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Singularly perturbed equations of degenerate type
Abstract
This work is devoted to the study of nonvariational, singularly perturbed elliptic equations of degenerate type. The governing operator is anisotropic and ellipticity degenerates along the set of critical points. The singular behavior is of order along ϵ-level layers , and a non-homogeneous source acts in the noncoincidence region . We obtain the precise geometric behavior of solutions near ϵ-level surfaces, by means of optimal regularity and sharp geometric nondegeneracy. We further investigate Hausdorff measure properties of ϵ-level surfaces. The analysis of the asymptotic limits as the ϵ parameter goes to zero is also carried out. The results obtained are new even if restricted to the uniformly elliptic, isotropic setting.
... Hence, we are led to consider reaction terms fulfilling (1.3) B 0 ≤ ζ ε (x,t) ≤ A ε χ (0,ε) (t) + B, ∀ (x,t) ∈ Ω × R + , for nonnegative constants A , B 0 , B ≥ 0. Notice that ζ ε ≡ 0 satisfies (1. 3). Nevertheless, we shall also impose the following non-degeneracy assumption in order to ensure that such a reaction term enjoys an authentic singular character: ...
... Finally, it is readily verifiable that the reaction term in (1.5) does fulfill (1. 3 ...
... The proof of Linear growth consists of combining the construction of an appropriate barrier function with the minimality of Perron solutions. Such an instrumental idea was first introduced in the last author's works [3] and [58] for the fully nonlinear scenario. ...
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 ε > 0 fixed, we seek a non-negative function u ε satisfying [|∇u ε | p + a(x)|∇u ε | q ] ∆u ε = ζ ε (x,u ε) in Ω, u ε (x) = g(x) on ∂ Ω, in the viscosity sense for suitable data p,q ∈ (0,∞), a, g, where ζ ε one behaves singularly of order O ε −1 near ε-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. Particularly, for a restricted class of non-linearities, we prove the finiteness of the (N − 1)-dimensional Hausdorff measure of level sets. At the end, we address a complete and in-deep analysis concerning the asymptotic limit as ε → 0 + , which is related to one-phase solutions of inhomogeneous non-linear free boundary problems in flame propagation and combustion theory. Finally, we also present some fundamental regularity tools in the theory of doubly degenerate fully nonlinear elliptic PDEs, which may have their own mathematical interest.
... the list of contributions is fairly diverse, including aspects such as existence/uniqueness issues, Harnack inequality, ABP estimates [30] and [27], Liouville type results [28], local Hölder and Lipschitz estimates, local gradient estimates [5], [17], [18], [19], and [44], as well as connections with a variety of free boundary problems of Bernoulli type [30], obstacle type [34] and [35], singular perturbation type [6], [27] and [58], and dead-core type [28], just to name a few. At this point, we must quote the series of Berindelli-Demengel's key works [17], [18] and [19], where local gradient regularity to (1.7) have been extended to up-to-the-boundary C 1,γ estimates in the presence of sufficiently regular domain and boundary datum. ...
... [30]); Singularly perturbed problems (cf. [6], [27] and [58]). ...
... In turn, the FBP considered in (5.10) appears as the limit of certain inhomogeneous singularly perturbed problems in the non-variational framework of high energy activation model in combustion and flame propagation theories ( cf. [23] and [65] for the stationary divergence setting and [6], [57] and [58] for related non-divergence topics), whose simplest model case is given by we seek a non-negative profile u ε (for each ε > 0 fixed) satisfying |∇u ε | pε(x) + a ε (x)|∇u ε | qε(x) ∆u ε = ζ ε (u ε ) + f ε (x) in Ω, u ε (x) = g(x) on ∂Ω, (5.11) in the viscosity sense, for suitable data p ε (·), q ε (·), a ε (·), g, where ζ ε (s) := 1 ε ζ s ε one behaves singularly of order o ε −1 near ε-layer surfaces. In such a scenario, existing solutions are globally (uniformly) Lipschitz continuous (see [27,Theorem 1.4] and [57, Theorem 1] for specific results) such that ∇u ε L ∞ (Ω) ≤ C(n, (pε) min , (qε)max, ζ L ∞ (Ω) , aε L ∞ (Ω) , g C 1,κ (∂Ω) , fε L ∞ (Ω) , Ω). ...
We establish the existence and sharp global regularity results ($C^{0, \gamma}$, $C^{0, 1}$ and $C^{1, \alpha}$ estimates) for a class of fully nonlinear elliptic PDEs with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function $\mathfrak{a}(\cdot)\ge 0$. The model case in question is given by
$$
\left\{
\begin{array}{rcrcl}
\left[|Du|^{p(x)}+\mathfrak{a}(x)|Du|^{q(x)}\right]\mathscr{M}_{\lambda, \Lambda}^{+}(D^2 u)& = & f(x) & \text{in} & \Omega \\
u(x) & = & g(x) & \text{on} & \partial \Omega.
\end{array}
\right.
$$
for a bounded, regular and open set $\Omega \subset \R^n$, and appropriate continuous data $p(\cdot), q(\cdot)$, $f(\cdot)$ and $g(\cdot)$. Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models and free boundary problems.
... Therefore, the diffusion process exhibit a non-uniformly elliptic and doubly degenerate feature, which mixes up two power-degenerate type operators (cf. [ART15], [ART17], [BD14], [BD15], [BDL19], [daSLR21], [daSV20], [daSV21], [IS12] and [IS16] for related regularity estimates and free boundary problems driven by second order operators with a single degeneracy law). ...
... In turn, the FBP considered in (1.1) also appears as the limit of certain inhomogeneous singularly perturbed problems in the non-variational context of high energy activation model in combustion and flame propagation theories (cf. [ART17], [RS15] and [RT11] for related topics), whose simplest mathematical model (in this case) is given by: for each ε > 0 fixed, we seek a non-negative profile u ε satisfying ...
... [LR18I]). Particularly, we may to apply our findings to limit profiles of inhomogeneous singular perturbation problems for fully nonlinear operators of degenerate type, which they were addressed in [ART17] by third author et al. ...
We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such non-linear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli's classification scheme: Flat and Lipschitz free boundaries are locally $C^{1, \beta}$ for some $0 < \beta (universal) < 1$.
... Finally, it is readily verifiable that the reaction term in (1.5) does fulfill (1. 3 ...
... The proof of Linear growth consists of combining the construction of an appropriate barrier function with the minimality of Perron solutions. Such an instrumental idea was first introduced in the last author's works [3] and [58] for the fully nonlinear scenario. ...
... In conclusion, it is worth highlighting that our findings extend regarding non-variational scenario, former results from [3], [55], [58] and [61], and to some extent, those from [20], [35], [51], [53], [54] and [63], concerning degenerate and variational models, by making using of different systematic approaches and techniques adjusted to the framework of fully non-linear models with non-homogeneous degeneracy. Moreover, they are new even for the toy model: ...
In our work we study non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In such a scenario, we establish the existence of 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. Moreover, for a restricted class of non-linearities, we prove the finiteness of the (N-1)-dimensional Hausdorff measure of level sets. We also address a complete analysis concerning the asymptotic limit as the singular parameter, which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory.
... has also attracted great attention in the past years, see [11,12,21,26,7,8,38] among several other works on this subject. By now we have a fairly good understanding of the underlying regularity theory for solutions of equation (6). ...
... Here we explore the structure of (2). In particular, we examine its scaling properties that allow us to work under the conditions in (8). Suppose that u solves the equation (2) and consider v : ...
... we ensure that v solves an equation in the same class as (2), and it is under the smallness regime prescribed in (8). Estimates proven for v gets transported to u by factors that depend explicitly on K and τ . ...
In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the $p(x)$-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class $C^{1,\kappa}$ for a universal constant $0< \kappa < 1$. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.
... Oper.) appears, for example, in singularly perturbed free boundary problems ruled by fully non-linear equations, where the Hessian of solutions blows-up through the phase transition. For this very reason, the limiting free boundary condition is ruled by the F # rather than F (cf. [1], [10] and [24] for some enlightening examples). ...
... convexity) is precisely when b, c, ζ F = 0. In a geometric view-point as b = c = 0, such a condition means that for each (x, ξ, ι) ∈ Ω × R n × R fixed, there exists a hyperplane which decomposes R × Sym(n) in two semi-spaces such that the graph of F (·, x) is always below this hyperplane (see [1], [10], [24] and references therein for some motivations and other details). ...
... We stress out that such a class of operators plays a crucial role in order to establish finiteness of (n − 1)−Hausdorff measure in several fully non-linear free boundary problems (cf. [1], [7], [8], [10] and [24] for some enlightening examples of such an application). ...
In this manuscript we establish global Sobolev \textit{a priori} estimates for $L^p-$viscosity
solutions of fully nonlinear elliptic equations as follows
$$
\left\{
\begin{array}{rclcl}
F(D^2u,Du,u,x) &=& f(x)& \mbox{in} & \Omega\\
u(x)&=& \varphi(x) & \mbox{on} & \partial \Omega
\end{array}
\right.
$$
by considering minimal integrability condition on the data, i.e.,
$f \in L^p(\Omega), \varphi \in W^{2, p}(\Omega)$ for $n<p<\infty$ and a regular domain
$\Omega \subset \R^n$, and relaxed structural assumptions (weaker than convexity) on the governing operator.
Our approach makes use of techniques from geometric tangential analysis,
which consists in transporting ``fine'' regularity estimates from a limiting operator,
the \textit{Recession} profile, associated to F to the original operator via compactness methods.
We devote special attention to the borderline case, i.e., when $f \in p-\text{BMO} \supsetneq L^{\infty}$.
In such a scenery, we show that solutions admit $\text{BMO}$ type estimates for their second derivatives.
... . They prove several analytical and geometrical properties of solutions (see also [14] for global regularity character and [12] for an approach with inhomogeneous forcing term). A non-variational setting of the problem was studied in [1], where the authors obtain existence and optimal regularity results for the class of fully nonlinear, anisotropic degenerate elliptic problems ...
... We state the following theorem independently of the (E ε ) context, since it may be of independent interest. For the proof we refer to [15] (see also [1]). Theorem 2.1. ...
... Geometrically, it means that for each x ∈ Ω fixed, there exists a hyperplane which decomposes R × Sym(n) in two semi-spaces such that the graph of F (x, ·) is always below this hyperplane. Moreover, by assuming F (x, 0) = 0, the assumption (5.2) means that the distance from the hyperplane to the graph of F goes to infinity for matrices with big enough norms (see [1] and [13]). ...
We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly non-degenerate and have porous level surfaces. Moreover, for some restricted cases we show the finiteness of the (n-1)-dimensional Hausdorff measure of level sets. The analysis of the asymptotic limits is carried out as well.
... which was studied in [1] and [5], where the authors obtain several analytical and geometrical properties for viscosity solutions and their free boundaries. ...
... Next result is a consequence of Theorem 3 as ε → 0 + . 1 A universal constant is one which depends only on dimension and the parameters A and B. ...
We study an inhomogeneous nonlinear singularly perturbed problems related to non-variational operators enjoying a doubly degenerate character with prescribed boundary condition. In this scenario, we prove that solutions are locally Lipschitz continuous, grow as a linear function and are strongly non-degenerate. At the end, an analysis of the asymptotic limits is carried.
... See for example [10,27,28] for linear operators, [7,22,23,25,48,51,52] for fully nonlinear operators and [42,43] for the p-Laplacian. See also [6]. ...
... (1. 6) In Theorem 1.1 the constantsε and α depend only on p min , p max and n (the dimension of the space). ...
We consider viscosity solutions to a one-phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We apply the tools developed in De Silva (2011) to prove that flat free boundaries are C1,α. Moreover, we obtain some new results for the operator under consideration that are of independent interest.
... which was studied in [1], where the authors obtain several analytical and geometrical properties for viscosity solutions and their free boundaries. ...
... By combining Corollary 1 and the Lipschitz regularity from Theorem 1 we obtain the following result. 1 A universal constant is one which depends only on dimension and the parameters A, B and C. Theorem 5 Let Ω Ω. Fix x 0 ∈ {u 0 > 0} ∩ Ω such that dist(x 0 , F(u 0 , Ω )) ≤ dist(Ω , ∂Ω). Then there exists a universal constant C > 0 such that C −1 .dist(x ...
We study an inhomogeneous singularly perturbed problem related to Infinity Laplacian operator with prescribed boundary condition. We prove that solutions are locally Lipschitz continuous, grow as a linear function and are strongly non-degenerate. An analysis of the asymptotic limits is carried at the end.
... Such a kind of information is crucial in a number of quantitative features for many free boundary problems (cf. [5], [22], [35] and [40]). Precisely, we prove that (near their free boundaries) solutions are "trapped" between the graph of two suitable multiples of dist(·, ∂ {u > 0}). ...
... 5, Section 6],[20, Theorem 1.3],[22, Section 5] and[35, Section 6] for similar insights). Moreover, by technical considerations we must suppose the following upper bound control: |b| ≤ mC −(γ+1) , where C > 0 is the universal constant from sharp gradient estimate (Theorem 1.4), and max 0, γ 2 ...
In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators (which can be either degenerate or singular when ``the gradient is small'') under strong absorption condition of the general form:
$$
\mathfrak{G}(x, D u, D^2 u) = f(u)\chi_{\{u>0\}} \quad \mbox{in} \quad \Omega,
$$
where the mapping $u \mapsto f(u)$ fails to decrease fast enough at the origin, so allowing that non-negative solutions may create plateau regions, that is, \textit{a priori} unknown subsets where a given solution vanishes identically. We establish improved geometric $C_{\text{loc}}^{\kappa}$ regularity along the set $\mathfrak{F}_0 = \partial \{u>0\} \cap \Omega$ (the free boundary of the model). The sharp value of $\displaystyle \kappa \gg 1$ is obtained explicitly, and depends only on structural parameters. Non-degeneracy, finiteness of the $(N-1)$-Hausdorff measure of the free boundary and among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings. The results proven in this article are new even for simple equation
$$
|Du|^{\gamma}\Delta u = \lambda_0u^{\mu}\chi_{\{u>0\}},
$$
for certain constants $\lambda_0>0$, $\mu \in [0, \gamma+1)$ and $0 \neq \gamma>-1$. Finally, this work extends previous ones by allowing deal with more general classes of operators, making use of different methods.
... Global counterpart of such local regularity results can be found in [10] for (x, t) = t p with p ≥ 0 and in [6] for (x, t) = t p(x) + a(x)t q(x) with 0 ≤ p(·) ≤ q(·). It is noteworthy that the regularity theory for viscosity solutions to (1.1) plays a crucial role in the investigation of the free boundary problems of singular perturbation type [3,7], of obstacle type [16,17], and of one-phase Bernoulli type [14]. ...
We provide the Alexandroff–Bakelman–Pucci estimate and global C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1, \alpha }$$\end{document}-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
... Existence of minimal solutions. As commented before, we shall deal with solutions for (1.1) which are limits of mininal Perron's solutions for (P ε ) as ε → 0. Note that the lack of monoticity in equation (P ε ) with respect to the variable u does not allow us to make use of a direct application of the classical Perron method, and so, following existence results as in [3], we derive existence of viscosity solutions for the problem (P ε ). For the sake of the reader's convenience, we mention the following result. ...
This work is devoted to the study of a novariational class of singular elliptic equations ruled by the infinity Laplacian. We provide optimal $C^{1,\beta}$ regularity along the singular free boundary, where in particular, the optimality is designed by the magnitude of the singularity. By virtue of a suitable penalization scheme and a construction of radial super-solutions, existence and non-degeneracy properties are provided where fine geometric consequences for the singular free boundary are further obtained.
... Regarding non-variational scenario, i.e., fully nonlinear models such as (1.1), their regularity properties have been the object of intense investigation over the last decade due to their proper connection to several issues arising in pure mathematics, as well as a variety of geometric and free boundary problems (see e.g. [3,4,[13][14][15][16]27,[34][35][36] and [44] for an extensive but incomplete list of the latest contributions). ...
We establish sharp geometric Holder regularity estimates for Gradient for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy. Such regularity estimates simplify and generalize, to some extent, earlier ones via a 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 problems in the theory of elliptic PDEs, which may have their own interest. We also deliver explicit examples where our results are sharp.
... Such a kind of information is crucial in a number of quantitative features for many free boundary problems (see e.g. [6], [22], [35] and [40]). Precisely, we prove that (near their free boundaries) solutions decay like an appropriated power of dist(·, ∂ {u > 0}). ...
In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value depending only on structural parameters. Non degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.
We establish the existence and sharp global regularity results (, and estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function . The model case in question is given by for a bounded, regular, and open set , and appropriate continuous data , , and . Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.
We consider a one-phase free boundary problem governed by doubly degenerate fully nonlinear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such nonlinear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli’s classification scheme: Flat and Lipschitz free boundaries are locally C1,β for some 0 < β(universal) < 1. Particularly, our findings are new even in the toy model $${{\cal G}_{p,q}}\left[u \right]: = \left[{{{\left| {\nabla u} \right|}^p} + \mathfrak{a}\left(x \right){{\left| {\nabla u} \right|}^q}} \right]\Delta u,\,\,\,\,\,{\rm{for}}\,\,0 < p < q {{\cal G}_{p,q}}\left[u \right]: = \left[{{{\left| {\nabla u} \right|}^p} + \mathfrak{a}\left(x \right){{\left| {\nabla u} \right|}^q}} \right]\Delta u,\,\,\,\,\,{\rm{for}}\,\,0 < p < q < \infty \,\,{\rm{and}}\,\,0 \le \mathfrak{a} \in {C^0}\left(\Omega \right) \infty \,\,{\rm{and}}\,\,0 \le \mathfrak{a} \in {C^0}\left(\Omega \right)$$We also bring to light other interesting doubly degenerate settings where our results still work. Finally, we present some key tools in the theory of degenerate fully nonlinear PDEs, which may have their own mathematical importance and applicability.
In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} regularity along the free boundary. We also show existence of solutions, nondegeneracy properties and fine geometric estimates for the free boundary.
In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets. We work under a proximity regime on the exponent governing the nonlinearity of the problem. We prove that solutions are locally of class C1−,12− along free boundary points x0 ∈ ∂{u > 0}. Our argument consists of importing information from the heat equation, through approximation and localization methods.
We consider viscosity solutions to a one-phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We apply the tools developed in \cite{D} to prove that flat free boundaries are $C^{1,\alpha}$. Moreover, we obtain some new results for the operator under consideration that are of independent interest.
In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the p(x)-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class C1,κ for a universal constant 0<κ<1. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.
We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
In this paper we deliver improved C
1, α
regularity estimates for solutions to fully nonlinear equations F(D
2u) = 0, based on asymptotic properties inherited from its recession function \(F^{\star }(M):=\lim \limits _{\mu \rightarrow 0}\mu F(\mu ^{-1}M)\).
We consider a large class of degenerate or singular operators F defined on ℝ×(ℝ N ) * ×S, where S denotes the space of symmetric matrices on ℝ n , F is continuous. We give a new definition of viscosity sub and super solutions for F(x,∇u,∇ 2 u)=0. We prove a comparison theorem between sup and supersolutions for F(x,∇u,∇ 2 u)=b(u) where b is an increasing function, and a Liouville type result.
We consider a function which is a viscosity solution of a uniformly elliptic
equation only at those points where the gradient is large. We prove that the
H\"older estimates and the Harnack inequality, as in the theory of Krylov and
Safonov, apply to these functions.
We establish new, optimal gradient continuity estimates for solutions to a
class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u,
D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the
\textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u)
:= \{X : \nabla u(X) = 0 \}$. The innovative feature of our main result
concerns its optimality -- the sharp, encoded smoothness aftereffects of the
operator. Such a quantitative information usually plays a decisive role in the
analysis of a number of analytic and geometric problems. Our result is new even
for the classical equation $|\nabla u | \cdot \Delta u = 1$. We further apply
these new estimates in the study of some well known problems in the theory of
elliptic PDEs.
In this work we develop a systematic geometric approach to study fully
nonlinear elliptic equations with singular absorption terms as well as their
related free boundary problems. The magnitude of the singularity is measured by
a negative parameter $(\gamma -1)$, for $0 < \gamma < 1$, which reflects on
lack of smoothness for an existing solution along the singular interface
between its positive and zero phases. We establish existence as well sharp
regularity properties of solutions. We further prove that minimal solutions are
non-degenerate and obtain fine geometric-measure properties of the free
boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp
Hausdorff estimates which imply local finiteness of the perimeter of the region
$\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of
$\mathfrak{F}$.
In the present paper, a class of fully non-linear elliptic equations are
considered, which are degenerate as the gradient becomes small. H\"older
estimates obtained by the first author (2011) are combined with new Lipschitz
estimates obtained through the Ishii-Lions method in order to get
$C^{1,\alpha}$ estimates for solutions of these equations.
This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D
2u) = f(X), based on the weakest and borderline integrability properties of the source function f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L
n
norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C
1,α
regularity estimates are also delivered when \({f\in L^{n+\varepsilon}}\) . The limiting upper borderline case, \({f \in L^\infty}\) , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that \({u \in C^{1,{\rm Log-Lip}}}\) , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the \({C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}\) norm of u based on the L
n-ε
norm of f, where ɛ is the Escauriaza universal constant. The exponent \({\frac{n-2\varepsilon}{n-\varepsilon}}\) is optimal. When the source function f lies in L
q
, n > q > n−ε, we also obtain the exact, improved sharp Hölder exponent of continuity.
We consider the Dirichlet boundary value problem for a singular elliptic PDE like F[u] = p(x)u
−μ
, where p, μ ≥ 0, in a bounded smooth domain of \({\mathbb{R}^n}\) . The nondivergence form operator F is assumed to be of Hamilton–Jacobi–Bellman or Isaacs type. We establish existence and regularity results for such equations.
We study existence of continuous weak (viscosity) solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.
In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it is possible to extend the concept of eigenvalue, this paper concerns the cases when the inf of the principal eigenvalues is positive i.e. when both the maximum and the minimum principle holds.
Free boundary problems are often approximated by regularizing ones. To obtain information about the solution to the original problem one tries to establish results for the approximating one which carry over in the limit. A useful reference is Friedman’s book [F].
This book provides a self-contained development of the regularity theory for solutions of fully nonlinear elliptic equations. Caffarelli and Cabré offer a detailed presentation of all techniques needed to extend the classical Schauder and Calderón-Zygmund regularity theories for linear elliptic equations to the fully nonlinear context.
The authors present the key ideas and prove all the results needed for the regularity theory of viscosity solutions of fully nonlinear equations. The book contains the study of convex fully nonlinear equations and fully nonlinear equations with variable coefficients. This book is suitable as a text for graduate courses in nonlinear elliptic partial differential equations.
In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζε(u) with ζε→δ0⋅∫ζ. We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F(D2u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.
In this paper Harnack's inequality is proved and the Hölder exponent is estimated for solutions of parabolic equations in nondivergence form with measurable coefficients. No assumptions are imposed on the smallness of scatter of the eigenvalues of the coefficient matrix for the second derivatives.
Bibliography: 9 titles.
We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu=Dj(aij(x)Diu)+biui+c(x)u=0 in {u>0}, 〈aij(x)∇u,∇u〉=2 on ∂{u>0}. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Luε=βε(uε), where βε is a suitable approximation of Dirac delta function δ0. A useful variational characterization to solutions of the above approximating problem is established and used to obtain important geometric properties that enable regularity of the free boundary. This theory has been developed in connection to a very recent line of research as an effort to study existence and regularity theory for free boundary problems with gradient dependence upon the penalization.RésuméNous développons une théorie variationnelle pour l'étude du problème de la régularité de la frontière libre pour des opérateurs elliptiques : Lu=Dj(aij(x)Diu)+biui+c(x)u=0 en {u>0}, 〈aij(x)∇u,∇u〉=2 en ∂{u>0}. Nous régularisons et approximons la frontière libre par une méthode de perturbation singulière de la forme : Luε=βε(uε), où βε est une approximation adaptée de la fonction delta de Dirac δ0. Une caractérisation variationnelle des solutions du problème d'approximation ci-dessus est établie et employée pour obtenir les propriétés géométriques importantes qui impliquent la régularité de la frontière libre. Cette théorie a été développée en connection avec une ligne très récente de recherche comme effort pour étudier la théorie d'existence et de régularité pour des problèmes de la frontière libre avec la dépendance de gradient sur la pénalisation.
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman–Isaacs equations from stochastic control problems. We establish an Alexandroff–Bakelman–Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations.
We prove the classical Alexandroff–Bakelman–Pucci estimate for fully nonlinear elliptic equations involving singular or degenerate operators having as models , where are the Pucci extremal operators with parameters 0<λ⩽Λ and α>−1. To cite this article: G. Dávila et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).RésuméNous prouvons l'inégalité classique d'Alexandroff–Bakelman–Pucci pour des équations elliptiques entièrement non linéaires avec des opérateurs singulières ou dégénérés ayant comme modèles où sont les opérateurs extremal de Pucci avec des paramètres 0<λ⩽Λ et α>−1. Pour citer cet article : G. Dávila et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).