Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Abstract

We consider the fully nonlinear equation with variable-exponent double phase type degeneraciesUnder some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques, we obtain the sharp local C 1,α regularity of viscosity solutions to such equations.

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

“…Our findings extend/generalize regarding nonvariational scenario, former results (Hölder gradient estimates) from [3, Theorem 3.1 and Corollary 3.2], [10, 11] and [30], and to some extent, those from [12, Theorem 1.1], [13, Theorem 2.1], [24, Theorem 1] and [27, Theorem 1.1] by making using of different approaches and techniques adapted to the general framework of the fully nonlinear nonhomogeneous degeneracy models.…”

“…Another important piece of information we need in our article concerns to the notion of stability of viscosity solutions. The following lemma will be instrumental in the proof of Lemma 4.1, whose proof can be found in [12] and [27, Lemma 3.2]. Lemma Let $$(g_j)_j$$ be a sequence of Lipschitz continuous functions such that $$g_j \rightarrow g_{\infty }$$.…”

Fully nonlinear free transmission problems with nonhomogeneous degeneracies

“…For this reason, all the bounding constants appearing in the preliminary compactness estimates are carefully tracked in order to guarantee that none of them depend on the coefficients, which is coherent with the findings of [12,22,26,32,35]. Within the smallness framework, delicate perturbation arguments allow us to build a tangential path connecting viscosity subsolutions/supersolutions of (1.9)-(1.10) to a viscosity solution of the limiting profile-a homogeneous problem of the form…”

“…This new model received lots of attention recently in the setting of free boundary problems, nonhomogeneous 1-laplacian equations or obstacle problems, cf. [20][21][22], while in [32] the authors carefully combine the approaches of [12,26] to derive local C 1;˛0 -regularity for the viscosity solution of the fully nonlinear equation with variable exponents and nonhomogeneous degeneracy OEjDuj p.x/ C a.x/jDuj q.x/ F .D 2 u/ D f .x/ in : (1.5)…”

Fully nonlinear free transmission problems with nonhomogeneous degeneracies

“…This new model received lots of attention recently in the setting of free boundary problems, nonhomogeneous ∞-laplacian equations or obstacle problems, cf. [20][21][22]; while in [32] the authors carefully combine the approaches of [11,26] to derive local C 1,α0 -regularity for viscosity solution of the fully nonlinear equation with variable exponents and nonhomogeneous degeneracy [|Du| p(x) + a(x)|Du| q(x) ]F (D 2 u) = f (x)…”