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The paper studies a finite element algorithm giving approximations to the minimum-norm solution of ill-posed problems of the form A f = g Af = g , where A is a bounded linear operator from one Hubert space to another. It is shown that the algorithm is norm convergent in the general case and an error bound is derived for the case where g is in the range of A A ∗ A{A^\ast } . As an example, the method has been applied to the problem of evaluating the second derivative f of a function g numerically.

An algorithm is developed for the numerical computation of minimum norm solutions of Fredholm integral equations of the first kind $Kf = g$. The algorithm is different from Tihonov’s method and avoids direct collocation. The method has the following geometrical interpretation: One is looking for a vector of minimum norm that lies in a given finite dimensional subspace and also in a given neighborhood of the set of all solutions of $Kf = g$. A convergence proof for the algorithm is presented and four examples are treated numerically.

Variational methods are applied to evaluate the least constant c in Sobolev’s inequality for the imbedding of the Sobolev space $H^1 (0,s)$, where $(0,s)$ is a real interval, into the space $C(0,s)$ of bounded continuous functions on $(0,s)$. The method yields the exact solution explicitly. It is shown that $c = \tanh ^{ - {1 / 2}} (s)( \cong 1.1459{\text{ for }}s = 1)$ and that equality holds for the hyperbolic cosine function.