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Geometric properties of solutions to the anisotropic \(p\)-Laplace equation in dimension two. (English) Zbl 1002.35044
The solutions to the following equation in the plane are considered: \[ (*) : \text{div}(|A\nabla u \cdot\nabla u|^{(p-2)/2} A\nabla u) = 0 \quad . \] Here \(A(x)\) denotes a uniformly elliptic, Lipschitz continuous, and symmetric \(2 \times 2\) matrix. The equation \((*)\) is the Euler equation for the variational integral \( J(u) = \int |A\nabla u \cdot\nabla u|^{p/2} dx \quad . \) It is proved that the critical points of the non-constant solutions to \((*)\) are isolated and the number of critical points are estimated in parallel to previous results for linear equations. Furthermore, it is shown that the local topological structure of the level lines of the solutions is the same as that of harmonic functions. As an application of the discreteness of critical points the authors obtain a strong comparison theorem: If two different solutions \(u^{1}\) and \(u^{2}\) to \((*)\) satisfy \(u^{1} \leq u^{2}\), then in fact \(u^{1} < u^{2}\). This generalizes a result of J. J. Manfredi who obtained it for the \(p-\)Laplace equation (also in \(2\) dimensions), i.e. for the case where \(A = I\), [Proc. Am. Math. Soc. 103, No.2, 473-479 (1988; Zbl 0658.35041)]. It is noted, that such a comparison principle is still unknown in higher dimensions – even for the \(p-\)Laplacian. A generalization of an observation (concerning harmonic functions) due to T. Radó is also proved in the much wider setting of equation \((*)\): Two solutions to \((*)\) always exist such that they induce a diffeomorphism from the unit disc onto a given bounded, convex, and simply connected domain in the plane and such that this diffeomorphism extends a given homeomorphism from the unit circle to the boundary of the given domain.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
30C62 Quasiconformal mappings in the complex plane
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
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