Convex variational problems. Linear, nearly linear and anisotropic growth conditions.

*(English)*Zbl 1033.49001
Lecture Notes in Mathematics 1818. Berlin: Springer (ISBN 3-540-40298-5/pbk). x, 217 p. (2003).

In this book, the author considers two, apparently quite different, types of convex variational problems: those with linear growth and those with superlinear but anisotropic growth. The unifying element of this presentation is that both types of problem fit into the framework of so-called \((s,\mu,q)\) growth (for constants \(s\geq1\), \(\mu\in \mathbb R\), and \(q>1\)). To explain this growth, we write the variational problem as finding a function that minimizes the integral
\[
\int_{\Omega} f(| Du| )\, dx
\]
(with \(u \in \mathbb R^N\) and \(\Omega \subset \mathbb R^n\)) over all competing functions in a suitable function space. We say that \(f\) is of \((s,\mu,q)\) growth if there is an \(N\)-function \(F\) satisfying a \(\Delta_2\) condition such that \(\liminf_{t\to\infty} F(t)/t^s >0\) and if there are positive constants \(c_1\), \(c_2\), \(\lambda\), and \(\Lambda\) such that
\[
\begin{gathered} f(Z) \geq c_1F(| Z| )-c_2, \\ \lambda (1+| Z| ^2)^{-\mu/2} \leq D^2f(Z)(Y,Y) \leq \Lambda (1+| Z| ^2)^{(q-2)/2} \end{gathered}
\]
for all \(Y\) and \(Z\) in \(\mathbb R^{nN}\). Details of the results depend on the relations between \(s\), \(\mu\) and \(q\). For example, if \(q<(2-\mu)+(2s)/n\), and if \(N=1\), then the minimizing function has HĂ¶lder continuous derivatives with any exponent less than \(1\). Of course, many other, sharper regularity results are included, such as partial regularity for problems with \(N>1\), and even full regularity for such vector problems if \(f\) has the form \(f(Z)=g(| Z| )\) for a suitable function \(g\). For problems with linear growth, uniqueness of a minimizer is not to be expected but an alternative result is proved, namely that the dual solution is unique, and this has consequences for solutions of the original problem.

The reviewer is surprised that his work on anisotropic problems [Ann. Sc., Norm. Sup. Pisa Cl. Sci., IV. Ser. 21, 498–522 (1994; Zbl 0839.35018)] was not included.

The reviewer is surprised that his work on anisotropic problems [Ann. Sc., Norm. Sup. Pisa Cl. Sci., IV. Ser. 21, 498–522 (1994; Zbl 0839.35018)] was not included.

Reviewer: Gary M. Lieberman (Ames)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49N60 | Regularity of solutions in optimal control |

49N15 | Duality theory (optimization) |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J20 | Variational methods for second-order elliptic equations |

35J50 | Variational methods for elliptic systems |