This monograph treats parametric minimal surfaces of codimension one in the Euclidean space $R^{n+1}$; more precisely, boundaries $\partial X$ of sets $X\subsetR^{n+1}$ with minimal perimeter, and also nonparametric surfaces with a graph $u$ corresponding to the solutions $u\:R^n\supset\Omega\toR$ of convex variational problems of the type $\int_\Omega F({\rm grad}\,u)\,dx\to\min$. The authors' principal results concerning parametric minimal surfaces are: the existence of sets $X$ with minimal perimeter, the smoothness of their reduced boundary $\partial^*X$ (De Giorgi's theorem) and the estimate $\dim(\partial X-\partial^*X)\le n-8$ on the singular set (Federer's theorem), which is based on the fact that every minimizing codimension one cone in $R^{n+1}$ is a hyperplane for $n\le6$ (Simon's theorem). The authors' main results on convex variational problems of the type $\int_\Omega F({\rm grad}\,u)\,dx\to\min$ are: the existence of solutions with prescribed boundary values on bounded convex domains $\Omega$ in the class of Lipschitz functions (Hilbert\mhy Haar solutions) and the regularity of these solutions, i.e., De Giorgi's theorem on the Hölder continuity of their derivative. Further topics concerning nonparametric minimal surfaces are: the interior a priori gradient estimate (following Trudinger's proof of this result originally due to Bombieri, De Giorgi and Miranda), the generalized Dirichlet problem for the minimal surface equation on bounded or unbounded Lipschitz domains, the theory initiated by Miranda of generalized solutions to the minimal surface equation which may take the values $+\infty$ or $-\infty$ on sets of positive $n$-dimensional measure, and the proof of the celebrated Bernstein theorem that minimal graphs defined over all of $R^n$ are hyperplanes for $n\le7$, as well as the counterexample by Bombieri, De Giorgi and Giusti in dimension $n=8$ based on the minimality of Simon's cone ${(x, y)\inR^4\timesR^4$: $|x|^2=|y|^2}$ in $R^8$. Much of this material is also covered in the recent book by \n E. Giusti\en [Minimal surfaces and functions of bounded variation, Birkhäuser, Basel, 1984; MR0775682 (87a:58041)]. However, the monograph under review contains additional topics and aims at greater generality. For example, the a priori gradient estimate is established for surfaces of prescribed mean curvature (not necessarily vanishing everywhere as in the minimal surface case) in order to be able to treat the capillary problem later on. Similarly, the De Giorgi regularity theory is formulated not only for strictly minimal sets but for sets which are almost minimal, in a certain sense, to include sets $X$ whose distributional mean boundary curvature is sufficiently integrable. (This extension of De Giorgi's theorem is originally due to Massari.) Additional topics include a purely differential geometric proof of the Bernstein theorem based on curvature estimates (first given by Schoen, Simon and Yau) and a proof of the optimal isoperimetric inequality in $R^n$ (using slicing theory and induction on $n$ to obtain some isoperimetric inequality and symmetrization with respect to hyperplanes to achieve the optimal constant and discuss equality). The exposition is very technical and very little motivation is provided. According to the preface, what the authors want to do here, is enter into the details of ideas and results about the codimension one case$\ldots$''. A reader not already aquainted with the subject might prefer Giusti's textbook as an introduction. However, the authors have taken care to keep the presentation self-contained so that even standard material from measure theory has been included. The principal value of the monograph under review lies in the fact that it gives a condensed but complete treatment of many important results about minimal (and more general) surfaces of codimension one and contains many of the authors' original ideas. The monograph will surely become a standard reference in the theory of minimal surfaces and surfaces of prescribed mean curvature.

### Minimal surfaces of codimension one

#### Abstract

This monograph treats parametric minimal surfaces of codimension one in the Euclidean space $R^{n+1}$; more precisely, boundaries $\partial X$ of sets $X\subsetR^{n+1}$ with minimal perimeter, and also nonparametric surfaces with a graph $u$ corresponding to the solutions $u\:R^n\supset\Omega\toR$ of convex variational problems of the type $\int_\Omega F({\rm grad}\,u)\,dx\to\min$. The authors' principal results concerning parametric minimal surfaces are: the existence of sets $X$ with minimal perimeter, the smoothness of their reduced boundary $\partial^*X$ (De Giorgi's theorem) and the estimate $\dim(\partial X-\partial^*X)\le n-8$ on the singular set (Federer's theorem), which is based on the fact that every minimizing codimension one cone in $R^{n+1}$ is a hyperplane for $n\le6$ (Simon's theorem). The authors' main results on convex variational problems of the type $\int_\Omega F({\rm grad}\,u)\,dx\to\min$ are: the existence of solutions with prescribed boundary values on bounded convex domains $\Omega$ in the class of Lipschitz functions (Hilbert\mhy Haar solutions) and the regularity of these solutions, i.e., De Giorgi's theorem on the Hölder continuity of their derivative. Further topics concerning nonparametric minimal surfaces are: the interior a priori gradient estimate (following Trudinger's proof of this result originally due to Bombieri, De Giorgi and Miranda), the generalized Dirichlet problem for the minimal surface equation on bounded or unbounded Lipschitz domains, the theory initiated by Miranda of generalized solutions to the minimal surface equation which may take the values $+\infty$ or $-\infty$ on sets of positive $n$-dimensional measure, and the proof of the celebrated Bernstein theorem that minimal graphs defined over all of $R^n$ are hyperplanes for $n\le7$, as well as the counterexample by Bombieri, De Giorgi and Giusti in dimension $n=8$ based on the minimality of Simon's cone ${(x, y)\inR^4\timesR^4$: $|x|^2=|y|^2}$ in $R^8$. Much of this material is also covered in the recent book by \n E. Giusti\en [Minimal surfaces and functions of bounded variation, Birkhäuser, Basel, 1984; MR0775682 (87a:58041)]. However, the monograph under review contains additional topics and aims at greater generality. For example, the a priori gradient estimate is established for surfaces of prescribed mean curvature (not necessarily vanishing everywhere as in the minimal surface case) in order to be able to treat the capillary problem later on. Similarly, the De Giorgi regularity theory is formulated not only for strictly minimal sets but for sets which are almost minimal, in a certain sense, to include sets $X$ whose distributional mean boundary curvature is sufficiently integrable. (This extension of De Giorgi's theorem is originally due to Massari.) Additional topics include a purely differential geometric proof of the Bernstein theorem based on curvature estimates (first given by Schoen, Simon and Yau) and a proof of the optimal isoperimetric inequality in $R^n$ (using slicing theory and induction on $n$ to obtain some isoperimetric inequality and symmetrization with respect to hyperplanes to achieve the optimal constant and discuss equality). The exposition is very technical and very little motivation is provided. According to the preface, what the authors want to do here, is enter into the details of ideas and results about the codimension one case$\ldots$''. A reader not already aquainted with the subject might prefer Giusti's textbook as an introduction. However, the authors have taken care to keep the presentation self-contained so that even standard material from measure theory has been included. The principal value of the monograph under review lies in the fact that it gives a condensed but complete treatment of many important results about minimal (and more general) surfaces of codimension one and contains many of the authors' original ideas. The monograph will surely become a standard reference in the theory of minimal surfaces and surfaces of prescribed mean curvature.
##### Scheda breve Scheda completa Scheda completa (DC)
1984
9780444868732
Elliptic partial differential equations; Minimnal surfaces
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/534077
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