In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to exhibit, possibly, the shortest and most elegant proofs and to give a complete and self-contained overview of the theory.
The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem $(P)$ (See below) is addressed. Secondly, we investigate the question of $H^s$ versus $C^0$- weighted minimizers of the functional associated to problem $(P)$ and then give applications to existence and multiplicity results.
We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the study of extremals of Poincaré inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Hölder continuity in time of order $(p-1)/p$ for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.
We demonstrate two proofs for the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-Δ_p u=0,\quad p>2,\quad 0<q<p-1. \] The first proof takes advantage of the expansion of positivity for the degenerate, parabolic $p$-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value pro…
We establish the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-Δ_p u=0,\quad 1<p<2,\quad 0<p-1<q. \] The proof exploits the space expansion of positivity for the singular, parabolic $p$-Laplacian and employs the method of intrinsic scaling by carefully balancing the double singularity.
In this manuscript we establish local Hölder regularity estimates for bounded solutions of a certain class of doubly degenerate evolution PDEs. By making use of intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In such a scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence for our findings and approach, we address a Liouville type resu…
We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(Dψ({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $ψ$ and $F$ are convex functions, and this is a model equation for more general doubly nonlinear evolutions that arise in the study of phase transitions in materials. We show that if ${\bf v}$ is a weak solution, then $D{\bf v}$ is locally Hölder continuous except for possibly on a lower dimensional subset of the domain of ${\bf v}$. Our proof is based on compactness properties of solutions, two integral identities and a fractional time derivative estimate for $D{\b…
We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ Dψ({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $ψ$ and $F$ are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of $F$ are Hölder continuous, we show that $D^2{\bf v}$ and ${\bf v}_t$ are locally Hölder continuous except for possibly on a lower dimensional subset of $U\times (0,T)$. Our approach relies on two integral identities, decay of the…
We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-Δ_p u=0,\quad p>1. \] The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain…
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
This paper considers a certain doubly singular parabolic equations with one singularity occurs in the time derivative, whose model is \begin{equation*} \partial_tβ(u)-\operatorname{div}|Du|^{p-2}Du\ni0,\qquad \text{in}\quad Ω\times(0,T)\end{equation*} where $Ω\subset\mathbb{R}^N$ and $N\geq3$. We show that the bounded weak solutions are locally continuous in the range $$2-ε_0\leq p<2,$$ provided $ε_0>0$ is small enough, and the continuity is stable as $p\to2$.
A singular perturbation problem from the artificial compressible system to the incompressible system is considered for a doubly diffusive convection when a Hopf bifurcation from the motionless state occurs in the incompressible system. It is proved that the Hopf bifurcation also occurs in the artificial compressible system for small singular perturbation parameter, called the artificial Mach number. The time periodic solution branch of the artificial compressible system is shown to converge to the corresponding bifurcating branch of the incompressible system in the singular limit of vanishing …
In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov-Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.
A singularly perturbed system for doubly diffusive convection equations, called the artificial compressible system, is considered on a two-dimensional infinite layer for a parameters range where the Hopf bifurcation occurs in the corresponding incompressible system. The spectrum of the linearized operator in a time periodic function space is investigated in detail near the bifurcation point when the singular perturbation parameter is small. The results of this paper are the basis of the study of the nonlinear Hopf bifurcation problem and the singular limit of the time periodic bifurcating solu…
This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a $p$-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Hölder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what ext…
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.
We prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift in divergence form involving maximal monotone graphs defined everywhere, on which neither growth nor continuity assumptions are imposed.
We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p-Poisson equation, for the porous medium equation and for the doubly nonlinear equation.
We study the partial regularity of minimizers for certain singular functionals in the calculus of variations, motivated by Ball and Majumdar's recent modification [BM] of the Landau-de Gennes energy functional.
Here we describe a simple and fundamental approach to the maximal L^p regularity of parabolic equations, which only uses the concept of singular integrals of Volterra type. Knowledge of analytic semigroups, R-boundedness or H^\infty-functional calculus are not required.
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 we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.
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.
In this article we prove that solutions of singular fully nonlinear partial differential equations are $C^{1,β}$. We also prove the simplicity of the principal eigenvalues for the Dirichlet Problem associated to these operators using that regularity, a strict comparison principle and Sard's theorem.
We study the regularity of the solutions to initial-boundary value problems for N-systems of the p-Laplacian type, in $n\geq 3$ space variables, with square-integrable external forces in the space-time cylinder. So, the ellipticity coefficient remains unbounded. The singular case $μ=0 $ is covered.
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.
We consider a family of singular Volterra integral equations that appear in the study of monotone travelling-wave solutions for a family of diffusion-convection-reaction equations involving the $p$-Laplacian operator. Our results extend the ones due to B.\,Gilding for the case $p=2$. The fact that $p\neq2$ modifies the nature of the singularity in the integral equation, and introduces the need to develop some new tools for the analysis. The results for the integral equation are then used to study the existence and properties of travelling-wave solutions for doubly nonlinear diffusion-reaction …
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 the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.
A general class of singular abstract Cauchy problems is considered which naturally arises in applications to certain Free Boundary Problems. Existence of an associated evolution operator characterizing its solutions is established and is subsequently used to derive optimal regularity results. The latter are well known to be important basic tools needed to deal with corresponding nonlinear Cauchy Problems such as those associated to Free Boundary Problems.
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $Φ$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{\infty}(Ω)$-regularity for positive solutions
Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping
University of Monastir（莫纳斯提尔大学）；Higher Institute of Applied Mathematics and Computer Science, University of Kairouan（凯鲁万大学应用数学与计算机科学高等研究院）；Florida International University（佛罗里达国际大学）
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case p=2. In this article we establish a priori estimates and regularity results for the $p$-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.
We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.
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 revisit and sharpen the results from our previous work, where we investigated the regularity of the singular set of the free boundary in the nonlinear obstacle problem. As in the work of Figalli-Serra on the classical obstacle problem, we show that each stratum can be further decomposed into a `good' part and an `anomalous' part, where the former is covered by $C^{1,1-}$ manifolds, and the latter is of lower dimension.
In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations $$\partial_t u-|Du|^γΔ_p^N u=f,$$ where $-1<γ<0$, $1<p<\infty$, and $f$ is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.
