Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications
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Calculus of Variations
Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications João Vitor da Silva1 · Gleydson C. Ricarte2 Received: 12 May 2020 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 1,β We establish sharp Cloc geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy, whose model equation is given by 2 |Du| p + a(x)|Du|q M+ λ, (D u) = f (x, u) in , for a bounded and open set ⊂ R N , and appropriate data p, q ∈ (0, ∞), a and f . Such regularity estimates simplify and generalize, to some extent, earlier ones via totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear models in the theory of elliptic PDEs, which may have their own interest. We also deliver explicit examples where our results are sharp. Mathematics Subject Classification 35B65 · 35J60 · 35J70 · 35R35
1 Introduction 1.β
In this work we shall derive sharp Cloc geometric regularity estimates for solutions of a class of nonlinear elliptic equations having a non-homogeneous double degeneracy, whose mathematical model is given by H(x, Du)F(x, D 2 u) = f (x, u) in ,
(1.1)
Communicated by N.Trudinger.
B
João Vitor da Silva [email protected] Gleydson C. Ricarte [email protected]
1
Departamento de Matemática Instituto de Ciências Exatas, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Brasília, DF CEP: 70910-900, Brazil
2
Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, Fortaleza, CE CEP: 60455-760, Brazil 0123456789().: V,-vol
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for a β ∈ (0, 1), a bounded and open set ⊂ R N , f ∈ C 0 ( × R) ∩ L ∞ ( × R), where F is assumed to be a second order, fully nonlinear (uniformly elliptic) operator, i.e., it is nonlinear in its highest (second) order derivatives. Throughout this work we will suppose the following structural conditions: (A0) (Continuity and normalization condition) Fixed x → F(x, ·) ∈ C 0 (Sym(N )) and F(·, O N ×N ) = 0. (A1) (Uniform ellipticity) For any pair of matrices X, Y ∈ Sym(N ) + M− λ, (X − Y) ≤ F(x, X) − F(x, Y) ≤ Mλ, (X − Y)
where M± λ, are the Pucci’s extremal operators given by M− ei + ei and M+ ei + λ ei λ, (X) := λ λ, (X ) := ei >0
ei 0
ei 0 such that x, x0 → F (x, x0 ) :=
|F(x, X) − F(x0 , X)| ≤ CF ω(|x − x0 |), X X∈Sym(N ) sup
X =0
which measures the oscillation of coefficients of F around x 0 . For simplicity purposes, we shall often write F (x, 0) = F (x). Finally, for notation purposes we define F (x, x0 ) ≤ CF , ∀ x, x0 ∈ , x = x0 . F C ω () := inf CF > 0 : ω(|x − x0 |) In our studies, we enforce that the diffusion properties
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