We simplify and give an alternative proof of hypoellipticity for generalizations of the singular sum of squares of complex vector fields studied by Kohn, with an appendix by Derridj and Tartakoff, in the Annals of Mathematics, vol. 162 no. 2, 2005, pp. 943-986. The main generalization is that the complex vector fields now come from domains of finite type. We also prove real analytic hypoellipticity and the optimality of our estimates.
We prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of $\mathbb{R}^n$ space. In particular the result applies also to some operators in Carnot groups.
In this paper we prove local analytic hypoellipticity for a degenerate sum of squares of complex vector fields generalizing those of Kohn in "Hypoellipticity and Loss of Derivatives". Kohn's article is to appear in the Annals of Mathematics with an appendix by Derridj and Tartakoff proving local analyticity in that case. Here we consider vector fields corresponding to a CR manifold of finite type so there is a kind of double degeneracy, and prove local analyticity using our balanced localization of high derivatives in the "missing" direction T. In a forthcoming paper, Bove, Derridj, Kohn and T…
We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classification results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non-degenerate anisotropic equations.
Singular degenerate differential operator equations are studied. The uniform separability of boundary value problems for degenerate elliptic equation and optimal regularity properties of Cauchy problem for degenerate parabolic equation are obtained. These problems have a numeros applications which occur in fluid mechanics, environmental engineering and atmospheric dispersion of pollutants.
We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
The boundary value problems for linear and nonlinear singular degenerate differential-operator equations are studied. We prove a well-posedeness of linear problem and optimal regularity result for the nonlinear problem which occur in fluid mechanics, environmental engineering and in the atmospheric dispersion of pollutants.
We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.
Sobolev-type regularity results are proved for solutions to a class of second order elliptic equations with a singular or degenerate weight, under non-homogeneous Neumann conditions. As an application a Pohozaev-type identity for weak solutions is derived.
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are expressed through generalized hypergeometric functions. Sufficient conditions are obtained under which the number of linearly independent solutions is equal to the order of the equation.
In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-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.
We consider non smooth general degenerate/singular parabolic equations in non divergence form with degeneracy and singularity occurring in the interior of the spatial domain, in presence of Dirichlet or Neumann boundary conditions. In particular, we consider well posedness of the problem and then we prove Carleman estimates for the associated adjoint problem.
We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge-Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.
We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an $L^1$-contraction result. In addition, we analyze systems where degenerate equations are coupled to semilinear reaction diffusion equations. This setting includes mathematical models for biofilm growth which are the motivation for our analysis.
We study existence and uniqueness of solutions to a class of nonlinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To this purpose some barrier functions are properly introduced and used.
In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Ampère type equations which may be degenerate and singular on the boundary of convex domains. We will establish a relation of the H$\ddot{o}$lder exponent for the solutions with the convexity for the domains.
We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.
This paper concerns Hopf's boundary point lemma, in certain $C^{1,Dini}$-type domains, for a class of singular/degenerate PDE-s, including $p$-Laplacian. Using geometric properties of levels sets for harmonic functions in convex rings, we construct sub-solutions to our equations that play the role of a barrier from below. By comparison principle we then conclude Hopf's lemma.
In this paper we obtain sharp Petrovskii criteria for the p-parabolic equation, both in the degenerate case p>2 and the singular case 1<p<2. We also give an example of an irregular boundary point at which there is a barrier, thus showing that regularity cannot be characterized by the existence of just one barrier.
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$Δu = -χ_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {\em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss.
We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.
We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear $p$-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide global-in-time error estimates on the difference between the solution of our problem and the solution of the corresponding first order problem.
In this note, we prove $\mathcal{C}^{1,γ}$ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of \cite{BDL1} concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.
We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure.
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate 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 degenerate elliptic equations.
We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.
We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^α(x,\partial{}B)Φ_p(\nabla u)) = b(x)f(u)$ in a ball B, where $f$ is normalized regularly varying at infinity with index $σ+1>p-1,\ p>1$. In particular, how the asymptotic behavior of the solution changes over the varying index and degeneracy and/ or singularity present in the equation. We also include the second order blow-up rate for the corresponding semilinear problem.
In this paper we extend some existence's results concerning the generalized eigenvalues for fully nonlinear operators singular or degenerate. We consider the radial case and we prove the existence of an infinite number of eigenvalues, simple and isolated. This completes the results obtained by the author with Isabeau Birindelli for the first eigenvalues in the radial case, and the results obtained for the Pucci's operator by Busca Esteban and Quaas and for the $p$-Laplace operator by Del Pino and Manasevich.
The existence of strong solutions to general class of strongly coupled parabolic systems will be discussed. These systems can be degenerate or singular as boundedness of theirs solutions are unavailable and not assummed. The results greatly improve those in a recent papers \cite{letrans,dleJFA,dleANS} as the systems can have quadratic growth in gradients. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA,dleANS} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Degenerate and singular…
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - Δ_x)^s$ for $s \in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($a=0$).
We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.
A general class of strongly coupled elliptic systems with quadratic growth in gradients is considered and the existence of their strong solutions is established. The results greatly improve those in a recent paper \cite{dleJFA} as the systems can be either degenerate or singular when their solutions become unbounded. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Examples in physical models will be provided.
The degenerate Cahn-Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn-Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second order equations which can be solved with standard packages. This system is endowed with an ener…
We consider the Wulff-type energy functional $$ \mathcal{W}_Ω(u) := \int_ΩB(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential $F(u)$ and we deduce several rigidity and symmetry properties.
We establish the interior $C^{1,α}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^γF(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left\{ \begin{aligned} u_t&=(1+|Du|^2)^{γ/2}F(D^2u) &&\text{in $Q_1$} \newline u&=φ&&\text{on $\partial_p Q_1$}. \end{aligned}\right. \end{equation*} Our approach utilizes the Bernstein method with approximations in view of difference quotient.
We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence o…
In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right hand side. \begin{equation}\label{01} \left\{ \begin{array}{lll} -\displaystyle\mbox{div}( a(x,u,\nabla u))&=\displaystyle\frac{f}{u^γ} & \mbox{ in } Ω\\ u&>0 &\mbox{ in }Ω\\ u&=0 &\mbox{ on } δΩ\end{array} \right. \end{equation} where $Ω$ is bounded open subset of $I\!\!R^{N}(N\geq2),$ $γ>0$ and $ f$ is a nonnegative function that belongs to some Lebesgue space.
We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along $S(u)$, $u$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $S(u)$, $Du$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. Our main, leading result fosters a new understanding on smoothness properties of solutions to degenerate or sin…
We prove interior Hölder estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$ u_t=|\nabla u|^κ\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ where $p\in (1,\infty)$ and $κ\in (1-p,\infty).$ This includes the from $L^\infty$ to $C^{1,α}$ regularity for parabolic $p$-Laplacian equations in both divergence form with $κ=0$, and non-divergence form with $κ=2-p$. This work is a continuation of a paper by the last two authors \cite{JS}.
In this paper we prove some existence and regularity results concerning parabolic equations dtu = F(D u, D2 u) + f(x,u) with some boundary conditions, on Omega times ]0,T[, where Omega is some bounded domain which possesses the cone property and $F$ is singular or degenerate, with some uniform ellipticity conditions.
In this paper we consider equations $-| \nabla u |^αF ( D^2 u) = |u|^{p-1} u $ in an annulus. $F$ is Fully Nonlinear Elliptic, $α$ is some real $> -1$ and $p > 1+ α$. The solutions are intended in the sense of the definition given by Birindelli and Demengel \cite{BD1}. We prove the existence of non trivial positive/negative radial solutions.
