Denote by $Δ$ the Laplacian and by $Δ_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {Δv Δ_\infty v } -\frac12[|D^2v|^2-(Δv)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results:
1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$|Du|^{\frac{p-γ}2}Du\in W^{1,2}_{\rm loc},$$ with $γ<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the known $W^{2,q}_{\rm loc}$-regularity of $p$-harmonic functions for some $q>2$.
2. When $n\ge 2$ and $p\in(1,2)\cup(2,3+\frac2{n-2})$, the viscosity solutions to parabolic normalized $p $-Laplace equation have the $W_{\rm loc}^{2,q}$-regularity in the spatial variable and the $W_{\rm loc}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when $n=2$ an open question in [17] is completely answered.
3. When $n\ge 1 $ and $p\in(1,2)\cup(2,3)$, the weak/viscosity solutions to parabolic $p $-Laplace equation have the $W_{\rm loc}^{2,2}$-regularity in the spatial variable and the $W_{\rm loc}^{1,2}$-regularity in the time variable. The range of $p$ (including $p=2$ from the classical result) here is sharp for the $W_{\rm loc}^{2,2}$-regularity.
In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.
For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
We prove interior Hölder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous $p$-Laplacian equation \[ u_t=|\nabla u|^{2-p} \mbox{ div} (|\nabla u|^{p-2}\nabla u), \] where $1<p<\infty$. This equation arises from tug-of-war-like stochastic games with noise. It can also be considered as the parabolic $p$-Laplacian equation in non divergence form.
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) .
In this paper we introduce a natural function class and prove the existence and uniqueness of both nonnegative renormalized solutions and entropy solutions for the fractional p-Laplacian parabolic problem with L^1 data. And moreover, we obtain the equivalence of renormalized solutions and entropy solutions and establish a comparison result.
We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $Ω\times \R^+$, where $Ω$ is domain in the Heisenberg group $\Hn$, and $2\le p \le 4$. The result continues to hold in the more general setting of contact sub-Riemannian manifolds.
A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic $p$-Laplacian system is locally Hölder continuous in the interior. However, the boundary regularity is not yet well understood. In this paper we prove a boundary $L^\infty$-estimate for the spatial gradient $Du$ of solutions to the parabolic $p$-Laplacian system \begin{equation*} \partial_t u - \Div \big(|Du|^{p-2}Du\big) = 0 \quad\mbox{in $Ω\times(0,T)$} \end{equation*} for $p\ge 2$, together with a quantitative estimate. In particular, this implies the global Lipschitz regularity of so…
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = Δ_\infty^h u$ where $Δ_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $Δ_\infty^h u = |Du|^{h-3} < D^2uDu,Du>$. We also derive the asymptotic behavior of $u$ for the problem posed in the whole space and for the Dirichlet problem with zero boundary conditions.
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.
In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional function f which can be chosen freely is proven. Then, Di Giorgi's method is applied using level sets which will lead to L-infinity-estimates on the gradient of the weak solution.
We consider the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of global attractors and we analyze convergence properties of the solutions as well of the attractors.
In this paper we study an evolution equation involving the normalized $p$-Laplacian and a bounded continuous source term. The normalized $p$-Laplacian is in non divergence form and arises for example from stochastic tug-of-war games with noise. We prove local $C^{α, \fracα{2}}$ regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.
In this paper, we prove boundary pointwise $C^{k,α}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in obstacle-type problems.
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
In this article, we investigate the initial and boundary blow-up problem for the $p$-Laplacian parabolic equation $u_t-Δ_p u=-b(x,t)f(u)$ over a smooth bounded domain $Ω$ of $\mathbb{R}^N$ with $N\ge2$, where $Δ_pu={\rm div}(|\nabla u|^{p-2}\nabla u)$ with $p>1$, and $f(u)$ is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.
We study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution. Moreover, we prove that if the obstacle is continuous, so is the solution.
We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.
We provide a detailed (and fully rigorous) derivation of several fundamental properties of bounded weak solutions to initial-value problems for general conservative 2nd-order parabolic equations with p-Laplacian diffusion and (arbitrary) bounded and integrable initial data.
In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov-Backelman-Pucci estimate with 0<σ<2. And we show a Harnack inequality, Hölder regularity, and C^{1,α}-regularity of the solutions by obtaining decay estimates of their level sets.
We obtain up to a flat boundary regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.
In a wide class of the so called Obstacle Problems of parabolic type it is shown how to improve the optimal regularity of the solution and as a consequence how to obtain space-time regularity of the corresponding free boundary.
We investigate strong and weak versions of maximum and comparison principles for a class of quasilinear parabolic equations with the $p$-Laplacian $$ \partial_t u - Δ_p u = λ|u|^{p-2} u + f(x,t) $$ under zero boundary and nonnegative initial conditions on a bounded cylindrical domain $Ω\times (0, T)$, $λ\in \mathbb{R}$, and $f \in L^\infty(Ω\times (0, T))$. Several related counterexamples are given.
Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.
We give a short, simple proof of maximal regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent.
This note presents a regularity result with proof for an initial-boundary value problem of a linear parabolic system involving curl of the unknown vector field, subjected to the boundary condition of prescribing the tangential component of the solution.
We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal operators with kernels not being absolutely continuous with respect to the Lebesgue measure.
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,α}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.
We establish Hölder estimates for the time derivative of solutions of fully non-linear parabolic equations that does not necessarily have $C^{2,α}$ estimates.
We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be $C^{2,1}$.
We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a (strictly positive) finite time.
This paper concerns the initial-boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then prove the existence, uniqueness and decay estimate of global solutions and the blowup phenomena of solutions with subcritical initial energy. Next, we extend parallelly these results to the critical initial energy. Lastly, the existence, uniqueness and asymptotic behavior of global solutions with supercritical initial energy are proved by further analyzing …
In this paper, we consider a non-local diffusion equation involving the fractional $p(x)$-Laplacian with nonlinearities of variable exponent type. Employing the sub-differential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.
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.
We consider Orlicz-growth generalization to evolutionary $p$-Laplacian and to the evolutionary symmetric $p$-Laplacian. We derive the spatial second-order Caccioppoli estimate for a local weak solution to these systems. The result is new even for the $p$-case.
We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form $u_t-Iu=f$, where the operator $I$ is not necessarily translation invariant. As a consequence of this and the regularity results for translation invariant case, we obtain $C^{1,α}$ interior estimates in space for non translation invariant operators under some hypothesis on the time regularity of the boundary data.
We establish Hölder estimates for the time derivative of solutions of non-local parabolic equations under mild assumptions for the boundary data. As a consequence we are able to extend the Evans-Krylov estimate for rough kernels to parabolic equations.
We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.
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 establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies a non-degeneracy condition. This is the optimal regularity for this problem.
We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.
In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, discontinuous paths in the dynamics of the stock's prices.
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.
