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- Title
A Vázquez-type strong minimum/maximum principle for partial trace operators.
- Authors
Kindu, Bukayaw; Mohammed, Ahmed; Tsegaw, Birilew
- Abstract
In this paper we study a Strong Minimum Principle of Vázquez type for partial trace operators with gradient terms. More explicitly, given an $ n $-tuple $ {\bf a} = (a_1, \cdots, a_n) $ of non-negative real numbers with $ a_n>0 $, we give sufficient conditions on a continuous function $ H:\mathbb{R}\times\mathbb{R}_0^+\to\mathbb{R} $ in order for non-negative viscosity supersolutions of$ \mathcal{P}_{\bf a}(D^2u) = H(u, |Du|) $in connected open subsets of $ \mathbb{R}^n $ that vanish at some point in $ \Omega $ to vanish identically in $ \Omega $. When $ H $ depends only on the gradient, the condition is also necessary. Here $ \mathcal{P}_{\bf a} $ belongs to a class of fully nonlinear degenerate elliptic operators that includes the min-max operator which is defined as the sum of the minimum and the maximum eigenvalues of the Hessian matrix. Under suitable conditions on $ H $ and $ {\bf a} = (a_1, \cdots, a_n) $, both a Strong Maximum Principle and a Compact Support Principle for subsolutions will also be investigated. Not only does our work cover a new class of degenerate operators and a wide class of Hamiltonians not investigated in the literature, but to the best of our knowledge, some of our results are new even when $ \mathcal{P}_{\bf a} $ reduces to the standard Laplacian.
- Subjects
ELLIPTIC operators; HESSIAN matrices; REAL numbers; MAXIMUM principles (Mathematics); CONTINUOUS functions; EIGENVALUES
- Publication
Discrete & Continuous Dynamical Systems - Series S, 2023, Vol 16, Issue 11, p1
- ISSN
1937-1632
- Publication type
Article
- DOI
10.3934/dcdss.2023115