We examine $ L^p $-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $ p_0 < p < d $, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from – and are inspired by – fundamental facts in the theory of $ L^p $-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione [
We prove the uniqueness property for a class of entire solutions to the equation
where $ sigma $ is a nonnegative locally finite measure in $ mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {rm div}, mathcal{A}(x, nabla u) $ is the $ mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 < p < infty $) on $ mathcal{A}(x, xi) $ ($ x, xi in mathbb{R}^n $); the model case $ mathcal{A}(x, xi) = xi | xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ Delta_p $ on $ mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem,
in the sub-natural growth case $ 0 < q < p-1 $, where $ mu, sigma $ are nonnegative locally finite measures in $ mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ mathcal{A}(x, xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.
We study minimizers of non-autonomous functionals
when $ varphi $ has generalized Orlicz growth. We consider the case where the upper growth rate of $ varphi $ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $ varphi $ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.
We obtain density estimates for the free boundaries of minimizers $ u ge 0 $ of the Alt-Phillips functional involving negative power potentials
These estimates remain uniform as the parameter $ gamma to 2 $. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $ gamma to 2 $. The results are based on the $ Gamma $-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional
considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.
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