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Uniqueness of solutions for some elliptic equations and systems in nearly star-shaped domains. (English) Zbl 0960.35035

The stability of the uniqueness of trivial solutions of certain elliptic boundary value problems is studied, where the star-shaped domain is small perturbed – so-called nearly star-shaped domain. Firstly, the constancy results of solutions for models in liquid crystals and harmonic maps are generalized to the case of nearly star-shaped domains (for the star-shaped domains see K. S. Chou and X.-P. Zhu [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 12, No. 1, 99-115 (1995; Zbl 0843.35027)]). The uniqueness of trivial solutions is proved also for the following Dirichlet problem: \[ \begin{cases} -\Delta u=|u|^{p-1}u\;& \text{in }\Omega , \\ u=0\;& \text{on }\partial \Omega , \end{cases} \] where \(p>(n+2)/(n-2)\), and \(\Omega \subset \mathbb{R}^{n}\) is a smooth and nearly star-shaped domain. In the sequel, the uniqueness of smooth solutions for the systems motivated by nonlinear elastostatics is proved, where the energy density is uniformly strictly quasiconvex and the domain is nearly star-shaped (for the case of star-shaped domains see R. J. Knops and C. A. Stuart [Arch. Ration. Mech. Anal. 86, 233-249 (1984; Zbl 0589.73017)]). Finally, the existence of nearly star-shaped domains is given.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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