This paper by Hadamard is often cited as being the source of the definition of well-posed and ill-posed problems.
However, it is in French so I cannot verify that claim.
Is there an English translation?
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Sign up to join this communityThis paper by Hadamard is often cited as being the source of the definition of well-posed and ill-posed problems.
However, it is in French so I cannot verify that claim.
Is there an English translation?
No, there is no "official" English translation, however, Google translate should work just fine, here is the translation of the first paragraph, without any corrections from my side:
The general question that we propose to study is the determination of the solutions of a linear equation to the derivatives partial ones of the second order (this is the case to which we shall confine ourselves) by data at the limits. In other words, we have to summarize the results obtained in relation to the following two questions:
1) What are the data required to determine a solution of a partial differential equation?
2) How can we calculate the solution according to the data provided?
These two questions are precisely those which Mr. Poincaré his important discourse On the Reports of Pure Analysis and Mathematical Physics (International Congress of Mathematicians, Zurich, 1897), cited as an example of the aid which Physics is capable of bringing to analysis. And, indeed, there is perhaps no case in which the conclusion which he has developed is more strikingly apparent:
• It is mathematical physics that shows us what problems we must ask ourselves.
• It is also what makes us foresee the solution.
(Incidentally, a nice description of how physics can help mathematics.)
Following a suggestion by Sylvain Jullien, I tried the translator at DeepL.com, on the same paragraph. Remarkably similar translation, actually, I presume they use pretty much the same neural network architecture as Google.
The general question that we propose to study is the determination of the solutions of a linear equation to derivatives partial second order data (this is the case to which we will confine ourselves) with data at the limits. In other words, we have to summarize the results achieved with respect to the following two-pronged question:
1) What is the evidence to determine a solution to an partial derivatives?
2) How to calculate the solution based on data so provided?
These two questions are precisely those which Mr Poincaré, in his important discourse On the reports of Pure Analysis and Mathematical Physics (International Congress of Mathematicians, Zurich, 1897) cited as an example of the help that Physics is likely to bring to Analysis. And, indeed, there may not be any case in which the conclusion that he came to is more striking:
• It is mathematical physics that shows us which problems we must face.
• It is also what makes us foresee the solution.
... and just for completeness, here is what Microsoft Translator produces (somewhat less to my liking):
The general question we propose to study is the determination of the solutions of a linear equation to the derivatives partial second order (this is the case we will confine ourselves to) by data to the limits. In other words, we have to sum up the results gained relativity to the following double question:
1) What are the data to determine a solution of a equation to partial derivatives?
2) How can the solution be calculated according to of the data so provided?
These two questions are precisely those that Mr. Poincaré, in his important speech on the Reports of Analyse pure and Mathematical physics (International Congress of Mathematicians, Zurich, 1897), cited as an example of the aid that physics is likely to bring to the analysis. And, indeed, there may be no cases in which more strikingly appears the conclusion that it has developed:
• It is the mathematical physics that shows us which problems we have to ask ourselves.
• She also makes us anticipate the solution.