Lipschitz regularity results for nonlinear strictly elliptic equations and applications

Abstract : Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.
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Olivier Ley, Vinh Duc Nguyen. Lipschitz regularity results for nonlinear strictly elliptic equations and applications. Journal of Differential Equations, Elsevier, 2017, 263 (7), pp.4324-4354. ⟨10.1016/j.jde.2017.05.020⟩. ⟨hal-01344438⟩

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