Hadamard’s example of ill-posed problems

E334044

classical construction in analysis counterexample example in partial differential equations ill-posed problem mathematical example

Hadamard’s example of ill-posed problems is a classical mathematical construction illustrating how small changes in input data can cause large, unstable changes in solutions, thereby violating the standard criteria for well-posedness in analysis and partial differential equations.

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All labels observed (3)

Label	Occurrences
Hadamard well-posedness	1
Hadamard’s definition of well-posedness	1
Hadamard’s example of ill-posed problems canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T3167255 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Hadamard’s example of ill-posed problems
Context triple: [Jacques Hadamard, knownFor, Hadamard’s example of ill-posed problems]

A. Hilbert’s twenty-second problem
Hilbert’s twenty-second problem is one of David Hilbert’s famous list of 23 problems, concerning the uniformization of analytic relations and the representation of multi-valued analytic functions by single-valued ones on suitable Riemann surfaces.
B. Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
C. The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
D. Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
E. Méthodes de calcul différentiel absolu et leurs applications
Méthodes de calcul différentiel absolu et leurs applications is a foundational mathematical work that systematically develops the theory of tensor calculus and its applications, laying groundwork later used in general relativity.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Hadamard’s example of ill-posed problems
Target entity description: Hadamard’s example of ill-posed problems is a classical mathematical construction illustrating how small changes in input data can cause large, unstable changes in solutions, thereby violating the standard criteria for well-posedness in analysis and partial differential equations.

A. Hilbert’s twenty-second problem
Hilbert’s twenty-second problem is one of David Hilbert’s famous list of 23 problems, concerning the uniformization of analytic relations and the representation of multi-valued analytic functions by single-valued ones on suitable Riemann surfaces.
B. Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
C. The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
D. Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
E. Méthodes de calcul différentiel absolu et leurs applications
Méthodes de calcul différentiel absolu et leurs applications is a foundational mathematical work that systematically develops the theory of tensor calculus and its applications, laying groundwork later used in general relativity.
F. None of above. chosen

Statements (38)

Predicate	Object
instanceOf	classical construction in analysis ⓘ counterexample ⓘ example in partial differential equations ⓘ ill-posed problem ⓘ mathematical example ⓘ
appearsIn	Hadamard’s work on the Cauchy problem ⓘ linked to: Lectures on Cauchy’s problem in linear partial differential equations classical literature on PDEs ⓘ
concerns	large changes in corresponding solutions ⓘ small perturbations in boundary or initial data ⓘ
context	mathematical analysis ⓘ theory of differential equations ⓘ
demonstrates	importance of continuous dependence on data ⓘ necessity of regularization methods ⓘ pathological behavior of solutions ⓘ
failsProperty	uniqueness or stability in the usual sense ⓘ
formalizes	concept of instability in PDE problems ⓘ
hasProperty	lack of stability ⓘ lack of well-posedness ⓘ non-continuous dependence of solution on data ⓘ
hasRole	paradigmatic counterexample in analysis ⓘ standard textbook example of ill-posedness ⓘ
illustrates	instability of solutions ⓘ sensitivity to small perturbations in data ⓘ violation of Hadamard’s criteria for well-posedness ⓘ
motivates	development of regularization techniques for inverse problems ⓘ study of stability conditions for PDEs ⓘ
namedAfter	Jacques Hadamard ⓘ
relatedTo	Cauchy problems for elliptic equations ⓘ Hadamard’s definition of well-posedness ⓘ linked to: Hadamard’s example of ill-posed problems backward continuation problems ⓘ
satisfies	existence of formal solutions ⓘ
usedAs	motivation for defining well-posed problems ⓘ
usedIn	functional analysis ⓘ inverse problems ⓘ regularization theory ⓘ theory of partial differential equations ⓘ
usedToTeach	difference between well-posed and ill-posed problems ⓘ role of continuity in solution operators ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jacques Hadamard → knownFor → Hadamard’s example of ill-posed problems ⓘ

Hadamard’s example of ill-posed problems → relatedTo → Hadamard’s definition of well-posedness ⓘ

linked to: Hadamard’s example of ill-posed problems

Lectures on Cauchy’s Problem in Linear Partial Differential Equations → associatedConcept → Hadamard well-posedness ⓘ

subject linked to: Lectures on Cauchy’s problem in linear partial differential equations