Search results for "p-Laplacian."
showing 10 items of 67 documents
Anisotropic -Laplacian equations when goes to
2010
Abstract In this paper we prove a stability result for an anisotropic elliptic problem. More precisely, we consider the Dirichlet problem for an anisotropic equation, which is as the p -Laplacian equation with respect to a group of variables and as the q -Laplacian equation with respect to the other variables ( 1 p q ), with datum f belonging to a suitable Lebesgue space. For this problem, we study the behaviour of the solutions as p goes to 1 , showing that they converge to a function u , which is almost everywhere finite, regardless of the size of the datum f . Moreover, we prove that this u is the unique solution of a limit problem having the 1-Laplacian operator with respect to the firs…
Nonlinear elliptic equations having a gradient term with natural growth
2006
Abstract In this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is: (1) { − Δ p u = g ( u ) | ∇ u | p + f , in Ω , u = 0 , on ∂ Ω . Here Ω is a bounded open set in R N ( N ⩾ 2 ), Δ p denotes the so-called p-Laplace operator ( p > 1 ) and g is a continuous real function. Given f ∈ L m ( Ω ) ( m > 1 ), we study under which growth conditions on g problem (1) admits a solution. If m ⩾ N / p , we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m > N p ; while if 1 m N / p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condit…
A non-homogeneous elliptic problem dealing with the level set formulation of the inverse mean curvature flow
2015
Abstract In the present paper we study the Dirichlet problem for the equation − div ( D u | D u | ) + | D u | = f in an unbounded domain Ω ⊂ R N , where the datum f is bounded and nonnegative. We point out that the only hypothesis assumed on ∂Ω is that of being Lipschitz-continuous. This problem is the non-homogeneous extension of the level set formulation of the inverse mean curvature flow in a Euclidean space. We introduce a suitable concept of weak solution, for which we prove existence, uniqueness and a comparison principle.
On the best Lipschitz extension problem for a discrete distance and the discrete ∞-Laplacian
2012
Abstract This paper concerns the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞-Laplacian problem, which arises as the dynamic programming formula for the value function of some e -tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum f by taking the limit as p → ∞ in a nonlocal p -Laplacian problem.
The p-Laplacian with respect to measures
2013
We introduce a definition for the $p$-Laplace operator on positive and finite Borel measures that satisfy an Adams-type embedding condition.
Singular quasilinear elliptic systems involving gradient terms
2019
Abstract In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.
Large solutions for nonlinear parabolic equations without absorption terms
2012
In this paper we give a suitable notion of entropy solution of parabolic $p-$laplacian type equations with $1\leq p<2$ which blows up at the boundary of the domain. We prove existence and uniqueness of this type of solutions when the initial data is locally integrable (for $1<p<2$) or integrable (for $p=1$; i.e the Total Variation Flow case).
Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles
2014
In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of −g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞ to a double obstacle problem (with obstacles g1, g2) for the p-Laplacian. In fact, under a certain natural constraint on the extra …
Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles
2012
Abstract We study a quasilinear elliptic problem depending on a parameter λ of the form − Δ p u = λ f ( u ) in Ω , u = 0 on ∂ Ω . We present a novel variational approach that allows us to obtain multiplicity, regularity and a priori estimate of solutions by assuming certain growth and sign conditions on f prescribed only near zero. More precisely, we describe an interval of parameters λ for which the problem under consideration admits at least three nontrivial solutions: two extremal constant-sign solutions and one sign-changing solution. Our approach is based on an abstract localization principle of critical points of functionals of the form E = Φ − λ Ψ on open sublevels Φ − 1 ( ] − ∞ , …
Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions
2019
In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.