0.2 Y. Achdou 0.0 0.2 0.4 0.6 0.8 Sobolev extension property 1.0

A new trace operator Haar wavelets on Γ New trace results Extensions Relationship with classical Sobolev spaces Theorem (Y.A., T. Deheuvels, N. Tchou, 2010) 1 If 0 < t < min( d dim H(Ξ) p, 1), then JLip t,p (Γ) = W t,p (Γ), 2 For all t (0, 1) and s > d dim H(Ξ) p, JLip t,p (Γ) W s,p (Γ). (recall that dim H (Ξ) = 0 or dim H (Ξ) = d/2) Corollary (Y.A., T. Deheuvels, N. Tchou, 2010) Define p 2 dim H (Ξ). 1 If 1 p < p, then l (W 1,p (Ω)) = W 2 d 1 p,p (Γ). 2 If p p, then l (W 1,p (Ω)) W s,p (Γ) for all s < d dimh(ξ) p The regularity of the wavelets shows that the result is sharp.

A new trace operator Haar wavelets on Γ New trace results Extensions Principle of the proof Estimate the W s,p (Γ)-norm of a function u JLip t,p by using Λ X0 X1 a partition of Γ and all its images by the similarities f σ the Haar wavelet decomposition of u discrete Hardy inequalities: γ R, p 1, C s.t., for any sequence of positive real numbers (c k ) k N, 2 γn p c k C 2 γn p c n if γ < 0, n N k n n N 2 γn p c k C 2 γn p c n if γ > 0. k n n N n N

A new trace operator Haar wavelets on Γ New trace results Extensions W 1,p -Extension property of Ω Theorem ( T. Deheuvels, 2011) It is possible to construct a linear extension operator P, which is continuous from W 1,p (Ω) to W 1,p (R 2 ) for all p < p. The following theorem a posteriori justifies the use of several notions of trace on Γ. Theorem (Y.A., T. Deheuvels, N. Tchou, 2011) For all p 1, every function u W 1,p (Ω) is strictly defined µ-almost everywhere on Γ, and ū Γ = l (u), µ-almost everywhere on Γ The proof uses the extension operator P constructed for p < p.

A new trace operator Haar wavelets on Γ New trace results Extensions Corollary (Y.A., T. Deheuvels, N. Tchou, 2011) With p defined by p = 2 dim H (Ξ), for all p < p, Ω is a W 1,p -extension domain, for all p > p, Ω is not a W 1,p -extension domain.

Boundary problems with Neumann conditions on Γ For g L 2 µ (Γ) and u H 1 2 (Γ 0 ),! w H 1 (Ω) s.t. w Γ 0 = u, and w v = gl (v)dµ, v V(Ω). ( ) Ω Γ where V(Ω) = {v H 1 (Ω) s.t. v Γ0 = 0}. Z 3 Γ Goal Let Z n be the ramified domain truncated after the n-th generation. The goal is to compute w Z n with no error due to domain truncation, by using transparent boundary conditions. Γ 0

A Dirichlet to Neumann operator Theorem (Harmonic lifting: energy decay) Call H(u) the solution to (*) with g = 0. There exists a constant ρ < 1 s.t. for all u H 1 2 (Γ 0 ), H(u) 2 ρ p Ω\Z p Ω H(u) 2, p 0. Z Γ Γ 0 3 Definition (Dirichlet to Neumann operator) Let T be the Dirichlet to Neumann operator: T u = H(u) n Γ0 T u, v = H(u) H(v). Ω

Transparent boundary conditions Main idea Self-similarity and scale invariance of the PDE imply that: if w = H(u) then w f1 (Ω) f 1 = H ( w f1 (Γ 0 ) f 1 ), w f2 (Ω) f 2 = H ( w f2 (Γ 0 ) f 2 ).

Transparent boundary conditions Main idea Self-similarity and scale invariance of the PDE imply that: if w = H(u) then w f1 (Ω) f 1 = H ( w f1 (Γ 0 ) f 1 ), w f2 (Ω) f 2 = H ( w f2 (Γ 0 ) f 2 ). If T is available, then w Y 0 can be computed by sol- -ving a boundary value problem in Y 0.

Solution in Z n with arbitrary small truncation error Algorithm If T is available, then for all n 0, one can find H(u) Z n by solving sequentially 1 + 2 + + 2 n 1 boundary value problems in Y 0 with nonlocal transparent conditions on f 1 (Γ 0 ) f 2 (Γ 0 ) involving T, and Dirichlet data on Γ 0 computed from the previous step.

Fixed point iterations for computing T Let O be the cone of self-adjoint, positive semi-definite, continuous linear operators from H 1 2 (Γ 0 ) to its dual, vanishing on the constants. Clearly T O. Take the map M : S O M(S) O, where M(S)u = w n Γ 0, with w = 0 in Y 0, w Γ 0 = u, w n f i (Γ 0 ) f i = 1 a S(w f i (Γ 0 ) f i ) i = 1, 2 w n = 0 on the rest of the boundary of Y 0 Theorem The operator T is the unique fixed point of M. Moreover, for all S O, M p (S) T Cρ p.

The case when g 0 The following strategy may be used for computing w Z n with an arbitrary accuracy: If g is Haar wavelet then from self-similarity, w Z n can be computed exactly by induction w.r.t. the wavelets, by solving a finite number of boundary value problems in Y 0 with nonhomogeneous transparent boundary conditions on f 1 (Γ 0 ) f 2 (Γ 0 ) involving the transparent condition operator T.

The error may be made as small as desired We obtain a method for approximating w Z n and error estimates (depending on the truncation in the wavelet expansion and of the approximation of T ) Theorem There exists ρ < 1 and a constant C independent of g s.t. n, P N with 1 n < P error (P ) H 1 (Z n 1 ) C 2 P ρ P n g L 2 µ, where the wavelet expansion of g is truncated at level P.

Geometrical construction The Neumann datum is a Haar wavelet Y. Achdou Sobolev extension property

The Neumann datum is cos(3πs/2)g 0 (s) 1 0.1 0.01 "compare_level0" "compare_level1" "compare_level2" "compare_level3" "compare_level4" 0.001 1e-04 1e-05 1e-06 1e-07 2 2.5 3 3.5 4 Expansion with wavelets of levels 5 (resp. 4, 2) Error in L 2 norm (logscale) in Y 0, Z 1,..., Z 5 w.r.t. the truncation of the wavelet expansion

Helmholtz equation u + k 2 u = 0 The operator is no longer scale invariant. Therefore, a whole sequence of operators T a 2p k is needed. But there is a backward induction relation: and T a 2p k T a 2(p 1) k T a 4 k T a 2 k T k. lim T a p 2p k = T 0. This observation allows one for designing an algorithm to solve the Helmholtz equation with an arbitrary small truncation error, except maybe for special frequencies (the eigenfrequencies of the pb).

Geometrical construction Example: eigenmodes of the Laplace operator with Neumann cond. on Γ0 0.2 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0.15 0.1 0.05 0-0.05-0.1-0.15 1 0 2 4 6 8 10 12 14 16-10 -8-6 -4-2 0 42 86 18 10-10 16-8 14-6 12-4 -2 10 0 8 2 6 4 Z 3. 6 4 8 2 10 0 Left: the third eigenmode, restricted to Right: the sixth 3 eigenmode, restricted to Z. Y. Achdou Sobolev extension property

w Geometrical construction Linear evolution equations (e.g. the wave equation) Strategy Compute a large number of (normalized) eigenmodes Compute the expansion of solution of the wave equation on the eigenmodes. 0.2 0.15 0.1 0.05 0-0.05-0.1-0.15 0 10 20 30 40 50 60 t "wave_equation" The solution of the wave equation vs. t at a given point of Ω, with a compactly supported u 0 and u 1 = 0