An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations

Abstract

We prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of R n space. In particular the result applies also to some operators in Carnot groups.1991 Mathematics Subject Classification. 35D40, 35B65, 35H20. Key words and phrases. Degenerate elliptic operators, Nonlinear elliptic operators, Carnot groups, viscosity solutions, Theorem on sums, Hölder regularity.The author is supported by MURST, Italy, and INDAM-GNAMPA project 2017: Regolarità delle soluzioni viscose per equazioni a derivate parziali non lineari degeneri.The author wishes to thank P. Mannucci for pointing out her joint papers [15] and [16].

Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

“…More general results for PDEs over Hörmander vector fields will appear in [6]. Our model equations of the form (1) are those that we call, in analogy to [15], uniformly subelliptic, cf (5) (see also [25,Definition 1.1]), whose prototype are First, we complete and extend the results initiated in [20,21,10] to fully nonlinear subelliptic equations in the Heisenberg group perturbed by semilinear terms. More precisely, we use a nonlinear degenerate Hadamard theorem [20,22,21] to derive Liouville properties for viscosity solutions to model nonlinear PDEs of the form…”

Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators

“…In particular we recall the fundamental result [20]. Concerning the regularity issues of viscosity solutions of degenerate equations closely related to ours, we refer to [8,13,14,15]. We conclude this introduction by pointing out that if we drop the assumption (H1), in the sense that a 1 = a N = 0, then there exist viscosity solutions of…”

“…In particular we recall the fundamental result [20]. Concerning the regularity issues of viscosity solutions of degenerate equations closely related to ours, we refer to [8,13,14,15].…”

Estimates for the $\infty$-Laplacian relative to vector fields

“…Among them, we wish to recall the following works [BGI18], [BGL17], [FV20b], [FG21], where the regularity of viscosity solutions of truncated operators has been studied. Moreover, always in the frame of a degenerate situation but in a non-commutative structures, we point out the results contained in [Fer20], [FV20a] and [Gof20].…”