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.
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 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.
We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover existence, uniqueness, and characterization of solutions (above a threshold representing the "infinity Bernoulli constant"), their regularity, and their relationship with the solutions to the interior Bernoulli problem for the $p$-laplacian.
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 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.
We prove the uniqueness for viscosity solutions of a differential equation involving the infinity-Laplacian with a variable exponent. A version of the Harnack's inequality is derived for this minimax problem.
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 investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to infinity. We characterize velocity fields, so that positive solutions decay or lift-off at spatial infinity as time tends to infinity. This addresses the question of stability of the zero solution for decaying perturbations.
This paper concerns with the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional $p_n$-Laplacian when $p_n\to\infty$ as a particular case, tough it could be extended to a function of the Hölder quotient of order $s$, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Hölder infinity Laplacian.
In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal $C^{1,α}$ regularity along the free boundary. We also show existence of solutions, nondegeneracy properties and fine geometric estimates for the free boundary.
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 …
We extend some theorems for the Infinity-Ground State and for the Infinity-Potential, known for convex polygons, to other domains in the plane, by applying Alexandroff's method to the curved boundary. A recent explicit solution disproves a conjecture.
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.
We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -Δ_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.
We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.
Here we generalize quasilinear parabolic $p-$Laplacian type equations to obtain the prototype equation as \[ u_t - \text{div} (g(|Du|)/ |Du| \cdot Du) = 0, \] where a nonnegative, increasing, and continuous function $g$ trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $1<g_0 \leq g_1 < \infty$. Through this generalization in the setting from Orlicz spaces, we provide a uniform proof with a single geometric setting that a bounded weak solution is locally Hölder continuous considering $1 < g_0 \leq g_1 \leq 2$ and $2 \leq g_0 \leq g_1 < \infty$ separately. By using…
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.
The aim of this paper is to establish global Calderón--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div} (|F|^{p-2}F)~\text{in}~Ω\times (0,T)\subset \mathbb{R}^{n+1}, $$ proving that $$F\in L^q\Rightarrow \nabla u\in L^q,$$ for any $q>\max\{p,\frac{n(2-p)}{2}\}$ and $p>1$. Acerbi and Mingione \cite{Acerbi07} proved this estimate in the case $p>\frac{2n}{n+2}$. In this article we settle the case $1<p\leq \frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm…
Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a certain regularity condition on the constraint is satisfied. If this regularity condition fails, then solutions obtained from game theory and $L^p$-approximation need not coincide.
In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in $B_1$ are universally Lipschitz continuous in $B_{1/2}$, which is the optimal regularity for the problem. We make a new use of the Ishii-Lions' method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.
In this article we find necessary and sufficient conditions for the strong maximum principle and compact support principle for non-negative solutions to the quasilinear elliptic inequalities $$Δ_\infty u + G(|Du|) - f(u)\,\leq 0\quad \text{in}\; \mathcal{O},$$ and $$Δ_\infty u + G(|Du|) - f(u)\,\geq 0\quad \text{in}\; \mathcal{O},$$ where $\mathcal{O}$ denotes the infinity Laplacian, $G$ is an appropriate continuous function and $f$ is a nondecreasing, continuous function with $f(0)=0$.
We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth.
We study general parabolic equations of the form $u_t = div A(x,t, u,D u) + div(|F|^{p-2} F)+ f$ whose principal part depends on the solution itself. The vector field $A$ is assumed to have small mean oscillation in $x$, measurable in $t$, Lipschitz continuous in $u$, and its growth in $Du$ is like the $p$-Laplace operator. We establish interior Calderón-Zygmund estimates for locally bounded weak solutions to the equations when $p>2n/(n+2)$. This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucia…
We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic $p$-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration \[ u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^β=0, \ p>2, β>0 \] The interface may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters $p,β, sign~b$, and asymptotics of the initial function near its support. In all cases, we prove the explicit formula for the interface and the local solutio…
We consider a degenerate parabolic equation associated with the fractional $% p $-Laplace operator $\left( -Δ\right) _{p}^{s}$\ ($p\geq 2$, $s\in \left( 0,1\right) $) and a monotone perturbation growing like $\left\vert s\right\vert ^{q-2}s,$ $q>p$ and with bad sign at infinity as $\left\vert s\right\vert \rightarrow \infty $. We show the existence of locally-defined strong solutions to the problem with any initial condition $u_{0}\in L^{r}(Ω)$ where $r\geq 2$ satisfies $r>N(q-p)/sp$. Then, we prove that finite time blow-up is possible for these problems in the range of parameters provided for…
Under the lack of variational structure and nondegeneracy, we investigate three notions of \textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and boundary Harnack inequality are proved to support our analysis. This is a continuation of our first work [3] and a contribution in the development of the theory of \textit{generalized principal eigenvalue} beside the works [8, 13, 12, 9, 29]. We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniquene…
This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results apply to the Pucci operator, the normalized $q$-Laplacians with $1<q\leq\infty$, the Finsler Laplacian and more general quasilinear operators.
In this paper, we study the long-time behaviour of solutions of Cauchy problem for the parabolic $p$-Laplacian equation with variable coefficients. Under mild conditions on the coefficient of the principal part and without upper growth restriction on the source function, we prove that this problem possesses a compact and invariant global attractor in $L^{2}(R^{n})$.
In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional $p$-Laplacian with logarithmic nonlinearity \begin{equation*}\label{eq}\left\{ \begin{array}{llc} u_{t}+(-Δ)^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log(|u|) & \text{in}\ & Ω,\;t>0 , u =0 & \text{in} & \mathbb{R}^{N}\backslash Ω,\;t > 0, u(x,0)=u_{0}(x), & \text{in} &Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^N \, ( N\geq 1)$ is a bounded domain with Lipschitz boundary and $2\leq p< \infty$. The local existence will be done by using the Galerkin approximations. By combining the po…
In this paper, we prove new Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + f(x, u)\,=\,0\quad \text{in}\; \mathbb{R}^d,$$ where $γ\in [0, 2]$ and $Δ^γ_\infty$ is a $(3-γ)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $\liminf_{|x|\to\infty}\lim_{s\to0}f(x,s)/s^{3-γ}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-γ}$ decreasing in $s$, we further obtain the u…
In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity solution of the mixed boundary value problem, $$ \{\begin{array}{ll} \displaystyle -Δ_{\infty}u(x)=0\quad & \text{in} Ω, \displaystyle \frac{\partial u}{\partial n}(x)= g (x)\quad & \text{on} Γ_N, \displaystyle u(x)= 0 \quad & \text{on} Γ_D. \end{array}. $$
We give a complete characterization, as "stadium-like domains", of convex subsets $Ω$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $Ω$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet probl…
In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t β(u) - \nabla \cdot α(x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $β: \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in β(0)$ and $ \nabla \cdot α(x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $β$. However…
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=Ω\times(0,T)$, $Σ_T=\partialΩ\times(0,T)$, $Ω$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partialΩ$, $λ\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(Ω)$, $u_0…
