Behavior of solutions of a nonlinear variational problem in a neighborhood of singular points of the boundary and at infinity

G. V. Grishina

N. E. Bauman Moscow State Technical University

Abstract: A study is made of the functions realizing a minimum for the functional
$$ \int_\Omega F(x,u,Du,\dots,D^mu)\,dx, $$
where $F$ has power order of growth with respect to $D^mu$. The rate of decrease is established for the $m$th-order derivatives of an extremal in the integral metric in a neighborhood of a singularity on the boundary of power cusp type and at infinity in domains having outside some ball the structure of a cylinder or layer, and also domains constricting or expanding at infinity in a power manner. Estimates are obtained under the assumption that homogeneous Dirichlet conditions or Neumann conditions are given on the indicated part of the boundary. The estimates depend on the geometry of the domain. The results obtained are new also for a broad class of nonlinear elliptic equations.

UDC: 517.9

MSC: Primary 49K30, 49Q20; Secondary 49S05, 35J65

Received: 11.10.1991

 English version: Russian Academy of Sciences. Sbornik. Mathematics, 1994, 78:2, 333–355

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