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Zbl 0812.35037
Bethuel, Fabrice; Ghidaglia, Jean-Michel
Regularity of solutions of certain elliptic equations in dimension two and the co-area formula. (Régularité des solutions de certaines équations elliptiques en dimension deux et formule de la co-aire.) (French)
[J] Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 1993, No.1, 36 p. (1993).

This is a survey paper. The authors discuss the connections between the co-area formula, the Hardy space ${\cal H}\sp 1 (\bbfR\sp 2)$, and the regularity question for weak solutions of certain nonlinear elliptic equations, resp. systems of such equations, in two dimensions. In general, if $\Delta\varphi =h$ and $h\in L\sp 1$ one cannot deduce regularity properties of the solution $\varphi$. If, however, $h$ satisfies in addition certain structural conditions one may obtain useful a-priori estimates for $\varphi$. This has been used in the context of weakly harmonic mappings as well as for solutions of the equations of prescribed mean curvature in two dimensions. In this paper, the authors first discuss a linear model problem. The next section contains generalizations and exhibits the connection with the Hardy space ${\cal H}\sp 1 (\bbfR\sp 2)$. The main part of the paper is devoted to the regularity question for weak $H$-surfaces. In this section several interesting partial improvements of earlier work by {\it E. Heinz} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1986, 15 S. (1986; Zbl 0627.35033)] are presented. A final section contains an application to the question of weak compactness of the set of solutions to the stationary Euler equations. The authors give the main ideas behind the theorems and present proofs in many cases. Only for those parts that are too technical the reader is referred to the original literature.
[M.Grüter (Saarbrücken)]

MSC 2000:: *35J60 Nonlinear elliptic equations
35B65 Smoothness of solutions of PDE

Keywords: co-area formula; Hardy space; equations of prescribed mean curvature

Citations: Zbl 0627.35033

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