Search results for "dirichlet"
showing 10 items of 197 documents
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…
Quasilinear degenerate parabolic equations in unbounded domains
2004
We prove the existence of bounded solutions of Cauchy-Dirichlet problem associated to a degenerate parabolic equation of second order in divergence form in unbounded domain.
Radial symmetry of minimizers to the weighted Dirichlet energy
2020
AbstractWe consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.
Positive solutions for singular double phase problems
2021
Abstract We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a p-Laplacian and of a weighted q-Laplacian ( q p ) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter λ > 0 , the equation has at least two positive solutions.
A Multiplicity result for a class of strongly indefinite asymptotically linear second order systems
2010
We prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the Maslov index of the linear Hamiltonian systems associated to the asymptotic limits of the given nonlinearity.
Stochastic Processes on Ends of Tree and Dirichlet Forms
2016
We present main ideas and compare two constructions of stochastic processes on the ends (leaves) of the trees with varying numbers of edges at the nods. In one of them the trees are represented by spaces of numerical sequences and the processes are obtained by solving a class of Chapman-Kolmogorov Equations. In the other the trees are described by the set of nodes and edges. To each node there is naturally associated a finite dimensional function space and the Dirichlet form on it. Having a class of Dirichlet forms at the nodes one can under certain conditions build a Dirichlet form on L2 space of funcions on the ends of the trees. We show that the state spaces of two approaches are homeomo…
Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory
2004
Let Ω be a bounded set in ℝN with boundary of class C1. We are interested in the problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$ (1) where ϕ ∈ L1(∂Ω), u0 ∈ L2(Ω) and a(x, ξ) = ∇ξ f(x, ξ, f being a function with linear growth in ‖ξ‖ as ‖ξ‖ → ∞. One of the classical examples is the nonparametric area integrand for which \( f(x,\xi ) = \sqrt {1 + \left\| \xi \right\|^2 } \). Prob…
On Some Properties of the Dirichlet Problem at Resonance
2008
Abstract The boundary value problem at resonance 𝑥″ + 𝑥 = 𝑞 sin 𝑡 + 𝑓(𝑡,𝑥,𝑥′), 𝑥(0) = 0, 𝑥(π) = 0, is considered, where 𝑓 : [0,π] × 𝑹2 → 𝑹 is a bounded Carathéodory function, 𝑞 is a parameter. We state the multiplicity results without assuming that 𝑓 has limits.
From user-generated data to data-driven innovation: A research agenda to understand user privacy in digital markets
2021
Abstract In recent years, strategies focused on data-driven innovation (DDI) have led to the emergence and development of new products and business models in the digital market. However, these advances have given rise to the development of sophisticated strategies for data management, predicting user behavior, or analyzing their actions. Accordingly, the large-scale analysis of user-generated data (UGD) has led to the emergence of user privacy concerns about how companies manage user data. Although there are some studies on data security, privacy protection, and data-driven strategies, a systematic review on the subject that would focus on both UGD and DDI as main concepts is lacking. There…