The Solvability of Differential Equations Nils Dencker Lund - - PowerPoint PPT Presentation

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators

The Solvability of Differential Equations

Nils Dencker

Lund University

7 March 2019

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators

1 Introduction

Definitions Solvability

2 Background

Principal type Lewy’s counterexample The bracket condition

3 The Nirenberg-Treves conjecture

Conditions Resolution

4 Non-principal type operators

Limit characteristics Subprincipal type

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Definitions Solvability

Introduction

Let x ∈ Rn, the results are local and generalize to manifolds. The complex derivative D = 1

i ∂ gives

P(x, D)u(x) = (2π)−n

• eix·ξP(x, ξ)ˆ

u(ξ) dξ u ∈ C ∞

0 (Rn)

Here P(x, ξ) is the symbol of the operator. P(x, D) is PDO if ξ → P(x, ξ) is polynomial in ξ, ΨDO if P(x, ξ) = pm(x, ξ) + pm−1(x, ξ) + . . . with pj homogeneous of degree j in ξ, m is the order, pm = p is the principal symbol and pm−1 = ps the subprincipal symbol of P. One can use this to localize in phase space (x, ξ) ∈ T ∗Rn with ΨDO, so called microlocal analysis.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Definitions Solvability

Local solvability

P is locally solvable near x0 if Pu = f has a local weak (distribution) solution u near x0 for all f ∈ C ∞ in a set of finite codimension, thus P has locally a finite cokernel. This is equivalent to a priori estimates for the L2 adjoint P∗ u(0) ≤ C(P∗u(N) + u(−n) + Au(0)) ∀ u ∈ C ∞

0 (Rn)

for some N where A = 0 near x0, and we use the L2 Sobolev norms. One can prove local non-solvability by constructing local approximate solutions to P∗u = 0, which are called pseudomodes. Observe that in the analytic category all non-degenerate PDO are locally solvable by the Cauchy-Kovalevsky theorem.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Principal type Lewy’s counterexample The bracket condition

Background

Constant coefficient PDO are locally solvable. (Ehrenpreis and Malgrange 1955) Variable coefficients: Principal symbol p(x, ξ) is invariant as a function of (x, ξ) ∈ T ∗Rn under changes of variables. Elliptic case: p(x, ξ) = 0 for ξ = 0 are solvable. (Lax-Milgram 1954) The Hamilton field of p: Hp = ∂ξp∂x − ∂xp∂ξ, then Hpp ≡ 0. Definition The operator P is of principal type if Hp does not have the radial direction ξ, ∂ξ when p = 0, thus dp = 0 when p = 0. Then P has simple characteristics, which is a generic condition for non-elliptic operators. Operators of principal type with real principal symbol are solvable. (H¨

• rmander’s thesis 1955)

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Principal type Lewy’s counterexample The bracket condition

The Lewy counterexample

Hans Lewy’s counterexample (1957) The operator P = Dx1 + iDx2 + i(x1 + ix2)Dx3 = Dx1 − x2Dx3 + i(Dx2 + x1Dx3) is not locally solvable anywhere in R3. P is the tangential Cauchy-Riemann operator on the boundary of the strictly pseudoconvex set { (z1, z2) : |z1|2 + 2 Im z2 < 0 } ⊂ C2 By the Cauchy-Kovalevsky theorem P is solvable for analytic functions, so P is solvable up to an arbitrarily small error near any given point by approximation.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Principal type Lewy’s counterexample The bracket condition

The bracket condition

Theorem Local solvability implies that { Re p, Im p } = HRe p Im p = 0 on p−1(0), where p is the principal symbol of P. (H¨

• rmander 1960)

Thus almost all non-elliptic linear PDO are not locally solvable. The principal symbol of [P, P∗] is 1

i { p, p } = −2 { Re p, Im p }, so a

vanishing bracket means that the operator is approximately normal. For ΨDO the condition is { Re p, Im p } ≤ 0 on p−1(0), but for PDO the bracket is odd in ξ so it has to vanish on p−1(0) (switch ξ ↔ −ξ). Non-zero bracket means that the principal symbol satisfies a stable topological winding number condition, for example p = τ ± it.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Principal type Lewy’s counterexample The bracket condition

Tangential Cauchy-Riemann operators

If Ω ⊂ Cn be defined by ̺(z) < 0, ̺ ∈ C ∞ real with ∂̺ = 0. Then Pjk = ∂zj̺∂zk − ∂zk̺∂zj j = k are the tangential Cauchy-Riemann operators on ∂Ω since Pjk̺ = 0. Ω is strictly pseudo-convex domain if ∂z∂z̺ v, v > 0 when ∂z̺ v = 0 and v = 0 For n = 2 and a complex vector field P = P12 in R3 this holds if 1) P, P and [P, P] are linearly independent ⇒ [P, P] = 0 2) Pz = 0 has two linearly independent solutions z1, z2 (coordinates). But almost all vector fields on R3 have a trivial kernel: the constants (Treves et al 1983). Thus there exist arbitrarily small perturbations of P that are not tangential Cauchy-Riemann operators.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Principal type Lewy’s counterexample The bracket condition

Pseudospectrum

The Schr¨

• dinger operator with potential V ∈ C ∞(Rn)

P(h) = −h2∆ + V (x) = p(x, hD) 0 < h ≤ 1 If z ∈ {|ξ|2 + V (x) : { Re p, Im p } = 2ξ, ∇ Im V (x) = 0} then (P(h) − z)−1 ≥ CNh−N ∀ N By Sard’s theorem, this holds for almost all values when Im V ≡ 0. Holds for any semiclassical ΨDO. (D., Sj¨

• strand and Zworski 2004).

Application: complex absorbing potential in quantum chemistry. This has been generalized to systems and eigenvalues of the principal symbol (D. 2008). The ”almost eigenvalues” are called pseudospectrum and the ”almost eigenfunctions” are called pseudomodes.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Principal type Lewy’s counterexample The bracket condition

Instability of the Cauchy problem

The Cauchy-Kovalevsky theorem gives local solvability for the Cauchy problem for quasilinear analytic vector fields with analytic data on a non-characteristic analytic initial surface. But almost all data has an arbitrarily small C ∞ perturbation so that the Cauchy problem has no C 2 solution. (Lerner, Morimoto and Xu 2010) Example: the non-homogeneous Burger’s equation ∂tu + u∂x1u = f (t, x, u) (t, x) ∈ R × Rn with analytic f has no C 2 solution for almost all non-analytic Cauchy data u(0, x). For example when the bracket [∂t + Re u ∂x, Im u ∂x] = 0 thus 2 Im u(0, x) Re ∂x1u(0, x) = Im f (0, x, u(0, x)).

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Conditions Resolution

The Nirenberg-Treves conditions

The bracket condition was not sufficient for solvability, so Nirenberg and Treves introduced other conditions on the principal symbol: Condition (Ψ): Im p does not change sign from − to + along the

• riented bicharacteristics of Re p. (Not a generic condition!)

Condition (P): no change of sign, i.e., both (Ψ) and (Ψ) hold. These conditions are invariant and gives that HRe p Im p ≤ 0 at p−1(0). Conditions (P) and (Ψ) are equivalent for PDO, but not for ΨDO. (Switch ξ ↔ −ξ, if the degree of p is even/odd, then Hp odd/even.) Example For Dt + if (t, x, Dx) with f real and first order, HRe p Im p = ∂tf (t, x, ξ) and condition (Ψ) means that t → f (t, x, ξ) cannot change sign from − to + for increasing t and fixed (x, ξ).

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Conditions Resolution

The Nirenberg-Treves conjecture (1969) A principal type ΨDO is locally solvable if and only if the principal symbol satisfies condition (Ψ). Nirenberg and Treves proved this for PDO with analytic coefficients. Condition (Ψ) is necessary for solvability for ΨDO. (H¨

• rmander 1980)

Condition (Ψ) is sufficient for solvability for ΨDO in two variables. (Lerner 1988) Theorem If P is a principal type ΨDO with principal symbol satisfying condition (Ψ) then P is locally solvable. (D. 2006) Has been generalized to systems of principal type with constant characteristics (D. 2011).

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Non-principal type operators

If the principal symbol pm vanishes of at least second order, the subprincipal symbol ps ∼ = pm−1 is an important invariant. Example Let P = D1D2 + B(x, D) where B is first order. The principal symbol ξ1ξ2 vanishes of second

• rder at the double characteristic set Σ2 = { ξ1 = ξ2 = 0 }.

The subprincipal symbol is the principal symbol b of B. If xj → Im b changes sign for j = 1, 2, then P is not solvable and if ± Im b > 0 then P is solvable (Mendoza–Uhlmann 1983-84). Corresponds to condition (Ψ) on the subprincipal symbol at the limits of the bicharacteristics of the principal symbol on Σ2. Conjecture: P is solvable if Im b does not change sign in (x1, x2).

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Limit characteristics

P with principal symbol p = τ 2 − t2|ξ|2 − |η|2 is effectively hyperbolic and locally solvable for any lower order terms. Thus condition (Ψ) on the subprincipal symbol cannot be necessary in general. Definition Manifold M is involutive if { p1, p2 } = 0 on M when pj = 0 on M, and Lagrangean if M is involutive of minimal dimension. When ξ = 0 in the effectively hyperbolic case, we find that Σ2 = { τ = t = |η| = 0 } is not involutive since { τ, t } = 1. Definition Γ ⊂ Σ2 is a limit bicharacteristic if there exist bicharacteristics Γj that converge to Γ as smooth curves parametrized on a fixed interval. Then the normalized Hamilton field |∇p|−1Hp ∈ C ∞ uniformly on Γj.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Assume the bicharacteristics Γj ∈ C ∞ uniformly, thus the normalized Hamilton field H

p = |∇p|−1Hp ∈ C ∞ uniformly on Γj, ∀j.

Definition A section of Lagrangean spaces L over Γ is grazing to Γ if T Γ ⊂ L ⊂ Tp−1(0) and the linearization of Hp is tangent to L. Curvature condition: there exist sections of grazing Lagrangean spaces Lj to Γj and C > 0 such that

• dH

p(w)

• Lj(w)
• ≤ C

w ∈ Γj ∀ j Gives uniform bounds on the curvature of p−1(0) on Lj and gives the evolution of Lj on Γj. It is necessary in order to get non-solvability. Bicharacteristics satisfying these conditions we call uniform and then limit bicharacteristics exist by compactness arguments.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Example Let p = τ − (A(t)x, x + 2B(t)x, ξ + C(t)ξ, ξ) /2 where A(t), B(t) and C(t) ∈ C ∞ are real n × n, and A(t) and C(t) are symmetric. Then p−1(0) = { τ = A(t)x, x/2 + B(t)x, ξ + C(t)ξ, ξ/2 } and the linearization of the Hamilton field Hp is ∂t + A(t)y + B∗(t)η, ∂η − B(t)y + C(t)η, ∂y Take L(t) = { (s, y, 0, E(t)y) } with symmetric E(t) ∈ C ∞ then L(t) is a Lagrangean space and is grazing if E ′(t) = A(t) + B(t)E(t) + E(t)B∗(t) + E(t)C(t)E(t) which is uniformly bounded by the curvature condition.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Assume condition (Ψ) on the subprincipal symbol ps does not hold

• n Γj when j → ∞ and that

min

∂Γj

• Im ps/|Hp| ds ≫ | log κj|

j → ∞ Lim(Ψ) for some starting point wj ∈ Γj, where minΓj |Hp| = κj → 0. Example If Im ps changes sign from − to + on the limit bicharacteristic, then the integral is ≥ C/κj ≫ | log κj|, j → ∞. Theorem If P has real principal symbol and { Γj }∞

j=1 is a family of uniform

bicharacteristics so that Lim(Ψ) is satisfied on Γj, then P is not locally solvable near any of the limit bicharacteristics. (D. 2016) This has been extended to complex principal symbols such that H

p

converges to a real vector field at Σ2. (D. 2016)

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Operators of subprincipal type

Assume p is complex and vanishes of order 2 ≤ k ≤ ∞ on involutive and non-radial Σ2, then the Hamilton fields give a symplectic foliation. Example Σ2 = { η = 0 } with leaves L =

• (x0, y, ξ0, 0) : y ∈ Rk

. Definition P is of subprincipal type if Hps

• Σ2 ⊆ TΣ2 transversal to L when ps = 0.

Examples ∂yps = 0 and ∂x,ξps = 0 when ps = 0 on Σ2 = { η = 0 }. ∆y + Dx is of subprincipal type, but ∆y + Dy is not. Definition The reduced subprincipal symbol ps,k = J k(p) + ps (ps if k = ∞) on N∗Σ2, where J k(p) is the k:th jet (or Taylor term) of p at Σ2. Remark If k < ∞ this is given by the blowup of the inhomogeneous refined principal symbol psub = p + ps at Σ2 = { η = 0 }.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Definition P satisfies condition Subk(Ψ) if ps,k satisfies condition (Ψ) on N∗Σ2/TL, i.e., on TΣ2/TL for fixed fiber η (η = 0 gives (Ψ) on ps). Example TΣ2/TL ∼ = { (w0; x, 0, ξ, 0) : w0 ∈ Σ2 } for Σ2 = { η = 0 }. Assume the blow-up of refined principal symbol qs,k(x, y; ξ, η) ∼ = s1−mpsub(x, y; s ξ, s

k−1 k η)

modulo O(s−1) is constant up to non-vanishing factors on the leaves of Σ2 (in y). Also need conditions on the vanishing of Hess p and the normal derivative ∂ηps,k in terms of ps,k when ps,k vanishes of infinite order. Theorem If P is subprincipal type on the non-radial involutive manifold Σ2 and P does not satisfy Subk(Ψ) then P is not locally solvable. (D. 2018)

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Counterexample to the Mendoza–Uhlmann conjecture

Let P = Dx1Dx2 + Dt + if (t, x, Dx) where f (t, x, ξ) real and of first order so that ∂xjf ≡ 0 for j = 1, 2. Then Lim(Ψ) is not satisfied since f is constant in (x1, x2), thus constant on the leaves of Σ2 and on the limit bicharacteristics. P is of subprincipal type, so P is not solvable if t → f (t, x, ξ) has a sign change from − to + of finite order on the double characteristics Σ2 = { ξ1 = ξ2 = 0 }. This gives a counterexample to the conjecture, the solvability of P also depends on the real part of the subprincipal symbol.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Examples

Example P = Dt + ia(t)∆x is not solvable if t → a(t) is real and changes sign from − to + as t increases. (Kannai (1971): a(t) = t.) P is of subprincipal type, k = 2 and the symbol τ + ia(t)|ξ|2 is constant

• n the leaves of Σ2 = { ξ = 0 } and condition Sub2(Ψ) does not hold.

Observe that |∂ξp| ∼ = | Hess p| ∼ = |a(t)| ∼ = |p| when τ = 0 and ξ = 0. Example P = Dt + i(Dx1Dx2 + tD2

x2) is solvable (Colombini, Pernazza

and Treves 2003). P is of subprincipal type, k = 2 and the symbol τ + i(ξ1ξ2 + tξ2

2) is

constant on the leaves of Σ2 = { ξ = 0 } and Sub2(Ψ) does not hold. Now |∂ξp| ∼ = | Hess p| ∼ = 1 ≫ |p| ∼ = |t| when τ = ξ1 = 0 and ξ2 = 0. Thus conditions on the vanishing of the normal derivative ∂ξp and Hess p are essential for non-solvability.

Nils Dencker SOLVABILITY

Introduction Background The Nirenberg-Treves conjecture Non-principal type operators Limit characteristics Subprincipal type

Open problems

Case when limit characteristics do not converge in C ∞, for example weakly hyperbolic operators. Complex limits of the normalized Hamilton fields. Solvability of weakly hyperbolic operators. Condition (Ψ) on the refined principal symbol p + ps in general. Systems of non-principal type. Non-linear differential operators, for example quasilinear vector fields.

Nils Dencker SOLVABILITY