Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

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

“…From the above discussion, there exists a viscosity solution $$u_\varepsilon$$ to () such that $$u_1\leqslant u_{\varepsilon }\leqslant u_2$$ and $$u_\varepsilon =g$$ on $$\partial \Omega$$. Since the sequence $$(u_\varepsilon )_{0<\varepsilon <1}$$ is uniformly bounded in $$\mathcal {C}^{0, \gamma }_{loc}(\Omega )$$ (see [20] and [23]), through a subsequence if necessary, we have that $$u_\varepsilon$$ converges to a function $$u_{\infty }$$ as $$\varepsilon \rightarrow 0$$. From standard stability results (see Lemma 2.1), we conclude that $$u_\infty$$ solves …”

“…Since the sequence (𝑢 𝜀 ) 0<𝜀<1 is uniformly bounded in  0,𝛾 𝑙𝑜𝑐 (Ω) (see [20] and [23]), through a subsequence if necessary, we have that 𝑢 𝜀 converges to a function 𝑢 ∞ as 𝜀 → 0. From standard stability results (see Lemma 2.1), we conclude that 𝑢 ∞ solves…”

“…Aftermath, da Silva-Ricarte in [21] improved De Filippis' result and addressed a variety of applications in nonlinear elliptic models and related free boundary problems. Concerning fully nonlinear models of (single) degenerate type (i.e., 𝔞 ≡ 0 in (1.7)) the list of contributions is fairly diverse, including aspects such as existence/uniqueness issues, Harnack inequality, Alexandroff-Bakelman-Pucci (ABP) estimates [20] and [9], Liouville-type results [19], local Hölder and Lipschitz estimates, local gradient estimates [3,[10][11][12], and [30], as well as connections with a variety of free boundary problems of Bernoulli type [20], obstacle type [23], singular perturbation type [4,9] and [37], and dead-core type [19], just to name a few.…”

“…Such a quantitative information plays an essential role in the development of several analytic and geometric problems, such as in the study of blow‐up procedures and related weak geometric and free boundary analysis (cf. [20] and [22] for related topics). This is the first nondegeneracy result for nonlinear degenerate models with variable exponent in the current literature.…”

Nonuniformly elliptic problems

“…These measures have eventually proven to be relevant in various issues, as for instance removability of singularities [50] and in symmetry problems [25,26]. Double phase degeneracies also appear in the setting of fully nonlinear problems, as first considered in [70] and then in [21,66,67,72]. After the initial contribution in [85], nonlocal double phase problems were considered in [39,40,200].…”