Search results for "Names"
showing 10 items of 6843 documents
Generalized dirichlet problem in nonlinear potential theory
1990
The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA(x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting.
An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities
2005
AbstractA multiplicity result for an eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities is obtained. The proof is based on a three critical points theorem for nondifferentiable functionals.
Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity
2008
AbstractThe existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.
The Dirichlet problem for the total variation flow
2001
Suppose that Ω is an open bounded domain with a Lipschitz boundary. The purpose of this chapter is to study the Dirichlet problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$ (5.1) where u0 ∈ L1(Ω) and ϕ ∈ L1 (∂Ω). This evolution equation is related to the gradient descent method used to solve the problem $$ \begin{gathered} Minimize \int {_\Omega \lef…
Radial solutions of Dirichlet problems with concave-convex nonlinearities
2011
Abstract We prove the existence of a double infinite sequence of radial solutions for a Dirichlet concave–convex problem associated with an elliptic equation in a ball of R n . We are interested in relaxing the classical positivity condition on the weights, by allowing the weights to vanish. The idea is to develop a topological method and to use the concept of rotation number. The solutions are characterized by their nodal properties.
The effects of convolution and gradient dependence on a parametric Dirichlet problem
2020
Our objective is to study a new type of Dirichlet boundary value problem consisting of a system of equations with parameters, where the reaction terms depend on both the solution and its gradient (i.e., they are convection terms) and incorporate the effects of convolutions. We present results on existence, uniqueness and dependence of solutions with respect to the parameters involving convolutions.
Positive solutions for parametric singular Dirichlet(p,q)-equations
2020
Abstract We consider a nonlinear elliptic Dirichlet problem driven by the ( p , q ) -Laplacian and a reaction consisting of a parametric singular term plus a Caratheodory perturbation f ( z , x ) which is ( p − 1 ) -linear as x → + ∞ . First we prove a bifurcation-type theorem describing in an exact way the changes in the set of positive solutions as the parameter λ > 0 moves. Subsequently, we focus on the solution multifunction and prove its continuity properties. Finally we prove the existence of a smallest (minimal) solution u λ ∗ and investigate the monotonicity and continuity properties of the map λ → u λ ∗ .
A Note on Riesz Bases of Eigenvectors of Certain Holomorphic Operator-Functions
2001
Abstract Operator-valued functions of the form A (λ) ≔ A − λ + Q(λ) with λ ↦ Q(λ)(A − μ)− 1 compact-valued and holomorphic on certain domains Ω ⊂ C are considered in separable Hilbert space. Assuming that the resolvent of A is compact, its eigenvalues are simple and the corresponding eigenvectors form a Riesz basis for H of finite defect, it is shown that under certain growth conditions on ‖Q(λ)(A − λ)− 1‖ the eigenvectors of A corresponding to a part of its spectrum also form a Riesz basis of finite defect. Applications are given to operator-valued functions of the form A (λ) = A − λ + B(λ − D)− 1C and to spectral problems in L2(0, 1) of the form −f″(x) + p(x, λ)f′(x) + q(x, λ)f(x) = λf(x…
Evolution Problems Associated to Linear Growth Functionals: The Dirichlet Problem
2003
Let Ω be a bounded set inIR N with Lipschitz continuous boundary ∂Ω. We are interested in the problem
Positive solutions of Dirichlet and homoclinic type for a class of singular equations
2018
Abstract We study a nonlinear singular boundary value problem and prove that, depending on a relationship between exponents of power terms, the problem has either solutions of Dirichlet type or homoclinic solutions. We make use of shooting techniques and lower and upper solutions.