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 ∈ R n , the results are local and generalize to manifolds. The complex derivative D = 1 i ∂ gives � P ( x , D ) u ( x ) = (2 π ) − n e ix · ξ P ( x , ξ )ˆ 0 ( R n ) u ∈ C ∞ u ( ξ ) d ξ 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 ) = p m ( x , ξ ) + p m − 1 ( x , ξ ) + . . . with p j homogeneous of degree j in ξ , m is the order , p m = p is the principal symbol and p m − 1 = p s the subprincipal symbol of P . One can use this to localize in phase space ( x , ξ ) ∈ T ∗ R n 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 x 0 if Pu = f has a local weak (distribution) solution u near x 0 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 L 2 adjoint P ∗ 0 ( R n ) � u � (0) ≤ C ( � P ∗ u � ( N ) + � u � ( − n ) + � Au � (0) ) ∀ u ∈ C ∞ for some N where A = 0 near x 0 , and we use the L 2 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 ∗ R n under changes of variables. Elliptic case: p ( x , ξ ) � = 0 for ξ � = 0 are solvable. (Lax-Milgram 1954) The Hamilton field of p: H p = ∂ ξ p ∂ x − ∂ x p ∂ ξ , then H p p ≡ 0. Definition The operator P is of principal type if H p 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¨ ormander’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 = D x 1 + iD x 2 + i ( x 1 + ix 2 ) D x 3 = D x 1 − x 2 D x 3 + i ( D x 2 + x 1 D x 3 ) is not locally solvable anywhere in R 3 . P is the tangential Cauchy-Riemann operator on the boundary of the strictly pseudoconvex set { ( z 1 , z 2 ) : | z 1 | 2 + 2 Im z 2 < 0 } ⊂ C 2 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 } = H Re p Im p = 0 on p − 1 (0) , where p is the principal symbol of P. (H¨ ormander 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 ⊂ C n be defined by ̺ ( z ) < 0, ̺ ∈ C ∞ real with ∂̺ � = 0. Then P jk = ∂ z j ̺∂ z k − ∂ z k ̺∂ z j j � = k are the tangential Cauchy-Riemann operators on ∂ Ω since P jk ̺ = 0. Ω is strictly pseudo-convex domain if � ∂ z ∂ z ̺ v , v � > 0 when ∂ z ̺ v = 0 and v � = 0 For n = 2 and a complex vector field P = P 12 in R 3 this holds if 1) P , P and [ P , P ] are linearly independent ⇒ [ P , P ] � = 0 2) Pz = 0 has two linearly independent solutions z 1 , z 2 (coordinates). But almost all vector fields on R 3 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 odinger operator with potential V ∈ C ∞ ( R n ) The Schr¨ P ( h ) = − h 2 ∆ + V ( x ) = p ( x , hD ) 0 < h ≤ 1 If z ∈ {| ξ | 2 + V ( x ) : { Re p , Im p } = 2 � ξ, ∇ Im V ( x ) � � = 0 } then � ( P ( h ) − z ) − 1 � ≥ C N h − N ∀ N By Sard’s theorem, this holds for almost all values when Im V �≡ 0. Holds for any semiclassical ΨDO. (D., Sj¨ ostrand 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 ( t , x ) ∈ R × R n ∂ t u + u ∂ x 1 u = f ( t , x , u ) 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 ∂ x 1 u (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 oriented 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 H Re 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 H p odd/even.) Example For D t + if ( t , x , D x ) with f real and first order, H Re p Im p = ∂ t f ( 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¨ ormander 1980) Condition (Ψ) is sufficient for solvability for ΨDO in two variables. (Lerner 1988) Theorem If P is a principal type Ψ DO 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