The notion of Well-Posedness for PDE problems goes back to Hadamard which is commonly phrased into three properties for solutions: 1) existence; 2) uniqueness; 3) continuous dependence. In the mid-1950s,

F. John introduced a notion of Well-Behaved PDE problems. This latter seems to be more relevant to numerical approximations, machine-learnings and data analysis. What may be more interesting is that Ill-Posed problems could be Well-Behaved and Well-posed ones may be Ill-Behaved; and these properties may not necessarily depend on only the types of equations but also on the specificity of problems and geometry of data involved.

In this lecture, I will illustrate these notions and properties of solutions through several examples for the simplest linear PDEs. For similar problems in highly heterogeneous, oscillating media (or with stochastic noise), there are some recent progress and many interesting open questions.

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