Boundedness of minimizers

Local Boundedness of Minimizers with Limit Growth Conditions

The energy-integral of the calculus of variations (1.1), (1.2) below has a limit behavior when q = np/(n − p), where p is the harmonic average of the exponents pi, i = 1, . . . , n. In fact, if q is larger than in the stated equality, counterexamples to the local boundedness and regularity of minimizers are known. In this paper we prove the local boundedness of minimizers (and also of quasi-min...

Minimizers and symmetric minimizers for problems with critical Sobolev exponent

In this paper we will be concerned with the existence and non-existence of constrained minimizers in Sobolev spaces D(R ), where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding D(R) →֒ L ∗ (R , Q) when Q is a non-negative, continuous, bounded function. However if Q has certain symmetry properties then all minim...

On the symmetry of minimizers

For a large class of variational problems we prove that minimizers are symmetric whenever they are C. AMS subject classifications. 35A15, 35B05, 35B65, 35H30, 35J20, 35J45, 35J50, 35J60.

Existence of Minimizers on Drops

Uniform Boundedness of Rational Points

One of the remarkable things about this theorem is the way in which it suggests that geometry informs arithmetic. The geometric genus g is a manifestly geometric condition, yet it is controlling what seems to be an arithmetic property. Why should the number of integral solutions to xn + yn = zn have anything to do with the shape of the complex solutions? You might argue that that the genus is e...