Table Of Content

SOME RESULTS ON EVOLUTION Contents 1. Introduction. 1 2. Solution of the parabolic problem. 4 2 1 2.1. Geometric properties of weak solutions. 4 0 2 n 2.2. Comparison principle. Walsh Lemma in unbounded domains. 12 a J 2.3. Existence of solutions and evolution. 13 1 1 2.4. Some geometric properties 19 ] V 3. Evolution of graphs 22 C . h 4. Limit for solutions 26 t a m References 29 [ 1 v 7 8 1. Introduction. 2 2 . Let z ,...,z be complex coordinates in Cn, n ≥ 2. Given a smooth function ̺ we set 1 1 n 0 2 1 |∂̺|2 = |̺α|2, : v 1≤α≤n X i X ̺ = ̺ ,̺ = ̺ ,1 ≤ α ≤ n. α zα α¯ z¯α r a Let M be a smooth hypersurface in Cn of local equation ̺ = 0. For every point p ∈ M let HT (M) ⊂ T (M) be the complex tangent hyperplane to M at p and ν = (̺ ,...,̺ ) p p ¯1 n¯ the normal vector to HT (M). p Let {E ,...,E } be an ortormal frame with origin at p and such that {E ,...,E } 1 n 1 n−1 is a framein HT (M) and ζ ,...,ζ the complex coordinates determined by {E ,...,E }. p 1 n 1 n The restriction to {ζ = 0} of the Levi form of ̺ is the intrinsec Levi form of M at p. Its n 1 2 SOME RESULTS ON EVOLUTION trace is n ̺ ̺ H(̺) = |∂̺|−1 δαβ¯− α β¯ ̺ |∂̺|2 αβ¯ α,β=1(cid:18) (cid:19) X at p. For n = 2, H is esentially the Levi operator. Let K be a compact subset of Cn, g : Cn → R a continuous function which is constant for |z| ≫ 0 and such that K = {g = 0}. Assume that v ∈ C0(Cn×R+) is a weak solution of the parabolic problem n v = δαβ − vαvβ¯ v in Ω×(0,+∞) t |∂v|2 αβ¯  α,β=1 (cid:16) (cid:17) (⋆) v = g P on Cn ×{0}    v = const for t ≫ 0.    Then the family {K} of the subsets K = {z ∈ Cn : v(z,t) = 0} (which actually t t≥0 t depends only on K) is called the evolution of K by H. Evolution of a compact subset K of C2 was introduced in [7], [8] where, after proving that the parabolic problem has a unique (weak) solution u, it was shown that if Ω is a bounded pseudoconvex domain of C2 with boundary of class C3, the evolution {Ω } of t t≥0 Ω is contained in Ω. Conversely, pseudoconcave points ”move out by evolution”, i.e. if Ω is not pseudoconvex then Ω 6⊆ Ω for some t > 0 (cfr. [9, Theorem 0.1]). The natural t problem of what kind of hull one can recover by evolution was investigated in [11]. In this paper we consider the evolution of a compact subset of Cn by H with a fixed part K∗ ⊆ K. Precisely, we study the following parabolic problem: n v = δαβ − vαvβ¯ v in Ω×(0,+∞) t |∂v|2 αβ¯  α,β=1 (cid:16) (cid:17) (P) P v = g on Ω×{0}    v(z,t) = g(z) for z ∈ bΩ×(0,+∞)   where Ω is a bounded strictly pseudoconvex domain in Cn such that K \K∗ ⊆ Ω, K∗ ⊆ bΩ SOME RESULTS ON EVOLUTION 3 and g : Ω → R is a continuous function such that g−1(0) = K. In Section 2 (see Theorems 2.6, 2.7) we will prove that a) the problem (P) has a unique (weak) solution v which is bounded and uniformly continuous in Ω×[0,+∞); b) ifg isaC2 function, thecorrespondingsolutionv of(P)isLipschitzonΩ×[0,+∞); c) the set X = (z,t) ∈ Ω×[0,+∞) : v(z,t) = 0 (cid:8) (cid:9) satisfies X ∩ Ω×{0} = K ×{0}, X ∩(bΩ×[0,+∞)) = K∗ ×[0,+∞) (cid:0) (cid:1) and it is actually independent of the choice of g and Ω. The family {E (K,K∗)} of compact subsets defined by t t≥0 E (K,K∗) = {z ∈ Cn : (z,t) ∈ X i.e. v(z,t) = 0}. t is then said to be the evolution of K with fixed part K∗ (by H). Of particular interest inthis setting isthe case when K is the graphM of a continuous function on the closure D of a bounded domain D in Cn−1 × R and K∗ = bM is the boundary bM of M. Generalizing the results of [10] for n = 2 we then prove the following theorem (see Theorem 4.4: if D is bounded, strictly pseudoconvex domain i.e. D ×iR is a strictly pseudoconvex domain in Cn then d) E (M,bM) is a graph for all t ≥ 0 (Theorem 3.1); t e) if bM is smooth and satisfies the compatibility conditions discovered in [2], then asymptotically E (M,bM) approaches, in the C0-topology the Levi flat hypersur- t face with boundary bM whose existence was proved in [2]. 4 SOME RESULTS ON EVOLUTION Let us mention that in the smooth case a parabolic initial value problem related to the flow of a real hypersurface of Cn by the trace of the Levi form is studied in a nice paper by Huisken and Klingenberg (cfr. [4]). 2. Solution of the parabolic problem. 2.1. Geometric properties of weak solutions. Let U ⊂ Cn × (0,+∞) be an open subset. An upper semicontinuous function v : U → [−∞,+∞) is said to be a (weak) subsolution of n v = H(v) = (δαβ¯−|∂v|−2v v )v . t α β¯ αβ¯ α,β=1 X if, for every (z0,t0) and a (viscosity) test function φ at (z0,t0) (i.e. φ is smooth near (z0,t0) and v −φ has a local maximum at (z0,t0)), one has φ (z0,t0) ≤ H(φ)(z0,t0) t if ∂φ(z0,t0) 6= 0 and n φ (z0,t0) ≤ δαβ¯−η η φ (z0,t0) t α β¯ αβ¯ αX,β=1(cid:16) (cid:17) for some η ∈ Cn with |η| ≤ 1, if ∂φ(z0,t0) = 0. A lower semicontinuous function v : U → (−∞,+∞] is said to be a (weak) supersolution if, for every (z0,t0) and a test function φ at (z0,t0) (i.e. φ is smooth near (z0,t0) and v −φ has a local minimum at (z0,t0)), one has φ (z0,t0) ≥ H(φ)(z0,t0) t if ∂φ(z0,t0) 6= 0 and n φ (z0,t0) ≥ δαβ¯−η η φ (z0,t0) t α β¯ αβ¯ αX,β=1(cid:16) (cid:17) for some η ∈ Cn with |η| ≤ 1, if ∂φ(z0,t0) = 0. SOME RESULTS ON EVOLUTION 5 Remark 2.1. Let A be an n × n hermitian matrix and η ∈ Cn with |η| ≤ 1. Then TrA > ηtAη provided A > 0. Conversely, if TrA > ηtAη for some η ∈ Cn with |η| ≤ 1 then A cannot be negative definite. In particular, from the above definition it follows that plurisubharmonic functions are (weak) subsolutions to v = H(v). t A (weak) solution is a continuous function which is both a subsolution and a supersolution. One checks that the following properties are true: 1) maximum (minimum) of a finite number of subsolutions (supersolutions) is a subsolution (supersolution); 2) if W′ ⊂ W ⊂ Cn × (0,+∞), W, W′ open and v : W → (−∞,+∞), v′ : W′ → [−∞,+∞) are subsolutions, such that for all ζ ∈ bW′ ∩W limsupv′(z) ≤ v(ζ) z→ζ then the function max (v(z),v′(z)) if z ∈ W′ w(z) = ( v(z) if z ∈ W \W′ is a subsolution in W; 3) translations of subsolutions (supersolutions) are subsolutions (supersolutions); i.e. if ζ ∈ Cn, h ∈ R is positive and vζ,h(z,t) = v(z+ζ,t+h) then vζ,h is a subsolution (supersolution) provided v is; 4) the limit of a decreasing sequence of subsolutions is a subsolution. Lemma 2.1. If ̺ : (a,b) → R is a continuous non decreasing function and v is a subsolution (or a supersolution) with the range of v in (a,b), then ̺◦v is a subsolution (or a supersolution, respectively). In particular, if v is a weak solution, then ̺◦v is a solution. Proof. There is a sequence of C∞ functions ̺ : (a,b) → R such that ̺′ (t) > 0, ̺ (t) ց n n n ̺(t), t ∈ (a,b), therefore it suffices to prove the lemma for ̺ = ̺ , for then ̺ ◦v ց ̺◦v n n 6 SOME RESULTS ON EVOLUTION and ̺◦v will be a subsolution due to 4). Let φ be a test function for ̺◦v. Then χ = ̺−1 is smooth and (strictly) increasing; since ψχ◦φ is a test function for v hence we have ψ (z0,t0) ≤ H(ψ)(z0,t0) t if ∂ψ(z0,t0) 6= 0 and n ψ (z0,t0) ≤ δαβ¯−η η ψ (z0,t0) t α β¯ αβ¯ αX,β=1(cid:16) (cid:17) for some η ∈ Cn with |η| ≤ 1, if ∂ψ(z0,t0) = 0. Consider now the case ∂ψ(z0,t0) 6= 0 and suppose, by a contradiction, that φ (z0,t0) > H(φ)(z0,t0). t Then ψ (z0,t0) = χ′(φ(z0,t0))φ (z0,t0) t t > χ′(φ(z0,t0))H(φ)(z0,t0) = H(ψ)(z0,t0) which is absurd. As for the case ∂ψ(z0,t0) = 0 it is enough to show the following: let W ⊂ Cn be open and ̺ : W → R a weak continuous solution of the inequality H(̺)(z) ≥ −h(z) where h : W → R+ is a continuous positive function. Suppose that χ is a continuous increasing function R → R with χ′ ∈ L∞(R) and 0 ≤ χ′ ≤ 1. Then H(χ◦̺)(z) ≥ −h(z), in the weak sense. We proceed as follows. Since χ can be approximated uniformly on compact subsets of R by smooth functions with the required properties, we may assume SOME RESULTS ON EVOLUTION 7 that χ : R → R, χ ∈ C∞(R), 0 < χ′(s) ≤ 1; hence χ−1 ∈ C∞(R). Let ψ be a smooth test function for H(χ◦̺) ≥ −h, i.e. ψ(z) ≥ (χ◦̺)(z) and ψ(z0) = (χ◦̺)(z0); then ψ∗ = χ−1 ◦̺ is a test function too, i.e. ψ⋆(z) ≥ ̺(z),ψ∗(z0) = ̺(z0). If ∂ψ(z0) 6= 0 we have ∂ψ∗(z0) 6= 0 and, by virtue of the hypothesis, H(ψ∗)(z0) ≥ −h(z0), hence H(ψ)(z) = H(χ◦ψ∗)(z0) = χ′(ψ∗(z0))H(ψ⋆)(z0) ≥ ≥ −χ′(ψ∗(z0))h(z0) > −h(z0). If ∂ψ(z0) = 0, then ∂ψ∗(z0) = 0 and there is a vector η ∈ Cn, |η| ≤ 1, with n δαβ¯−η η φ (z0,t0) ≥ −h(z0). α β¯ αβ¯ αX,β=1(cid:16) (cid:17) Now we observe that, since ψ∗(z0) = 0, 1 ≤ α ≤ n α n n δαβ¯−η η φ (z0,t0) = χ′(ψ∗(z0))φ (z0,t0) δαβ¯−η η φ (z0,t0) ≥ α β¯ αβ¯ t α β¯ αβ¯ αX,β=1(cid:16) (cid:17) αX,β=1(cid:16) (cid:17) −χ′(ψ∗(z0))h(z0) ≥ −h(z0). This ends the proof. (cid:3) In the sequel we will use the following Proposition 2.2. Let {v } be a family of weak subsolution of v = H(v) and assume α α∈A t that v = supv is locally bounded from above. Then the upper semicontinuous regulariza- α α∈A tion of v v∗(z,t) = limsup v(z′,t′). (z′,t′)→(z,t) is a weak subsolution. 8 SOME RESULTS ON EVOLUTION Proof. We first prove the following: let B ⋐ W be a ball of radius r centered at w0 = (z0,t0) and φ be such that (v − φ)(w0) > (v − φ)(w) for w ∈ B \ w0. Then there is a sequence wν → w0 and indices α ∈ A such that for every ν the function v −φ has a ν αν maximum at wν (relative to B). We may assume that (v −φ)(w0) = 0. For every ν ∈ N such that 1/ν ≤ r let −δ = max (v −φ)(w) : 1/ν ≤ r|w−w0| ≤ r . ν (cid:8) (cid:9) Since v − φ has a strict maximum (=0) at w0 (relative to B), −δ < 0 i.e. δ > 0. By ν ν definition of regularization (w,s) ∈ B ×[−∞,+∞) : s ≤ (v∗ −φ)(w) n o is the closure of (w,s) ∈ B ×[−∞,+∞) : s ≤ (v −φ)(w) . α α∈A [ (cid:8) (cid:9) Thus, for every ν there is a point (wν,sν) ∈ B ×R and α ∈ A such that ν 1 sν ≤ (v −φ)(wν) ≤ 0, |wν −w0|+sν ≤ min(δ ,1/ν); αν 2 ν in particular 1 1 |wν −w0| ≤ , − δ (v −φ)(wν) ≤ 0. ν 2 ν αν Let now wν denote any of the maximum points of (v −φ) . Since αν |B 1 (v −φ)(wν) ≥ − δ > −δ αν 2 ν ν ≥ max (v −φ)(w) : ν−1 ≤ |w−w0| ≤ r n o > max (v −φ)(w) : ν−1 ≤ |w−w0| ≤ r αν n o we conclude that |wν −w0| ≤ ν−1 i.e. wν → w0. In order to prove that v∗ is a weak subsolution let φ ∈ C∞(B) and suppose that v∗−φ has a maximum at w0 = (z0,t0) with ∂φ(z0,t0) 6= 0. Let φ (w) = φ(w)+ε|w−w0|2; then ε ∂φ(z0,t0) 6= 0, φ has a strict maximum at w0 so, in view of what already proved, there ε SOME RESULTS ON EVOLUTION 9 are point wν = (zν,tν) → w0 = (z0,t0) and α ∈ A such that (v −φ ) have maximum ν αν ε at wν with ∂φ (zν,tn) 6= 0 and ε ∂φ H(φ )(zν,tν) ≥ ε(zν,tν). ε ∂t Letting ν → +∞, we get ∂φ H(φ )(z0,t0) ≥ ε(z0,t0) ε ∂t and then with ε → 0 ∂φ H(φ )(z0,t0) ≥ ε(z0,t0). ε ∂t The proof if ∂φ(z0,t0) = 0 is similar. (cid:3) Finally, in order to prove the independence of the evolution of the pair (K,K∗) on Ω (see Introduction, c)) we discuss a local maximum property of the level sets of a weak solution v. For an open set V in Cn ×(0,+∞) set P (V) = ψ ∈ C2(V) : ψ ≤ H(ψ) . H t n o Let Z be a locally closed subset of V. We say that Z has local maximum property (relative to P ) if for every open set V ⋐ Cn×(0,+∞) such that V ∩Z is closed and V is compact, H and for every ψ ∈ P (V′) where V′ is a neighbourhood of V it holds: H max ψ = max ψ. V∩Z bV∩Z Lemma 2.3. Let W ⊆ Cn×(0,+∞) be open, v : W → R a weak solution of the v = H(v) t and Z = {v = 0}. Then a) Z has local maximum property; b) for every c > 0, Zc = (z,t) ∈ Z : t ≤ c has local maximum property. (cid:8) (cid:9) 10 SOME RESULTS ON EVOLUTION Proof. We first prove the following. Let v be a weak supersolution of v = H(v) in W. t Fix a point (z0,t0) ∈ W and a neighbourhood V ⊂ W of (z0,t0). Let φ ∈ C2(V) be such that φ(z0,t0) = v(z0,t0) = c and (1) (z,t) ∈ V : φ(z,t) > c ⊆ (z,t) ∈ V : v(z,t) > c . o n o (cid:8) Then φ (z0,t0) ≥ H(φ)(z0,t0) t if ∂φ(z0,t0) 6= 0 and n φ (z0,t0) ≥ δαβ¯−η η φ (z0,t0) t α β¯ αβ¯ αX,β=1(cid:16) (cid:17) for some η ∈ Cn with |η| ≤ 1, if ∂φ(z0,t0) = 0. Observe that, if there exists a non-decreasing continuous function ̺ : R → R such that ̺(c) = c and φ(z,t) ≤ (̺◦u)(z,t) on a neighbourhood of (z0,t0), then ̺◦u is still a weak supersolution, so the conclusions concerning φ are immediate. In order to construct ̺ let N be a compact neighbourhood of (z0,t0) such that N ⊂ V ⊂ W. Set ̺ (s) = c for s ≤ c. For every s satisfying 1 c ≤ s ≤ s := sup v(z,t) : (z,t) ∈ NBig} ∞ n let R = (z,t) : (z,t) ∈ N : v(z,t) ≤ s . s n o Since v is lower semicontinuous, the Rs’s are compact and Rs ⊂ Rs′ if s ≤ s′. For c ≤ s ≤ s we then define ∞ ̺ (s) = max{φ(z,t) : (z,t) ∈ R }. 1 s Clearly, ̺ is a non decreasing upper semicontinuous function, s 7→ R being an upper 1 s semicontinuous correspondence. Moreover, φ(z,t) ≤ (̺ ◦ u)(z,t). Indeed, assume for a contradiction that φ(z,t) > (̺ ◦ u)(z,t). If φ(z,t) > (̺ ◦ u)(z,t) this is impossible as