Search results for "-Laplacian"
showing 10 items of 103 documents
The convective eigenvalues of the one–dimensional p–Laplacian as p → 1
2020
Abstract This paper studies the limit behavior as p → 1 of the eigenvalue problem { − ( | u x | p − 2 u x ) x − c | u x | p − 2 u x = λ | u | p − 2 u , 0 x 1 , u ( 0 ) = u ( 1 ) = 0 . We point out that explicit expressions for both the eigenvalues λ n and associated eigenfunctions are not available (see [16] ). In spite of this hindrance, we obtain the precise values of the limits lim p → 1 + λ n . In addition, a complete description of the limit profiles of the eigenfunctions is accomplished. Moreover, the formal limit problem as p → 1 is also addressed. The results extend known features for the special case c = 0 ( [6] , [28] ).
Optimal mass transportation for costs given by Finsler distances via p-Laplacian approximations
2016
Abstract In this paper we approximate a Kantorovich potential and a transport density for the mass transport problem of two measures (with the transport cost given by a Finsler distance), by taking limits, as p goes to infinity, to a family of variational problems of p-Laplacian type. We characterize the Euler–Lagrange equation associated to the variational Kantorovich problem. We also obtain different characterizations of the Kantorovich potentials and a Benamou–Brenier formula for the transport problem.
Critical points in open sublevels and multiple solutions for parameter-depending quasilinear elliptic equations
2014
We investigate the existence of multiple nontrivial solutions of a quasilinear elliptic Dirichlet problem depending on a parameter $\lambda>0$ of the form $$ -\Delta_pu=\lambda f(u)\quad\mbox{in }\ \Omega,\quad u=0\quad\mbox{on }\ \partial\Omega, $$ where $\Omega\subset \mathbb{R}^N$ is a bounded domain, $\Delta_p$, $1 < p < +\infty$, is the $p$-Laplacian, and $f: \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying a subcritical growth condition. More precisely, we establish a variational approach that when combined with differential inequality techniques, allows us to explicitly describe intervals for the parameter $\lambda$ for which the problem under consideration admits nontri…
Stress concentration for closely located inclusions in nonlinear perfect conductivity problems
2019
We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We prove optimal $L^\infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.
Singular solutions to a quasilinear ODE
2005
In this paper, we prove the existence of infinitely many radial solutions having a singular behaviour at the origin for a superlinear problem of the form $-\Delta_pu=|u|^{\delta-1}u$ in $B(0,1)\setminus\{0\}\subset\mathbb R^N$,\, $u=0$ for $|x|=1$, where $N>p>1$ and $\delta>p-1$. Solutions are characterized by their nodal properties. The case $\delta+1 <\frac{Np}{N-p}$ is treated. The study of the singularity is based on some energy considerations and takes into account the classification of the behaviour of the possible solutions available in the literature. By following a shooting approach, we are able to deduce the main multiplicity result from some estimates on the rotation numbers asso…
Turán type inequalities for generalized inverse trigonometric functions
2013
In this paper we study the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and its dependence on the parameter $p$, and we present a Tur\'an type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Tur\'an type inequality for a series considered by Ramanujan, involving the digamma function.
PDE triangular Bézier surfaces: Harmonic, biharmonic and isotropic surfaces
2011
We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces. © 2010 Elsevier B.V. All rights reserved.
Calderón problem for the p-Laplace equation : First order derivative of conductivity on the boundary
2016
We recover the gradient of a scalar conductivity defined on a smooth bounded open set in Rd from the Dirichlet to Neumann map arising from the p-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when p 6 = 2. In the p = 2 case boundary determination plays a role in several methods for recovering the conductivity in the interior. peerReviewed
A model of capillary phenomena in RN with subcritical growth
2020
This paper deals with the nonlinear Dirichlet problem of capillary phenomena involving an equation driven by the p-Laplacian-like di¤erential operator in RN. We prove the existence of at least one nontrivial nonnegative weak solution, when the reaction term satisfies a sub-critical growth condition and the potential term has certain regularities. We apply the energy functional method and weaker compactness conditions.
Fractional p-Laplacian evolution equations
2016
Abstract In this paper we study the fractional p-Laplacian evolution equation given by u t ( t , x ) = ∫ A 1 | x − y | N + s p | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y for x ∈ Ω , t > 0 , 0 s 1 , p ≥ 1 . In a bounded domain Ω we deal with the Dirichlet problem by taking A = R N and u = 0 in R N ∖ Ω , and the Neumann problem by taking A = Ω . We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u ( y ) − u ( x ) | u ( y ) − u ( x ) | when u ( y ) = u ( x ) . We also consider the Cauchy problem in the whole R N by taking A = Ω = R N . We find existence and uniqueness of strong solutions for each of the above mentioned problem…