Krylov's Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with Quadratic Growth on the Gradient

Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations

“…Proof of the global estimate (1.8). Again, as a simple adaptation of the proofs in the appendices of [11]- [12] shows, to obtain (1.8) it is sufficient to prove that…”

“…This can be proved by Caffarelli's perturbation method [13], [28], more precisely, by seeing the problem as "tangential" to [26,Lemma 4.1]. Results of similar vein, even in larger generality (for unbounded coefficients and L p -viscosity solutions), have recently appeared in [12] for the so-called uniformly elliptic S * -class where the optimal boundary exponent depends also on α F , as well as in the recent preprints [22], [18]. We thus assume Theorem 1.2 for γ = 0 might be known to the experts; however, since this particular case is interesting in itself and deserves a quotable source, and since we use it in the proof of the full Theorem 1.2 and strive to make this work self-contained, we give a complete proof in Section 2.…”

“…Much more general versions of this result can be envisioned, featuring for instance VMO-like second-order coefficients instead of continuous, unbounded measurable first-and zero-order coefficients, L p -viscosity solutions, power growth in the gradient, less regularity of the boundary, C 1,Dini -regularity type results. We refer to [12], [22], [18], and the extensive list of references in these works for similar regularity results of more general nature.…”

“…(2.13) (see for instance the proofs in the appendices of [11], [12]). We recall that by the celebrated [13, Theorem 2], if F (D 2 u(y), y) = f (y) in B 1 (x), then for any τ < α F we have the interior estimate…”

Regularity estimates for fully nonlinear elliptic PDEs with general Hamiltonian terms and unbounded ingredients

“…Over the past years, fully nonlinear operators with nonlinear gradient growth have been widely recognized and investigated, in particular due to their importance in applications. For instance, several research interests include free boundary problems of obstacle type [27], Phragmén-Lindelöf type results [47]; multiplicity of solutions for nonproper equations with natural growth in the gradient [38]; stochastic homogenization of Hamilton-Jacobi models [1], Hölder estimates for degenerate PDEs with coercive Hamiltonians [11], and boundary regularity for equations and differential inequalities [5], among others.…”