Abstract
This work is devoted to studying non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In its simplest form, for each \varepsilon>0 fixed, we seek a non-negative function u^{\varepsilon} satisfying \begin{cases} \left[|\nabla u^{\varepsilon}|^p + \mathfrak{a}(x)|\nabla u^{\varepsilon}|^q \right] \Delta u^{\varepsilon} = \zeta_{\varepsilon}(x, u^{\varepsilon}) & \operatorname{in } \Omega,\\ u^{\varepsilon}(x) = g(x) & \operatorname{on } \partial \Omega, \end{cases} in the viscosity sense for suitable data p, q \in (0, \infty) , \mathfrak{a} and g , where \zeta_{\varepsilon} behaves singularly as \text{O} (\varepsilon^{-1}) near \varepsilon -level surfaces. In such a context, we establish the existence of certain solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. In particular, for a restricted class of nonlinearities, we prove the finiteness of the (N-1) -dimensional Hausdorff measure of level sets. We also address a complete and in-deep analysis concerning the asymptotic limit as \varepsilon \to 0^{+} , which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory. Finally, we present some fundamental regularity tools in the theory of doubly degenerate fully nonlinear elliptic PDEs, which may have their own mathematical interest.
Highlights
In this manuscript we shall develop an approach to study sharp and geometric estimates of one-phase solutions to singularly perturbed problems having a nonhomogeneous double degeneracy, whose mathematical model is given by: Fixed a parameter ε ∈ (0, 1), we would like to find (Pε) uε ≥ 0 viscosity solution toH (x, ∇uε )F(x, D2uε ) = ζε (x, uε ) in Ω uε(x) = g(x) on ∂ Ω, for a bounded and open set Ω ⊂ RN, where 0 ≤ g ∈ C0(∂ Ω), and F is a second order, fully non-linear operator, i.e., non-linear in its highest derivatives.We will focus our attention to reaction-diffusion models with singular behavior of order O ε−1 near ε-level layers, i.e. {uε ∼ ε}
Under the appropriated hypothesis on data, we show that, for ε → 0+, the family of solutions {uε}ε>0 to (Pε) are asymptotic approximations to a one-phase solution u0 of an inhomogeneous non-linear free boundary problem, which arises in the mathematical formulation of some issues in flame propagation and combustion theory
We will start with the definition of the viscosity solution to
Summary
In this manuscript we shall develop an approach to study (locally) sharp and geometric estimates of one-phase solutions to singularly perturbed problems having a nonhomogeneous double degeneracy, whose mathematical model is given by: Fixed a parameter ε ∈ (0, 1), we would like to find (Pε) uε ≥ 0 viscosity solution to. We will focus our attention to reaction-diffusion models with singular behavior of order O ε−1 near ε-level layers, i.e. In our research, the reaction term, i.e. ζε : Ω×R+ → R+, represents the singular perturbation of the model. In this point, we are interested in a singular behaviour of order. We shall impose the following non-degeneracy assumption in order to ensure that such a reaction term enjoys an authentic singular character:. Simpler cases covered by our analysis are singular reaction terms built up as a multiple of the approximation of unity plus a uniform bounded function (1.5).
Published Version
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