Scientists / Open questions

Lars Hörmander

Partial differential equations; microlocal analysis

1962 Fields Medal

28 questions

Sharp lifespan for the three-dimensional small-data fully nonlinear wave equation $\Box u+G(u',u'')=0$ (Hörmander, 1989)
Open problems from Hörmander's 'Quadratic hyperbolic operators': direct microlocal analysis of the forward fundamental solution of the quadratic hyperbolic model
Open core of Hörmander's hyperbolic systems with double characteristics: complex stability without the real-determinant hypothesis, classification in general size and dimension, extremal counts, codimension beyond the interpolation regime, and well-posedness for scalar-type double characteristics
Hörmander's open issues on non-uniqueness for the Cauchy problem: strictly convex support geometry, the Hölder regime $0<\lambda\le 1$ , and the elliptic case
Sharpness of Hörmander's condition $a_q$ ( $Z(q)$ ): necessity, characterization of closed range, and minimal boundary regularity for the $L^2$ $\bar\partial$ -theory on bounded domains
Mixed directional derivative estimates for solutions of hypoelliptic systems of differential equations with constant coefficients, and their extension beyond the hypoelliptic class
Half-space smoothness of solutions and the geometric characterization of strong P-convexity for constant-coefficient differential operators (Hörmander, 1970)
Off-diagonal singularities of the spectral function of elliptic systems with multiple eigenvalues of the principal symbol
An explicit algebraic or geometric characterization of Hörmander's subelliptic test estimate for first-order systems beyond the two-dimensional case
A general paradifferential/paracomposition framework extending Hörmander's Nash-Moser method to composition operators
Density of the futures of null solutions for spatially tempered linear constant-coefficient PDEs (Sasane 2021, Section 7)
Necessary and sufficient conditions for hypoellipticity of second-order operators with smooth variable coefficients
Caustic-time Mehler formulas for inhomogeneous quadratic Hamiltonians and the exponential image of the metaplectic semigroup (Hörmander 1995)
Closing the sufficiency–necessity gap in Hörmander's generalized Rellich theorem for constant-coefficient operators: minimal cone hypotheses, semi-algebraicity removal, and borderline decay exponents
Sharp asymptotic lifespan of small-data solutions to quasilinear wave equations (the John–Hörmander conjecture): unrestricted data in two and three space dimensions
Carleman estimates and uniqueness of the Cauchy problem for differential operators with nonsingular characteristics: closing the pseudo-convexity gap between necessary and sufficient conditions
Geometric characterization of the singular supports of solutions of general constant-coefficient partial differential equations and of P-convexity for singular supports
Uniqueness and smallness gaps in Hörmander's local theory for the nonlinear Molodensky problem
The maximal analytic continuation domain across a plane boundary for solutions of general elliptic boundary value problems (Hörmander's $\Omega^*$ problem)
Open ends of Hörmander's 1955 operator-theoretic program for general partial differential operators: in-the-large solvability, curved-boundary Cauchy spaces, and the fate of the comparison, spectral-asymptotic, and weak-strong extension questions
Sufficiency of the characteristic-cubic obstruction for P-convexity with respect to a covector in the indefinite-curvature case (converse of Hörmander's Theorem 6.6)
Systematic classification of polynomial families yielding distributional defect identities for second-order operators
A diffeomorphism-invariant, effectively checkable class of hypoelliptic variable-coefficient operators: the open core of the limitations of Hörmander's 1958 constant-strength theorem
The unresolved cases of Hörmander's Cauchy problem for hyperbolic operators with double characteristics: the equality case of the Ivrii–Petkov condition, spectral transition of the Hamilton map, and tangent bicharacteristics

Back to open problems

Sharp lifespan for the three-dimensional small-data fully nonlinear wave equation □u+G(u′,u′′)=0\Box u+G(u',u'')=0□u+G(u′,u′′)=0 (Hörmander, 1989)

Open problems from Hörmander's 'Quadratic hyperbolic operators': direct microlocal analysis of the forward fundamental solution of the quadratic hyperbolic model

Open core of Hörmander's hyperbolic systems with double characteristics: complex stability without the real-determinant hypothesis, classification in general size and dimension, extremal counts, codimension beyond the interpolation regime, and well-posedness for scalar-type double characteristics

Hörmander's open issues on non-uniqueness for the Cauchy problem: strictly convex support geometry, the Hölder regime 0<λ≤10<\lambda\le 10<λ≤1, and the elliptic case

Sharpness of Hörmander's condition aqa_qaq​ (Z(q)Z(q)Z(q)): necessity, characterization of closed range, and minimal boundary regularity for the L2L^2L2 ∂ˉ\bar\partial∂ˉ-theory on bounded domains

Mixed directional derivative estimates for solutions of hypoelliptic systems of differential equations with constant coefficients, and their extension beyond the hypoelliptic class

Half-space smoothness of solutions and the geometric characterization of strong P-convexity for constant-coefficient differential operators (Hörmander, 1970)

Off-diagonal singularities of the spectral function of elliptic systems with multiple eigenvalues of the principal symbol

An explicit algebraic or geometric characterization of Hörmander's subelliptic test estimate for first-order systems beyond the two-dimensional case

A general paradifferential/paracomposition framework extending Hörmander's Nash-Moser method to composition operators

Density of the futures of null solutions for spatially tempered linear constant-coefficient PDEs (Sasane 2021, Section 7)

Necessary and sufficient conditions for hypoellipticity of second-order operators with smooth variable coefficients

Caustic-time Mehler formulas for inhomogeneous quadratic Hamiltonians and the exponential image of the metaplectic semigroup (Hörmander 1995)

Closing the sufficiency–necessity gap in Hörmander's generalized Rellich theorem for constant-coefficient operators: minimal cone hypotheses, semi-algebraicity removal, and borderline decay exponents

Sharp asymptotic lifespan of small-data solutions to quasilinear wave equations (the John–Hörmander conjecture): unrestricted data in two and three space dimensions

Carleman estimates and uniqueness of the Cauchy problem for differential operators with nonsingular characteristics: closing the pseudo-convexity gap between necessary and sufficient conditions

Geometric characterization of the singular supports of solutions of general constant-coefficient partial differential equations and of P-convexity for singular supports

Uniqueness and smallness gaps in Hörmander's local theory for the nonlinear Molodensky problem

The maximal analytic continuation domain across a plane boundary for solutions of general elliptic boundary value problems (Hörmander's Ω∗\Omega^*Ω∗ problem)

Open ends of Hörmander's 1955 operator-theoretic program for general partial differential operators: in-the-large solvability, curved-boundary Cauchy spaces, and the fate of the comparison, spectral-asymptotic, and weak-strong extension questions

Sufficiency of the characteristic-cubic obstruction for P-convexity with respect to a covector in the indefinite-curvature case (converse of Hörmander's Theorem 6.6)

Systematic classification of polynomial families yielding distributional defect identities for second-order operators

A diffeomorphism-invariant, effectively checkable class of hypoelliptic variable-coefficient operators: the open core of the limitations of Hörmander's 1958 constant-strength theorem

The unresolved cases of Hörmander's Cauchy problem for hyperbolic operators with double characteristics: the equality case of the Ivrii–Petkov condition, spectral transition of the Hamilton map, and tangent bicharacteristics

Sharp lifespan for the three-dimensional small-data fully nonlinear wave equation $\Box u+G(u',u'')=0$ (Hörmander, 1989)

Hörmander's open issues on non-uniqueness for the Cauchy problem: strictly convex support geometry, the Hölder regime $0<\lambda\le 1$ , and the elliptic case

Sharpness of Hörmander's condition $a_q$ ( $Z(q)$ ): necessity, characterization of closed range, and minimal boundary regularity for the $L^2$ $\bar\partial$ -theory on bounded domains

The maximal analytic continuation domain across a plane boundary for solutions of general elliptic boundary value problems (Hörmander's $\Omega^*$ problem)