Correction to: “The Stefan problem in several space variables”
Avner Friedman · Transactions of the American Mathematical Society · 1969
The paper appeared in this Journal, 133 (1968), 51-87. We wish here to correct a few errors. 1. p. 59. The sentence beginning with It then converges on line 5 from the bottom is unjustified and, in any case, should be omitted. The assertion in the last sentence of this page is still true. In fact, for almost any point (x, t) of the set A where u =0 we have: Vm(X, t) -O 0. This readily implies: a(O -0) ? lim inf am(vm(x, t)) ? lim sup am(v..(vx t)) ? a(O + 0). Since also am(vm) . on A, it easily follows that a(O -0) _ /(x, t) ? a(O + 0) a.e. on A. 2. p. 60. The second sentence from the top is erroneous. However, the assertion 8(x, t)=c(x, u(x, t)) a.e. on B (B the set where u=#0) is still valid since lim Cm(X, u'm(X, t)) = c(x, u(x, t)) for each (x, t) E B and since cm(x, vm) & 8. 3. p. 60. The argument leading from (2.24) to (2.25) is erroneous. Also the conclusion (2.27) is unjustified. But the assertion (2.20) of Corollary 1 is still true and can be proved as follows: Since am(vm(x, t)) -? a(u(x, t)) a.e. in QT, it follows that for almost all J e [0, T] am(r'm(x, o)) -? a(u(x, a)) a.e. in G. The formula preceding (2.20) (on p. 60) holds also if Q1 is replaced by G x (oJ, T), if f is replaced by fT, and if f G(O) a(h) is replaced by f G(a) am(vm(x, a)). Taking m -oo we get (2.20) for all a, E Z, where Z'-[0, t ] -Z has measure zero. For s E Z', redefine a(u(x, s)) as the weak limit in L2(G) of a sequence {a(u(x, (Jm))} where Urm E Z. Call this redefined function y(x, s). Now redefine u(x, s) such that a(u(x, s)) = y(x, s) if y(x, s) 0, and u(x, s) = 0 if -a a y(x, s) _ 0. Since (2.20) holds for all (J = Urm, it also holds for J = s. This completes the proof of (2.20) (when u is redefined on the set G x Z'). Since (2.26) is clearly valid, the proof of (2.21) remains unchanged. 4. p. 63. The assertion, in the second paragraph, that the sequences {um}, {c(x, u,,,)} and {c(x, um)um} are a.e. convergent is unjustified. Therefore, what is constructed is not a weak solution in the sense of the definition of p. 54, but in some other weaker sense. (The uniqueness part of Theorem 4 is valid for weak solutions in the sense of the definition of p. 54.) This result, on the existence of a weak solution, is not used anywhere later on in the paper.