p-Laplacian Equation Research Articles

On the p-Laplacian evolution equation in metric measure spaces

The p-Laplacian evolution equation in metric measure spaces has been studied as the gradient flow in L2 of the p-Cheeger energy (for 1<p<∞). In this paper, using the first-order differential structure on a metric measure space introduced by Gigli, we characterise the subdifferential in L2 of the p-Cheeger energy. This gives rise to a new definition of the p-Laplacian operator in metric measure spaces, which allows us to work with this operator in more detail. In this way, we introduce a new notion of solutions to the p-Laplacian evolution equation in metric measure spaces. For p=1, we obtain a Green-Gauss formula similar to the one by Anzellotti for Euclidean spaces, and use it to characterise the 1-Laplacian operator and study the total variation flow. We also study the asymptotic behaviour of the solutions of the p-Laplacian evolution equation, showing that for 1≤p<2 we have finite extinction time.

Strong convergence rate of the averaging principle for a class of slow–fast stochastic evolution equations

ABSTRACT We prove a strong convergence rate of the averaging principle for general two-time-scales stochastic evolution equations driven by cylindrical Wiener processes. In particular, our general result can be used to deal with a large class of quasi-linear stochastic partial differential equations, such as stochastic reaction–diffusion equations, stochastic p-Laplace equations, stochastic porous media equations, and so on.

Grow-up of weak solutions in a [formula omitted]-Laplacian pseudo-parabolic problem

This paper deals with a p-Laplacian pseudo-parabolic equation with a homogeneous Dirichlet boundary condition in bounded domains. By using potential well method and the logarithmic Sobolev inequality, we study the grow-up phenomena and the bounds of growth rate, which are classified by the energy functional and the Nehari functional for the initial data.

The stochastic p-Laplace equation on ℝ d

We show well-posedness of the p-Laplace evolution equation on R d with square integrable random initial data for arbitrary 1 < p < ∞ and arbitrary space dimension d ∈ N . The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the p-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of 1 < p < ∞ and also depends on the space dimension d. We propose a framework of functional spaces which is independent of Sobolev space embeddings and space dimension. For additive noise, we show existence using a time discretization. Then, a fixed-point argument yields the result for multiplicative noise.

Nehari Manifold for Weighted Singular Fractional p-Laplace Equations

Nehari Manifold for Weighted Singular Fractional p-Laplace Equations

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-Laplace equations

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

In this article, we consider mixed local and nonlocal Sobolev (q,p)-inequalities with extremal in the case 0<q<1<p<∞. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal p-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular p-Laplace and mixed local and nonlocal p-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.

Global well-posedness for pseudo-parabolic p-Laplacian equation with singular potential and logarithmic nonlinearity

The main goal of this work is to investigate the initial boundary value problem for a class of pseudo-parabolic p-Laplacian equations with singular potential and logarithmic nonlinearity. First of all, we prove the local existence of weak solutions. Second, we show the existence of the global solution and the weak solution converging to the stationary solution when the time tends to infinity, and we show blow-up phenomena of solutions with the initial energy less than the mountain pass level d by using the potential well method. Finally, we parallelly stretch all the conclusions for the subcritical case to the critical case.

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast p-Laplacian evolution equation ut=Δpu on the whole Euclidean space, in the so-called “good fast diffusion range” 2NN+1<p<2. It is well-known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression.We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation.We characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. The GHP is then used as a tool to analyze the fine asymptotic behaviour for large times. For initial data that satisfy the same integral condition, we prove that the corresponding solutions behave like the Barenblatt with the same mass, uniformly in relative error.When the integral tail condition is not satisfied we show that both the GHP and the uniform convergence in relative error, do not hold anymore, and we provide also explicit counterexamples. We then prove a “generalized GHP”, that is, pointwise upper and lower bounds in terms of explicit profiles with a tail different from B. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients.To the best of our knowledge, analogous issues for the linear heat equation p=2, do not possess such clear answers, only partial results are known.

The Cauchy problem for a parabolic p-Laplacian equation with combined nonlinearities

This article studies the Cauchy problem for the evolution p-Laplacian equation ut−Δpu=λum+μ|∇u|qur in RN×(0,T). The local existence, global existence and nonexistence of solutions are investigated. In particular, for the case λ>0 and μ>0, we obtain an optimal Fujita-type result, which demonstrates the positive gradient term brings about the discontinuity phenomenon of the critical exponent. For the case λ>0 and μ<0, the existence and nonexistence of global solutions are also discussed.

The p-Laplacian in thin channels with locally periodic roughness and different scales* *Nakasato partially supported by CNPq 141675/2015-2 (Brazil), INCT/CAPES 88887.507830/2020-00 (Brazil) and Croatian Science Foundation under the project AsAn (IP-2018-01-2735). Pereira partially supported by CNPq 308950/2020-8, FAPESP 2020/04813-0 and 2020/14075-6 (Brazil), PID2019-103860GB-I00 MICINN (Spain) and Croatian Science Foundation under the project MultiFM IP-2019-04-1140 (Croatia).

In this work we analyse the asymptotic behaviour of the solutions of the p-Laplacian equation with homogeneous Neumann boundary conditions posed in bounded thin domains as and for some α > 0. We take a smooth function , L-periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter ɛ goes to zero and we determine the limit regime for three case: α < 1, α = 1 and α > 1.

Failure of Fatou type theorems for solutions to PDE of p-Laplace type in domains with flat boundaries

Let denote Euclidean n space and given k a positive integer let be a k-dimensional plane with If we first study the Martin boundary problem for solutions to the p-Laplace equation (called p-harmonic functions) in relative to We then use the results from our study to extend the work of Wolff on the failure of Fatou type theorems for p-harmonic functions in to p-harmonic functions in when Finally, we discuss generalizations of our work to solutions of p-Laplace type PDE (called -harmonic functions).

Positive Solutions for a Class of Fractional p-Laplacian Equation with Critical Sobolev Exponent and Decaying Potentials

Positive Solutions for a Class of Fractional p-Laplacian Equation with Critical Sobolev Exponent and Decaying Potentials

Erratum to: Multiple Positive Solutions for One Dimensional Third Order p-Laplacian Equations with Integral Boundary Conditions

Erratum to: Multiple Positive Solutions for One Dimensional Third Order p-Laplacian Equations with Integral Boundary Conditions

Gradient estimates for singular parabolic p-Laplace type equations with measure data

We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic p-Laplace equation $$u_t-\Delta _p u=\mu $$ with $$p\in (1,2)$$ . The case when $$p\in \big (2-\frac{1}{n+1},2\big )$$ were studied in Kuusi and Mingione (Ann Sc Norm Super Pisa Cl Sci 5 12(4):755–822, 2013). In this paper, we extend the results in Kuusi and Mingione (2013) to the open case when $$p\in \big (\frac{2n}{n+1},2-\frac{1}{n+1}\big ]$$ if $$n\ge 2$$ and $$p\in (\frac{5}{4}, \frac{3}{2}]$$ if $$n=1$$ . More specifically, in a more singular range of p as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential.

Calderón-Zygmund theory for non-convolution type nonlocal equations with continuous coefficient

Given \(2\le p<\infty \), \(s\in (0, 1)\) and \(t\in (1, 2s)\), we establish interior \(W^{t,p}\) Calderón-Zygmund estimates for solutions of nonlocal equations of the form $$\begin{aligned} \int _{\Omega } \int _{\Omega } K\left( x,|x-y|,\frac{x-y}{|x-y|}\right) \frac{(u(x)-u(y))(\varphi (x)-\varphi (y))}{|x-y|^{n+2s}} dx dy = g[\varphi ], \quad \forall \phi \in C_c^{\infty }(\Omega ) \end{aligned}$$where \(\Omega \subset \mathbb {R}^{n}\) is an open set. Here we assume K is bounded, nonnegative and continuous in the first entry – and ellipticity is ensured by assuming that K is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on manifolds, but the structure of the equation is general enough that it also applies to the certain fractional p-Laplace equations around points where \(u \in C^1\) and \(|\nabla u| \ne 0\).

Existence and multiplicity of solutions for perturbed fractional p-Laplacian equations with critical nonlinearity in N

In this paper, we consider the existence and multiplicity of solutions for the following perturbed fractional p-Laplacian equation Under some mild conditions on V, A and h, we show that the problem has at least one positive weak solution provided , and for any , it has pairs of solutions if , where and are sufficiently small positive numbers. Moreover, these solutions in as

On the second-order regularity of solutions to the parabolic p-Laplace equation

In this paper, we study the second-order Sobolev regularity of solutions to the parabolic p-Laplace equation. For any p-parabolic function u, we show that D(left| Duright| ^{frac{p-2+s}{2}}Du) exists as a function and belongs to L^{2}_{text {loc}} with s>-1 and 1<p<infty . The range of s is sharp.

Existence of multiple positive solutions for a class of infinite-point singular p-Laplacian fractional differential equation with singular source terms

Based on properties of Green’s function and by Avery–Peterson fixed point theorem, the existence of multiple positive solutions are obtained for singular p-Laplacian fractional differential equation with infinite-point boundary conditions, and an example is given to demonstrate the validity of our main results.

Positive solutions to mixed fractional p-Laplacian boundary value problems

Abstract In this paper, we discuss the existence and uniqueness of a positive solution for a p-Laplacian differential equation containing left and right Caputo derivatives. By the help of the Guo–Krasnoselskii theorem, we prove the existence of at least one positive solution. The existence of a unique positive solution is established under the assumption that the corresponding operator is α-concave and increasing. Numerical examples are given to check the obtained results.