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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