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On some properties of the positive m-Hessian operators in C 2(Ω)

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Abstract

We consider the p-Hessian operator T p [u] := tr p u xx on the set \({\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}\) and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in \({C^2(\bar{\Omega})}\) for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals I p [u; f] in \({C^2(\bar{\Omega})}\) and we derive some applications.

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  1. Mathematics and Mechanics Faculty, St.-Petersburg State University, St.-Petersburg, 199034, Russia

    N. M. Ivochkina

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  1. N. M. Ivochkina
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Correspondence to N. M. Ivochkina.

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To Professor Yvonne Choquet-Bruhat

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Ivochkina, N.M. On some properties of the positive m-Hessian operators in C 2(Ω). J. Fixed Point Theory Appl. 14, 79–90 (2013). https://doi.org/10.1007/s11784-013-0151-2

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