×

Nonlinear partial differential equations and free boundaries. Volume 1: Elliptic equations. (English) Zbl 0595.35100

It is well known that the literature on free boundary problems (f.b.p. in the sequel) for p.d.e. has increased tremendously during the last decade, so that the project of writing a book on this subject is as frightening as useful. Although some books have appeared in the last few years [see C. M. Elliott and J. R. Ockendon, ”Weak and variational methods for moving boundary problems” (1982; Zbl 0476.35080), A. Friedman, ”Variational principles and free-boundary problems” (1982; Zbl 0564.49002), J. Crank, ”Free and moving boundary problems”, Oxford (1984; Zbl 0547.35001)], there is such a variety of topics and of mathematical aspects in this area, that a volume like this on elliptic equations (and the one which is to come on hyperbolic and parabolic equations) with its rather specialized character is certainly welcome.
The book collects a considerable amount of results, mainly focussed on problems of the type \[ (1)\quad -\Delta_ pu+f(u)=g(x),\quad in\quad \Omega \subset {\mathbb{R}}^ N;\quad (2)\quad u=h(x),\quad on\quad \partial \Omega, \] where \(\Delta_ pu=div(| \nabla u|^{p-2}\nabla u)\), \(p>1\). For \(p=2\) equation (1) includes the nonlinear diffusion equation - \(\Delta\) \(\phi\) (u)\(+f(u)=g(x)\) for monotone \(\phi\).
The existence of a free boundary is generally associated with the fact that u vanishes identically over some set of nonzero measure. If N(u) and S(u) denote the null set and the support of u respectively, the f.b. is defined to be \(\partial S(u)\cap \partial N(u).\)
The material is divided in four chapters.
Chapt. 1: The f.b. in the Dirichlet problem for second order elliptic quasilinear equations. Assuming f(u) is nondecreasing, the tool of sub- and super-solutions is used to obtain conditions for existence or nonexistence of an f.b.. Information on qualitative properties of the f.b. is obtained using the technique of symmetric rearrangement and the behaviour of u near the f.b. is investigated as well as the regularity of the f.b..
Chapt. 2: The f.b. in other second order nonlinear problems. Some of the previous results are extended to equations or systems in which the nonlinear term f(u) is nonmonotone. A number of other cases (multivalued equations, f(u) singular near \(u=0\), nonisotropic equations, etc.) is considered.
Chapt. 3: Existence and location of the f.b. by means of energy methods. Information about the support of u is obtained via energy estimates as an alternative approach to methods based on comparison arguments.
Chapt. 4: The general theory for second order nonlinear elliptic equations: a particular overview. This overview has the aim of fitting the whole subject in the framework of the general theory of second order nonlinear elliptic equations. This effort provides a unifying point of view.
The bibliography is remarkably rich and up-to-date.
Reviewer: A.Fasano

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
76A05 Non-Newtonian fluids
35J60 Nonlinear elliptic equations