##
**Introduction to regularity theory for nonlinear elliptic systems.**
*(English)*
Zbl 0786.35001

Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser Verlag. viii, 131 p. (1993).

The book provides a wide and self-contained introduction to the regularity theory for minimizers of regular integrals in the calculus of variations and for solutions of nonlinear elliptic systems.

First of all, in chapter 1, Sobolev spaces are introduced and their main properties are proved, later the Hilbert space approach to boundary value problems for linear elliptic systems is presented together with Gårding’s inequality. The direct methods of the calculus of variations are then described and the main lower semicontinuity theorem both in the convex and in the quasiconvex case, are quoted.

In chapter 2 the study of the regularity for the solutions of elliptic systems is started, the tools used to this aim are a priori estimates of Caccioppoli type and the difference-quotient method. Thus \(H^{k,2}\)- interior and boundary regularity for weak solutions of linear elliptic systems are proved. \(H^{2,2}\)-interior regularity results are also proved for scalar minimizers of some nonlinear functionals.

In chapter 3 Schauder interior and boundary (with Dirichlet conditions) estimates are derived for linear elliptic systems in divergence form and in the nonvariational case. The estimates are obtained without using potential theory but following the ideas of Morrey and Campanato. To do this Morrey and Campanato spaces are introduced and studied. Hölder continuity for vector minimizers of some nonlinear functionals is also proved.

In chapter 4 the \(L^ p\)-theory for linear systems with continuous coefficients is presented in the variational and in the nonvariational cases. The theory is studied without making use of singular integrals but by using an interpolation theorem of Stampacchia. To do this some facts on the integrability of functions are first recalled: weak estimates, Marcinkiewicz interpolation theorem, Hardy-Littlewood maximal functions and maximal theorem, Lebesgue differentiation theorem and Calderon- Zygmund covering argument. Then BMO spaces are introduced and John- Nirenberg lemma together with Stampacchia interpolation theorem are proved. By these results \(L^ p\)-estimates for the first derivatives of the solutions of linear elliptic systems in divergence form and for the second derivatives of the solutions of linear elliptic systems not in divergence form are derived.

In chapter 5 the problem of the interior regularity for scalar minimizers and solutions of second order elliptic equations is studied. Such problem is treated by introducing and studying De Giorgi classes and then proving the well known De Giorgi theorem about Hölder continuity for solutions of second order elliptic equations with bounded measurable coefficients. The same result is also proved by using Moser iteration technique, moreover Harnack inequality for solutions of second order elliptic equations with bounded measurable coefficients is also proved. The problem of establishing Harnack inequality for functions in De Giorgi classes is also studied and the affirmative result by Di Benedetto and Trudinger is proved. Later the Giaquinta-Giusti notion of quasi-minima is introduced and Hölder continuity and Harnack inequality for them are proved. Finally the Giaquinta-Giusti result on Hölder continuity of the first derivatives for minimizers of multiple integrals of the calculus of variations is proved.

In the last chapter, starting from the counterexample of De Giorgi showing that an uniformly elliptic quadratic functional with bounded measurable coefficients can have an unbounded minimizer in \(H^{1,2} (\Omega;\mathbb{R}^ N)\), some partial \(C^{1,\alpha}\)-regularity results for vector minimizers of functionals with quadratic growth, or weak solutions of elliptic systems are proved. Moreover the singular set of minimizers for some quadratic functionals is studied.

First of all, in chapter 1, Sobolev spaces are introduced and their main properties are proved, later the Hilbert space approach to boundary value problems for linear elliptic systems is presented together with Gårding’s inequality. The direct methods of the calculus of variations are then described and the main lower semicontinuity theorem both in the convex and in the quasiconvex case, are quoted.

In chapter 2 the study of the regularity for the solutions of elliptic systems is started, the tools used to this aim are a priori estimates of Caccioppoli type and the difference-quotient method. Thus \(H^{k,2}\)- interior and boundary regularity for weak solutions of linear elliptic systems are proved. \(H^{2,2}\)-interior regularity results are also proved for scalar minimizers of some nonlinear functionals.

In chapter 3 Schauder interior and boundary (with Dirichlet conditions) estimates are derived for linear elliptic systems in divergence form and in the nonvariational case. The estimates are obtained without using potential theory but following the ideas of Morrey and Campanato. To do this Morrey and Campanato spaces are introduced and studied. Hölder continuity for vector minimizers of some nonlinear functionals is also proved.

In chapter 4 the \(L^ p\)-theory for linear systems with continuous coefficients is presented in the variational and in the nonvariational cases. The theory is studied without making use of singular integrals but by using an interpolation theorem of Stampacchia. To do this some facts on the integrability of functions are first recalled: weak estimates, Marcinkiewicz interpolation theorem, Hardy-Littlewood maximal functions and maximal theorem, Lebesgue differentiation theorem and Calderon- Zygmund covering argument. Then BMO spaces are introduced and John- Nirenberg lemma together with Stampacchia interpolation theorem are proved. By these results \(L^ p\)-estimates for the first derivatives of the solutions of linear elliptic systems in divergence form and for the second derivatives of the solutions of linear elliptic systems not in divergence form are derived.

In chapter 5 the problem of the interior regularity for scalar minimizers and solutions of second order elliptic equations is studied. Such problem is treated by introducing and studying De Giorgi classes and then proving the well known De Giorgi theorem about Hölder continuity for solutions of second order elliptic equations with bounded measurable coefficients. The same result is also proved by using Moser iteration technique, moreover Harnack inequality for solutions of second order elliptic equations with bounded measurable coefficients is also proved. The problem of establishing Harnack inequality for functions in De Giorgi classes is also studied and the affirmative result by Di Benedetto and Trudinger is proved. Later the Giaquinta-Giusti notion of quasi-minima is introduced and Hölder continuity and Harnack inequality for them are proved. Finally the Giaquinta-Giusti result on Hölder continuity of the first derivatives for minimizers of multiple integrals of the calculus of variations is proved.

In the last chapter, starting from the counterexample of De Giorgi showing that an uniformly elliptic quadratic functional with bounded measurable coefficients can have an unbounded minimizer in \(H^{1,2} (\Omega;\mathbb{R}^ N)\), some partial \(C^{1,\alpha}\)-regularity results for vector minimizers of functionals with quadratic growth, or weak solutions of elliptic systems are proved. Moreover the singular set of minimizers for some quadratic functionals is studied.

Reviewer: R.De Arcangelis (Napoli)

### MSC:

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

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

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

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

35J60 | Nonlinear elliptic equations |

35J50 | Variational methods for elliptic systems |

35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |