Abstract
Abstract We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.
Highlights
Introduction and main resultsA growing attention has been received by the Hessian partial trace operators in the last few decades
I=1 where λi is the eigenvalue of the Hessian matrix, in non-decreasing order, and a =in=1 is an n-tuple of numbers ai ≥ 0 such that a = max1≤i≤nai > 0
See [11] for a time-dependent version. Such operators share a number of qualitative properties with uniformly elliptic operators
Summary
A growing attention has been received by the Hessian partial trace operators in the last few decades. In [14, Section 3.2] the existence of solutions was proved for general weighted Hessian partial trace operators in a bounded domain with a suitable convexity assumption derived from [10]. We show existence and uniqueness for weighted partial trace operators with extremal eigenvalues in domains satisfying a uniform exterior cone property.
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