×

One-dimensional symmetry for solutions of quasilinear equations in \(\mathbb R^2\). (English) Zbl 1115.35045

The authors studies the solutions of the two-dimensional quasilinear equation \(\text{div}(a|\nabla u|\nabla u)+f(u)=0\) satisfying \(|\nabla u|>0\). He proves that certain growth condition on the argument of \(\nabla u\) guarantees that \(u\) is one-dimensional. As a consequence, it is obtained that any solution having one positive derivative is one-dimensional. This result provides a proof of a conjecture of De Giorgi (see E. De Giorgi [Convergence problems for functionals and operators, Recent methods in non-linear analysis, Proc. Int. Meet., Rome 1978, 131–188 (1979; Zbl 0405.49001)]) in dimension 2.
Reviewer: Pavel Rehak (Brno)

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs

Citations:

Zbl 0405.49001
PDFBibTeX XMLCite
Full Text: EuDML