1.3. STRATEGIES FOR STUDYING PDE 7 many diverse examples set forth in §1.2—this is no easy task. And indeed the very question of what it means to “solve” a given PDE can be subtle, depending in large part on the particular structure of the problem at hand. 1.3.1. Well-posed problems, classical solutions. The informal notion of a well-posed problem captures many of the desir- able features of what it means to solve a PDE. We say that a given problem for a partial differential equation is well-posed if (i) the problem in fact has a solution (ii) this solution is unique and (iii) the solution depends continuously on the data given in the problem. The last condition is particularly important for problems arising from physical applications: we would prefer that our (unique) solution changes only a little when the conditions specifying the problem change a little. (For many problems, on the other hand, uniqueness is not to be expected. In these cases the primary mathematical tasks are to classify and to character- ize the solutions.) Now clearly it would be desirable to “solve” PDE in such a way that (i)–(iii) hold. But notice that we still have not carefully defined what we mean by a “solution”. Should we ask, for example, that a “solution” u must be real analytic or at least infinitely differentiable? This might be desirable, but perhaps we are asking too much. Maybe it would be wiser to require a solution of a PDE of order k to be at least k times continuously differentiable. Then at least all the derivatives which appear in the statement of the PDE will exist and be continuous, although maybe certain higher derivatives will not exist. Let us informally call a solution with this much smoothness a classical solution of the PDE: this is certainly the most obvious notion of solution. So by solving a partial differential equation in the classical sense we mean if possible to write down a formula for a classical solution satisfying (i)–(iii) above, or at least to show such a solution exists, and to deduce various of its properties. 1.3.2. Weak solutions and regularity. But can we achieve this? The answer is that certain specific partial differential equations (e.g. Laplace’s equation) can be solved in the classical sense, but many others, if not most others, cannot. Consider for instance
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